08DritschelRovnyak.dvi


 영운 포
 6 days ago
 Views:
Transcription
1 Operator Theory: Advances and Applications, Vol 207, c 2010 Birkhäuser Verlag Basel/Switzerland The Operator FejérRiesz Theorem Michael A Dritschel and James Rovnyak To the memory of Paul Richard Halmos Abstract The FejérRiesz theorem has inspired numerous generalizations in one and several variables, and for matrix and operatorvalued functions This paper is a survey of some old and recent topics that center around Rosenblum s operator generalization of the classical FejérRiesz theorem Mathematics Subject Classification (2000) Primary 47A68; Secondary 60G25, 47A56, 47B35, 42A05, 32A70, 30E99 Keywords Trigonometric polynomial, FejérRiesz theorem, spectral factorization, Schur complement, noncommutative polynomial, Toeplitz operator, shift operator 1 Introduction The classical FejérRiesz factorization theorem gives the form of a nonnegative trigonometric polynomial on the real line, or, equivalently, a Laurent polynomial that is nonnegative on the unit circle For the statement, we write D = {z : z < 1} and T = {ζ : ζ =1} for the open unit disk and unit circle in the complex plane FejérRiesz Theorem A Laurent polynomial q(z) = m k= m q kz k which has complex coefficients and satisfies q(ζ) 0 for all ζ T can be written q(ζ) = p(ζ) 2, ζ T, for some polynomial p(z) =p 0 + p 1 z + + p m z m,andp(z) can be chosen to have no zeros in D The original sources are Fejér [22] and Riesz [47] The proof is elementary and consists in showing that the roots of q(z) occur in pairs z j and 1/ z j with z j 1 Then the required polynomial p(z) is the product of the factors z z j adjusted by a suitable multiplicative constant c Details appear in many places; see, eg, [28, p 20], [34, p 235], or [60, p 26]
2 224 MA Dritschel and J Rovnyak The FejérRiesz theorem arises naturally in spectral theory, the theory of orthogonal polynomials, prediction theory, moment problems, and systems and control theory Applications often require generalizations to functions more general than Laurent polynomials, and, more than that, to functions whose values are matrices or operators on a Hilbert space The spectral factorization problem is to write a given nonnegative matrix or operatorvalued function F on the unit circle in the form F = G G where G has an analytic extension to the unit disk (in a suitably interpreted sense) The focal point of our survey is the special case of a Laurent polynomial with operator coefficients The operator FejérRiesz theorem (Theorem 21) obtains a conclusion similar to the classical result for a Laurent polynomial whose coefficients are Hilbert space operators: if Q j, j = m,,m, are Hilbert space operators such that m Q(ζ) = Q k ζ k 0, ζ T, (11) k= m then there is a polynomial P (z) =P 0 +P 1 z + +P m z m with operator coefficients such that Q(ζ) =P (ζ) P (ζ), ζ T (12) This was first proved in full generality in 1968 by Marvin Rosenblum [49] The proof uses Toeplitz operators and a method of Lowdenslager, and it is a fine example of operator theory in the spirit of Paul Halmos Rosenblum s proof is reproduced in 2 Part of the fascination of the operator FejérRiesz theorem is that it can be stated in a purely algebraic way The hypothesis (11) on Q(z) isequivalenttothe statement that an associated Toeplitz matrix is nonnegative The conclusion (12) is equivalent to 2m + 1 nonlinear equations whose unknowns are the coefficients P 0,P 1,,P m of P (z) Can it be that this system of equations can be solved by an algebraic procedure? The answer is, yes, and this is a recent development The iterative procedure uses the notion of a Schur complement and is outlined in 3 There is a surprising connection between Rosenblum s proof of the operator FejérRiesz theorem and spectral factorization The problem of spectral factorization is formulated precisely in 4, using Hardy class notions A scalar prototype is Szegő s theorem (Theorem 41) on the representation of a positive integrable and logintegrable function w on the unit circle in the form h 2 for some H 2 function h The operator and matrix counterparts of Szegő s theorem, Theorems 45 and 47, have been known for many years and go back to fundamental work in the 1940s and 1950s which was motivated by applications in prediction theory (see the historical notes at the end of 4) We present a proof that is new to the authors and we suspect not widely known It is based on Theorem 43, which traces its origins to Rosenblum s implementation of the Lowdenslager method In 4 wealso state without proof some special results that hold in the matrix case The method of Schur complements points the way to an approach to multivariable factorization problems, which is the subject of 5 Even in the scalar case,
3 The Operator FejérRiesz Theorem 225 the obvious first ideas for multivariable generalizations of the FejérRiesz theorem are false by wellknown examples Part of the problem has to do with what one might think are natural restrictions on degrees In fact, the restrictions on degrees are not so natural after all When they are removed, we can prove a result, Theorem 51, that can be viewed as a generalization of the operator FejérRiesz theorem in the strictly positive case We also look at the problem of outer factorization, at least in some restricted settings In recent years there has been increasing interest in noncommutative function theory, especially in the context of functions of freely noncommuting variables In 6 we consider noncommutative analogues of the dtorus, and corresponding notions of nonnegative trigonometric polynomials In the freely noncommutative setting, there is a very nice version of the FejérRiesz theorem (Theorem 61) In a somewhat more general noncommutative setting, which also happens to cover the commutative case as well, we have a version of Theorem 51 for strictly positive polynomials (Theorem 62) Our survey does not aim for completeness in any area In particular, our bibliography represents only a selection from the literature The authors regret and apologize for omissions 2 The operator FejérRiesz theorem In this section we give the proof of the operator FejérRiesz theorem by Rosenblum [49] The general theorem had precursors A finitedimensional version was given by Rosenblatt [48], an infinitedimensional special case by Gohberg [26] We follow standard conventions for Hilbert spaces and operators If A is an operator, A is its adjoint Norms of vectors and operators are written Except where noted, no assumption is made on the dimension of a Hilbert space, and nonseparable Hilbert spaces are allowed Theorem 21 (Operator FejérRiesz Theorem) Let Q(z) = m k= m Q kz k be a Laurent polynomial with coefficients in L(G) for some Hilbert space G IfQ(ζ) 0 for all ζ T, then Q(ζ) =P (ζ) P (ζ), ζ T, (21) for some polynomial P (z) =P 0 + P 1 z + + P m z m with coefficients in L(G) The polynomial P (z) can be chosen to be outer The definition of an outer polynomial will be given later; in the scalar case, a polynomial is outer if and only if it has no zeros in D The proof uses (unilateral) shift and Toeplitz operators (see [11] and [29]) By a shift operator here we mean an isometry S on a Hilbert space H such that the unitary component of S in its Wold decomposition is trivial With natural identifications, we can write H = G G for some Hilbert space G and S(h 0,h 1,)=(0,h 0,h 1,)
4 226 MA Dritschel and J Rovnyak when the elements of H are written in sequence form Suppose that such a shift S is chosen and fixed If T,A L(H), we say that T is Toeplitz if S TS = T,and that A is analytic if AS = SA An analytic operator A is said to be outer if ran A is a subspace of H of the form F F for some closed subspace F of G As block operator matrices, Toeplitz and analytic operators have the forms T 0 T 1 T 2 A T 1 T 0 T 1 A 1 A 0 0 T = T 2 T 1 T, A = 0 A 2 A 1 A (22) 0 Here { E 0 S j TE 0 G, j 0, T j = (23) E0 TS j E 0 G, j < 0, where E 0 g =(g, 0, 0,) is the natural embedding of G into H For examples, consider Laurent and analytic polynomials Q(z) = m k= m Q kz k and P (z) = P 0 + P 1 z + + P m z m with coefficients in L(G) Set Q j =0for j >mand P j =0forj>m Then the formulas Q 0 Q 1 Q 2 P Q 1 Q 0 Q 1 P 1 P 0 0 T Q = Q 2 Q 1 Q, T P = 0 P 2 P 1 P (24) 0 define bounded operators on H Boundedness follows from the identity Q(ζ)f(ζ),g(ζ) G dσ(ζ) = Q j k f k,g j G, (25) T k,j=0 where σ is normalized Lebesgue measure on T and f(ζ) =f 0 + f 1 ζ + f 2 ζ 2 + and g(ζ) =g 0 + g 1 ζ + g 2 ζ 2 + have coefficients in G, all but finitely many of which are zero The operator T Q is Toeplitz, and T P is analytic Moreover, Q(ζ) 0 for all ζ T if and only if T Q 0; Q(ζ) =P (ζ) P (ζ) for all ζ T if and only if T Q = TP T P Definition 22 We say that the polynomial P (z) is outer if the analytic Toeplitz operator A = T P is outer In view of the example (24), the main problem is to write a given nonnegative Toeplitz operator T in the form T = A A,whereA is analytic We also want to know that if T = T Q for a Laurent polynomial Q, thenwecanchoosea = T P for an outer analytic polynomial P of the same degree Lemmas 23 and 24 reduce the problem to showing that a certain isometry is a shift operator
5 The Operator FejérRiesz Theorem 227 Lemma 23 (Lowdenslager s Criterion) Let H be a Hilbert space, and let S L(H) be a shift operator Let T L(H) be Toeplitz relative to S as defined above, and suppose that T 0 LetH T be the closure of the range of T 1/2 in the inner product of H Then there is an isometry S T mapping H T into itself such that S T T 1/2 f = T 1/2 Sf, f H In order that T = A A for some analytic operator A L(H), it is necessary and sufficient that S T is a shift operator In this case, A can be chosen to be outer Proof The existence of the isometry S T follows from the identity S TS = T, which implies that T 1/2 Sf and T 1/2 f have the same norms for any f H If S T is a shift operator, we can view H T as a direct sum H T = G T G T with S T (h 0,h 1,)=(0,h 0,h 1,) Here dim G T dim G Toseethis,notice that a short argument shows that T 1/2 ST and S T 1/2 agree on H T, and therefore T 1/2 (ker ST ) ker S The dimension inequality then follows because T 1/2 is onetoone on the closure of its range Therefore we may choose an isometry V from G T into G Define an isometry W on H T into H by W (h 0,h 1,)=(Vh 0,Vh 1,) Define A L(H) by mapping H into H T via T 1/2 and then H T into H via W : Af = WT 1/2 f, f H Straightforward arguments show that A is analytic, outer, and T = A A Conversely, suppose that T = A A where A L(H) is analytic Define an isometry W on H T into H by WT 1/2 f = Af, f H ThenWS T = SW, and hence ST n = W S n W for all n 1 Since the powers of S tend strongly to zero, so do the powers of ST, and therefore S T is a shift operator Lemma 24 In Lemma 23, lett = T Q be given by (24) for a Laurent polynomial Q(z) of degree m IfT = A A where A L(H) is analytic and outer, then A = T P for some outer analytic polynomial P (z) of degree m Proof Let Q(z) = m k= m Q kz k Recall that Q j =0for j >m By (23) applied to A, what we must show is that S j AE 0 =0forallj>m It is sufficient to show that S m+1 AE 0 = 0 By (23) applied to T,sinceT = A A and A is analytic, E 0 A S j AE 0 = E 0 Sj TE 0 = Q j =0, j > m It follows that ran S m+1 AE 0 ran AS k E 0 for all k 0, and therefore ran S m+1 AE 0 ran A (26) Since A is outer, ran A reduces S, and so rans m+1 AE 0 S m+1 AE 0 = 0 by (26), and the result follows ran A Therefore The proof of the operator FejérRiesz theorem is now easily completed
6 228 MA Dritschel and J Rovnyak Proof of Theorem 21 Define T = T Q as in (24) Lemmas 23 and 24 reduce the problem to showing that the isometry S T is a shift operator It is sufficient to show that ST n f 0 for every f in H T Claim: If f = T 1/2 h where h H has the form h = (h 0,,h r, 0,), then ST n f = 0 for all sufficiently large n For if u H and n is any positive integer, then ST n f,t 1/2 u = f,st n T 1/2 u = T 1/2 h, T 1/2 S n u = H T H T H Th,Sn u H By the definition of T = T Q, Th has only a finite number of nonzero entries (depending on m and r), and the first n entries of S n u are zero (irrespective of u) The claim follows from the arbitrariness of u In view of the claim, ST n f 0 for a dense set of vectors in H T, and hence by approximation this holds for all f in H T ThusS T is a shift operator, and the result follows A more general result is proved in the original version of Theorem 21 in [49] There it is only required that Q(z)g is a Laurent polynomial for a dense set of g in G (the degrees of these polynomials can be unbounded) We have omitted an accompanying uniqueness statement: the outer polynomial P (z) in Theorem 21 canbechosensuchthatp (0) 0, and then it is unique See [2] and [50] 3 Method of Schur complements We outline now a completely different proof of the operator FejérRiesz theorem The proof is due to Dritschel and Woerdeman [19] and is based on the notion of a Schur complement The procedure constructs the outer polynomial P (z) = P 0 + P 1 z + + P m z m one coefficient at a time A somewhat different use of Schur complements in the operator FejérRiesz theorem appears in Dritschel [18] The method in [18] plays a role in the multivariable theory, which is taken up in 5 We shall explain the main steps of the construction assuming the validity of two lemmas Full details are given in [19] and also in the forthcoming book [3] by Bakonyi and Woerdeman The authors thank Mihaly Bakonyi and Hugo Woerdeman for advance copies of key parts of [3], which has been helpful for our exposition The book [3] includes many additional results not discussed here Definition 31 Let H be a Hilbert space Suppose T L(H), T 0 Let K be a closed subspace of H, andletp K L(H, K) be orthogonal projection of H onto K Then (see Appendix, Lemma A2) there is a unique operator S L(K), S 0, such that (i) T PK SP K 0; (ii) if S L(K), S 0, and T PK SP K 0, then S S We write S = S(T,K) andcalls the Schur complement of T supported on K
7 The Operator FejérRiesz Theorem 229 Schur complements satisfy an inheritance property, namely, if K K + H, then S(T,K )=S(S(T,K + ), K ) If T isspecifiedinmatrixform, ( ) A B T = : K K B C K K, then S = S(T,K) is the largest nonnegative operator in L(K) such that ( ) A S B 0 B C The condition T 0 is equivalent to the existence of a contraction G L(K, K ) such that B = C 1 2 GA 1 2 (Appendix, Lemma A1) In this case, G can be chosen so that it maps ran A into ran C and is zero on the orthogonal complement of ran A, and then S = A 1 2 (I G G)A 1 2 When C is invertible, this reduces to the familiar formula S = A B C 1 B Lemma 32 Let M L(H), M 0, and suppose that ( ) A B M = : K K K K B C for some closed subspace K of H (1) If S(M,K) =P P and C = R R for some P L(K) and R L(K ),then there is a unique X L(K, K ) such that ( P X M = )( P 0 0 R X R ) and ran X ran R (31) (2) Conversely, if (31) holds for some operators P, X, R, thens(m,k) =P P We omit the proof and refer the reader to [3] or [19] for details Proof of Theorem 21 using Schur complements Let Q(z) = m k= m Q kz k satisfy Q(ζ) 0 for all ζ T We shall recursively construct the coefficients of an outer polynomial P (z) =P 0 + P 1 z + + P m z m such that Q(ζ) =P (ζ) P (ζ), ζ T Write H = G G and G n = G G with n summands As before, set Q k =0for k >m, and define T Q L(H) by Q 0 Q 1 Q 2 Q 1 Q 0 Q 1 T Q = Q 2 Q 1 Q 0 For each k =0, 1, 2,, define S(k) =S(T Q, G k+1 ),
8 230 MA Dritschel and J Rovnyak which we interpret as the Schur complement of T Q on the first k + 1 summands of H = G G ThusS(k) isa(k +1) (k + 1) block operator matrix satisfying S(S(k), G j+1 )=S(j), 0 j<k<, (32) by the inheritance property of Schur complements Lemma 33 For each k =0, 1, 2,, ( ) Y 0 Y1 Y k+1 Y 1 S(k +1)= S(k) Yk+1 for some operators Y 0,Y 1,,Y k+1 in L(G) Fork m 1, ( Y0 Y 1 Y k+1 ) = ( Q0 Q 1 Q k 1 ) Again see [3] or [19] for details Granting Lemmas 32 and 33, we can proceed with the construction Construction of P 0,P 1 Choose P 0 = S(0) 1 2 Using Lemma 33, write ( ) Y0 Y S(1) = 1 Y1 S(0) In Lemma 32(1) take M = S(1) and use the factorizations S(S(1), G 1 ) (32) = S(0) = P 0 P 0 and S(0) = P 0 P 0 Choose P 1 = X where X L(G) is the operator produced by Lemma 32(1) Then ( )( ) P S(1) = 0 P1 P0 0 0 P0 and ran P P 1 P 1 ran P 0 (33) 0 Construction of P 2 Next use Lemma 33 to write ( ) Y 0 Y1 Y 2 S(2) = ( ) Y, S(1) 1 Y 2 and apply Lemma 32(1) to M = S(2) with the factorizations S(S(2), G 1 ) (32) = S(0) = P0 P 0, ( )( ) P S(1) = 0 P1 P0 0 0 P0 P 1 P 0
9 The Operator FejérRiesz Theorem 231 This yields operators X 1,X 2 L(G) such that S(2) = P 0 X1 X2 0 P0 P1 P X 1 P 0 0, (34) 0 0 P0 X 2 P 1 P 0 ( ) ( ) X1 P0 0 ran ran (35) X 2 P 1 P 0 In fact, X 1 = P 1 To see this, notice that we can rewrite (34) as ( ) P X )( P 0 S(2) = 0 R X R, ( ) P0 0 ( ) P =, X = X2 P X 1 P 1, R = P0 0 By (33) and (35), ran P 1 ran P 0 and ran X 2 ran P 0, and therefore ran X ran P 0 Hence by Lemma 32(2), ( )( ) S(S(2), G 2 )= P P P = 0 X1 P0 0 0 P0 (36) X 1 P 0 Comparing this with ( )( S(S(2), G 2 ) (32) = S(1) (33) P = 0 P1 P0 0 P0 P 1 ) 0, P 0 (37) we get P0 P 1 = P0 X 1 By (35), ran X 1 ran P 0, and therefore X 1 = P 1 Now choose P 2 = X 2 to obtain S(2) = P 0 P1 P2 0 P0 P1 P P 1 P 0 0, (38) 0 0 P0 P 2 P 1 P 0 ( ) ( ) P1 P0 0 ran ran (39) P 2 P 1 P 0 Inductive step We continue in the same way for all k =1, 2, 3,Atthekth stage, the procedure produces operators P 0,,P k such that P0 Pk P 0 0 S(k) =, (310) 0 P0 P k P 0 P 1 P 0 0 ran ran (311) P k P k P 0
10 232 MA Dritschel and J Rovnyak By Lemma 33, in the case k m, ( Q 0 Q 1 Q m 0 0 ) Q 1 S(k) = Q m (312) 0 S(k 1) 0 The zeros appear here when k>m, and their presence leads to the conclusion that P k =0fork>m We set then P (z) =P 0 + P 1 z + + P m z m Comparing (310) and (312) in the case k = m, we deduce 2m +1relations which are equivalent to the identity Q(ζ) =P (ζ) P (ζ), ζ T Final step: P (z) is outer Define T P as in (24) With natural identifications, ( ) P P 1 T P = P 2 T P (313) The relations (311), combined with the fact that P k =0forallk>m,implythat P 1 ran P 2 ran T P Hence for any g G, a sequence f n can be found such that P 1 T P f n P 2 g Then by (313), T P ( g f n ) ( ) P0 g 0 It follows that ran T P contains every vector (P 0 g, 0, 0,)withg L(G), and hence ran T P ran P 0 ran P 0 The reverse inclusion holds because by (311), the ranges of P 1,P 2, are all contained in ran P 0 ThusP(z)isouter
11 4 Spectral factorization The Operator FejérRiesz Theorem 233 The problem of spectral factorization is to write a nonnegative operatorvalued function F on the unit circle in the form F = G G where G is analytic (in a sense made precise below) The terminology comes from prediction theory, where the nonnegative function F plays the role of a spectral density for a multidimensional stationary stochastic process The problem may be viewed as a generalization of a classical theorem of Szegő from Hardy class theory and the theory of orthogonal polynomials (see Hoffman [35, p 56] and Szegő [62, Chapter X]) We write H p and L p for the standard Hardy and Lebesgue spaces for the unit disk and unit circle See Duren [20] Recall that σ is normalized Lebesgue measure on the unit circle T Theorem 41 (Szegő s Theorem) Let w L 1 satisfy w 0 ae on T and log w(ζ) dσ > T Then w = h 2 ae on T for some h H 2,andh can be chosen to be an outer function Operator and matrix generalizations of Szegő s theorem are stated in Theorems 45 and 47 below Some vectorial function theory is needed to formulate these and other results We assume familiarity with basic concepts but recall a few definitions For details, see, eg, [30, 61] and [50, Chapter 4] In this section, G denotes a separable Hilbert space Functions f and F on the unit circle with values in G and L(G), respectively, are called weakly measurable if f(ζ),v and F (ζ)u, v are measurable for all u, v G Nontangential limits for analytic functions on the unit disk are taken in the strong (norm) topology for vectorvalued functions, and in the strong operator topology for operatorvalued functions We fix notation as follows: (i) We write L 2 G and L L(G) for the standard Lebesgue spaces of weakly measurable functions on the unit circle with values in G and L(G) (ii) Let HG 2 and H L(G) be the analogous Hardy classes of analytic functions on the unit disk We identify elements of these spaces with their nontangential boundary functions, and so the spaces may alternatively be viewed as subspaces of L 2 G and L L(G) (iii) Let N + L(G) be the space of all analytic functions F on the unit disk such that ϕf belongs to HL(G) for some bounded scalar outer function ϕ The elements of N + L(G) are also identified with their nontangential boundary functions A function F HL(G) is called outer if FH2 G is dense in H2 F for some closed subspace F of G A function F N + L(G) is outer if there is a bounded scalar outer function ϕ such that ϕf HL(G) and ϕf is outer in the sense just defined The definition of an outer function given here is consistent with the previously defined notion for polynomials in 2
12 234 MA Dritschel and J Rovnyak A function A HL(G) is called inner if multiplication by A on H2 G is a partial isometry In this case, the initial space of multiplication by A is a subspace of H 2 G of the form H 2 F where F is a closed subspace of G Toprovethis,noticethatboth the kernel of multiplication by A and the set on which it is isometric are invariant under multiplication by z Therefore the initial space of multiplication by A is a reducing subspace for multiplication by z, and so it has the form H 2 F where F is a closed subspace of G (see [29, p 106] and [50, p 96]) Every F HL(G) has an innerouter factorization F = AG, wherea is an inner function and G is an outer function This factorization can be chosen such that the isometric set HF 2 for multiplication by A on H2 G coincides with the closure of the range of multiplication by G The innerouter factorization is extended in an obvious way to functions F N + L(G) Details are given, for example, in [50, Chapter 5] The main problem of this section can now be interpreted more precisely: Factorization Problem Given a nonnegative weakly measurable function F on T, find a function G in N + L(G) such that F = G G ae on T If such a function exists, we say that F is factorable If a factorization exists, the factor G can be chosen to be outer by the innerouter factorization Moreover, an outer factor G can be chosen such that G(0) 0, and then it is unique [50, p 101] By the definition of N + L(G), a necessary condition for F to be factorable is that log + F (ζ) dσ <, (41) T where log + x is zero or log x according as 0 x 1or1<x<, andsoweonly need consider functions which satisfy (41) In fact, in proofs we can usually reduce to the bounded case by considering F/ ϕ 2 for a suitable scalar outer function ϕ The following result is another view of Lowdenslager s criterion, which we deduce from Lemma 23 A direct proof is given in [61, pp ] Lemma 42 Suppose F L L(G) and F 0 ae on T LetK F be the closure of F 1 2 HG 2 in L2 G,andletS F be the isometry multiplication by ζ on K F ThenF is factorable if and only if S F is a shift operator, that is, ζ n F 1 2 HG 2 = {0} (42) n=0 Proof In Lemma 23 take H = HG 2 viewed as a subspace of L2 G,andletS be multiplication by ζ on H Define T L(H) bytf = PFf, f H, wherep is the projection from L 2 G onto H2 G One sees easily that T is a nonnegative Toeplitz operator, and so we can define H T and an isometry S T as in Lemma 23 In fact, S T is unitarily equivalent to S F via the natural isomorphism W : H T K F such that W (T 1 2 f) =F 1 2 f for every f in H ThusS F is a shift operator if and only if S T is a shift operator, and by Lemma 23 this is the same as saying that T = A A where A L(H) is analytic, or equivalently F is factorable [50, p 110]
13 The Operator FejérRiesz Theorem 235 We obtain a very useful sufficient condition for factorability Theorem 43 Suppose F L L(G) and F 0 ae For F to be factorable, it is sufficient that there exists a function ψ in L L(G) such that (i) ψf HL(G) ; (ii) for all ζ T except at most a set of measure zero, ψ(ζ) F (ζ)g is onetoone If these conditions are met and F = G G ae with G outer, then ψg HL(G) Theorem 43 appears in Rosenblum [49] with ψ(ζ) = ζ m (viewed as an operatorvalued function) The case of an arbitrary inner function was proved and applied in a variety of ways by Rosenblum and Rovnyak [50, 51] V I Matsaev first showed that more general functions ψ can be used Matsaev s result is evidently unpublished, but versions were given by DZ Arov [1, Lemma to Theorem 4] and AS Markus [41, Theorem 343 on p 199] Theorem 43 includes all of these versions We do not know if the conditions (i) and (ii) in Theorem 43 are necessary for factorability It is not hard to see that they are necessary in the simple cases dim G =1anddimG = 2 (for the latter case, one can use [50, Example 1, p 125]) The general case, however, is open Proof of Theorem 43 Let F satisfy (i) and (ii) Define a subspace M of L 2 G by M = ζ n F 1 2 HG 2 = ζ n F 1 2 HG 2 n=0 We show that M = {0} By(i), ψf M = ψf 2 ζ n F 1 2 HG 2 ζ n ψfhg 2 ζ n HG 2 = {0} (43) n=0 n=0 Thus ψf 1 2 M = {0} Nowifg M, thenψf 1 2 g =0aeby(43)HenceF 1 2 g =0 ae by (ii) By the definition of M, g F 1 2 HG 2, and standard arguments show from this that g(ζ) F (ζ) 1 2 G ae Therefore g = 0 ae It follows that M = {0}, and so F is factorable by Lemma 42 Let F = G G ae with G outer We prove that ψg HL(G) by showing that ψg HG 2 H2 G SinceG is outer, GH2 G = H2 F for some closed subspace F of G By (i), ψg (GHG 2 )=ψfh2 G H2 G Therefore ψg HF 2 H2 G Suppose f H2 G F, and consider any h L2 G Then G f,h L 2 = f(ζ),g(ζ)h(ζ) G G dσ =0, T because ran G(ζ) F ae Thus ψg f = 0 ae It follows that ψg HG 2 H2 G,and therefore ψg HL(G) n=0 n=0
14 236 MA Dritschel and J Rovnyak For a simple application of Theorem 43, suppose that F is a Laurent polynomial of degree m, andchooseψ to be ζ m I In short order, this yields another proof of the operator FejérRiesz theorem (Theorem 21) Another application is a theorem of Sarason [55, p 198], which generalizes the factorization of a scalarvalued function in H 1 as a product of two functions in H 2 (see [35, p 56]) Theorem 44 Every G in N + L(G) can be written G = G 1G 2,whereG 1 and G 2 belong to N + L(G) and G 2 G 2 =[G G] 1/2 and G 1 G 1 = G 2 G 2 ae Proof Suppose first that G HL(G) Foreachζ T, write G(ζ) =U(ζ)[G (ζ)g(ζ)] 1 2, where U(ζ) is a partial isometry with initial space ran [G (ζ)g(ζ)] 1 2 Itcanbe shown that U is weakly measurable We apply Theorem 43 with F =[G G] 1 2 and ψ = U Conditions (i) and (ii) of Theorem 43 are obviously satisfied, and so we obtain an outer function G 2 HL(G) such that G 2 G 2 =[G G] 1/2 ae and UG 2 HL(G) SetG 1 = UG 2ByconstructionG 1 HL(G), G = U(G G) 1 2 =(UG 2 )G 2 = G 1 G 2, and G 1 G 1 = G 2 U UG 2 = G 2G 2 ae The result follows when G H L(G) The general case follows on applying what has just been shown to ϕ 2 G,where ϕ is a scalarvalued outer function such that ϕ 2 G HL(G) The standard operator generalization of Szegő s theorem also follows from Theorem 43 Theorem 45 Let F be a weakly measurable function on T having invertible values in L(G) satisfying F 0 ae If log + F (ζ) dσ < and log + F (ζ) 1 dσ <, T then F is factorable Proof Since log + F (ζ) is integrable, we can choose a scalar outer function ϕ 1 such that F 1 = F/ ϕ 1 2 L L(G) Since log + F (ζ) 1 is integrable, so is log + F 1 (ζ) 1 Hence there is a bounded scalar outer function ϕ such that T ϕf 1 1 L L(G)
15 The Operator FejérRiesz Theorem 237 We apply Theorem 43 to F 1 with ψ = ϕf1 1 Condition (i) is satisfied because ψf 1 = ϕi Condition (ii) holds because the values of ψ are invertible ae Thus F 1 is factorable, and hence so is F Theorem 43 has a halfplane version, the scalar inner case of which is given in [50, p 117] This has an application to the following generalization of Akhiezer s theorem on factoring entire functions [50, Chapter 6] Theorem 46 Let F be an entire function of exponential type τ, having values in L(G), such that F (x) 0 for all real x and log + F (t) 1+t 2 dt < Then F (x) =G(x) G(x) for all real x where G is an entire function with values in L(G) such that exp( iτz/2)g is of exponential type τ/2 and the restriction of G to the upper halfplane is an outer function Matrix case We end this section by quoting a few results for matrixvalued functions The matrix setting is more concrete, and one can do more Statements often require invertibility assumptions We give no details and leave it to the interested reader to consult other sources for further information Our previous definitions and results transfer in an obvious way to matrixvalued functions For this we choose G = C r for some positive integer r and identify operators on C r with r r matrices The operator norm of a matrix is denoted WewriteL r r,h r r in place of L L(G),H L(G) and for the norms on these spaces Theorem 45 is more commonly stated in a different form for matrixvalued functions Theorem 47 Suppose that F is an r r measurable matrixvalued function having invertible values on T such that F 0 ae and log + F is integrable Then F is factorable if and only if log det F is integrable Recall that when F is factorable, there is a unique outer G such that F = G G and G(0) 0 It makes sense to inquire about the continuity properties of the mapping Φ: F G with respect to various norms For example, see Jacob and Partington [37] We cite one recent result in this area Theorem 48 (Barclay [5]) Let F, F n, n =1, 2,,ber r measurable matrixvalued functions on T having invertible values ae and integrable norms Suppose that F = G G and F n = G n G n,whereg, G n are r r matrixvalued outer functions such that G(0) 0 and G n (0) 0, n =1, 2,Then if and only if lim n T G(ζ) G n (ζ) 2 dσ =0
16 238 MA Dritschel and J Rovnyak (i) lim F (ζ) F n (ζ) dσ =0,and n T (ii) the family of functions {log det F n } n=0 is uniformly integrable A family of functions {ϕ α } α A L 1 is uniformly integrable if for every ε>0 there is a δ>0 such that E ϕ α dσ < ε for all α A whenever σ(e) <δsee[5] for additional references and similar results in other norms A theorem of Bourgain [9] characterizes all functions on the unit circle which are products hg with g, h H : Afunctionf L has the form f = hg where g, h H if and only if log f is integrable This resolves a problem of Douglas and Rudin [17] The problem is more delicate than spectral factorization; when f = 1 ae, the factorization cannot be achieved in general with inner functions Bourgain s theorem was recently generalized to matrixvalued functions Theorem 49 (Barclay [4, 6]) Suppose F L r r has invertible values ae Then F has the form F = H G ae for some G, H in Hr r if and only if log det F is integrable In this case, for every ε>0 such a factorization can be found with G H < F + ε The proof of Theorem 49 in [6] is long and technical In fact, Barclay proves an L p version of this result for all p, 1 p Another type of generalization is factorization with indices We quote one result to illustrate this notion Theorem 410 Let F be an r r matrixvalued function with rational entries Assume that F has no poles on T and that det F (ζ) 0for all ζ in T Then there exist integers κ 1 κ 2 κ r such that F (z) =F (z)diag {z κ1,,z κr }F + (z), where F ± are r r matrixvalued functions with rational entries such that (i) F + (z) has no poles for z 1 and det F + (z) 0for z 1; (ii) F (z) has no poles for z 1 including z = and det F (z) 0for z 1 including z = The case in which F is nonnegative on T can be handled using the operator FejérRiesz theorem (the indices are all zero in this case) The general case is given in Gohberg, Goldberg, and Kaashoek [27, pp ] This is a large subject that includes, for example, general theories of factorization in Bart, Gohberg, Kaashoek, and Ran [7] and Clancey and Gohberg [13] Historical remarks Historical accounts of spectral factorization appear in [2, 30, 50, 52, 61] Briefly, the problem of factoring nonnegative matrixvalued functions on the unit circle rose to prominence in the prediction theory of multivariate stationary stochastic processes The first results of this theory were announced by Zasuhin [65] without complete proofs; proofs were supplied by MG Kreĭn in lectures Modern accounts
17 The Operator FejérRiesz Theorem 239 of prediction theory and matrix generalizations of Szegő s theorem are based on fundamental papers of Helson and Lowdenslager [31, 32], and Wiener and Masani [63, 64] The general case of Theorem 45 is due to Devinatz [15]; other proofs are given in [16, 30, 50] For an engineering view and computational methods, see [38, Chapter 8] and [56] The original source for Lowdenslager s Criterion (Lemmas 23 and 42) is [40]; an error in [40] was corrected by Douglas [16] There is a generalization, given by SzNagy and Foias [61, pp ], in which the isometry may have a nontrivial unitary component and the shift component yields a maximal factorable summand Lowdenslager s Criterion is used in the construction of canonical models of operators by de Branges [10] See also Constantinescu [14] for an adaptation to Toeplitz kernels and additional references 5 Multivariable theory It is natural to wonder to what extent the results for one variable carry over to several variables Various interpretations of several variables are possible The most straightforward is to consider Laurent polynomials in complex variables z 1,,z d that are nonnegative on the dtorus T d The method of Schur complements in 3 suggests an approach to the factorization problem for such polynomials Care is needed, however, since the first conjectures for a multivariable FejérRiesz theorem that might come to mind are false, as explained below Multivariable generalizations of the FejérRiesz theorem are thus necessarily weaker than the onevariable result One difficulty has to do with degrees, and if the condition on degrees is relaxed, there is a neat result in the strictly positive case (Theorem 51) By a Laurent polynomial in z =(z 1,,z d ) we understand an expression Q(z) = m 1 k 1= m 1 m d k d = m d Q k1,,k d z k1 1 zk d d (51) We assume that the coefficients belong to L(G), where G is a Hilbert space With obvious interpretations, the scalar case is included By an analytic polynomial with coefficients in L(G) we mean an analogous expression, of the form P (z) = m 1 k 1=0 m d k d =0 P k1,,k d z k1 1 zk d d (52) The numbers m 1,,m d in (51) and (52) are upper bounds for the degrees of the polynomials in z 1,,z d, which we define as the smallest values of m 1,,m d that can be used in the representations (51) and (52) Suppose that Q(z) has the form (51) and satisfies Q(ζ) 0 for all ζ T d,thatis,forallζ =(ζ 1,,ζ d )with ζ 1 = = ζ d = 1 Already in the scalar case, one cannot always find an analytic polynomial P (z) such that Q(ζ) = P (ζ) P (ζ), ζ T d This was first explicitly shown by Lebow and Schreiber [39] There are also difficulties in writing Q(ζ) = r j=1 P j(ζ) P j (ζ), ζ T d,forsome
18 240 MA Dritschel and J Rovnyak finite set of analytic polynomials, at least if one requires that the degrees of the analytic polynomials do not exceed those of Q(z) as in the onevariable case (see Naftalevich and Schreiber [44], Rudin [53], and Sakhnovich [54, 36])The example in [44] is based on a Cayley transform of a version of a real polynomial over R 2 called Motzkin s polynomial, which was the first explicit example of a nonnegative polynomial in R d, d>1, which is not a sum of squares of polynomials What is not mentioned in these sources is that if we loosen the restriction on degrees, the polynomial in [44] can be written as a sum of squares (see [19]) Nevertheless, for three or more variables, very general results of Scheiderer [57] imply that there exist nonnegative, but not strictly positive, polynomials which cannot be expressed as such finite sums regardless of degrees Theorem 51 Let Q(z) be a Laurent polynomial in z =(z 1,,z d ) with coefficients in L(G) for some Hilbert space G Suppose that there is a δ>0 such that Q(ζ) δi for all ζ T d Then Q(ζ) = r P j (ζ) P j (ζ), ζ T d, (53) j=1 for some analytic polynomials P 1 (z),,p r (z) in z =(z 1,,z d ) which have coefficients in L(G) Furthermore, for any fixed k, the representation (53) can be chosen such that the degree of each analytic polynomial in z k is no more than the degree of Q(z) in z k The scalar case of Theorem 51 follows by a theorem of Schmüdgen [58], which states that strictly positive polynomials over compact semialgebraic sets in R n (that is, sets which are expressible in terms of finitely many polynomial inequalities) can be written as weighted sums of squares, where the weights are the polynomials used to define the semialgebraic set (see also [12]); the proof is nonconstructive On the other hand, the proof we sketch using Schur complements covers the operatorvalued case, and it gives an algorithm for finding the solution One can also give estimates for the degrees of the polynomials involved, though we have not stated these We provetheorem51 for the cased = 2, following Dritschel [18] The general case is similar The argument mimics the method of Schur complements, especially in its original form used in [18] In place of Toeplitz matrices whose entries are operators, in the case of two variables we use Toeplitz matrices whose entries are themselves Toeplitz matrices The fact that the first level Toeplitz blocks are infinite in size causes problems, and so we truncate these blocks to finite size Then everything goes through, but instead of factoring the original polynomial Q(z), the result is a factorization of polynomials Q (N) (z) thatareclosetoq(z) When Q(ζ) δi on T d for some δ>0, there is enough wiggle room to factor Q(z) itself We isolate the main steps in a lemma
19 The Operator FejérRiesz Theorem 241 Lemma 52 Let Q(z) = m 1 j= m 1 m 2 k= m 2 Q jk z j 1 zk 2 be a Laurent polynomial with coefficients in L(G) such that Q(ζ) 0 for all ζ =(ζ 1,ζ 2 ) in T 2 Set Q (N) (z) = m 1 j= m 1 m 2 N +1 k N +1 k= m 2 Then for each N m 2, there are analytic polynomials F l (z) = m 1 N j=0 k=0 with coefficients in L(G) such that Q (N) (ζ) = Q jk z j 1 zk 2 F (l) jk zj 1 zk 2, l =0,,N, (54) N F l (ζ) F l (ζ), ζ T 2 (55) l=0 Proof Write m 1 ( m2 Q(z) = Q jk z2 )z k j1 = m 1 R j (z 2 ) z j 1, j= m 1 k= m 2 j= m 1 andextendallsumstorunfrom to by setting Q jk =0andR j (z 2 )=0if j >m 1 or k >m 2 Introduce a Toeplitz matrix T whose entries are the Toeplitz matrices T j corresponding to the Laurent polynomials R j (z 2 ), that is, T 0 T 1 T 2 Q j0 Q j, 1 Q j, 2 T 1 T 0 T 1 Q j1 Q j0 Q j, 1 T = T 2 T 1 T, T j = 0 Q j2 Q j1 Q, j0 j =0, ±1, ±2,NoticethatT is finitely banded, since T j =0for j >m 1 The identity (25) has the following generalization: Th,h = T q p h p,h q = Q(ζ)h(ζ),h(ζ) G dσ 2 (ζ) T 2 p=0 q=0 Here ζ =(ζ 1,ζ 2 )anddσ 2 (ζ) =dσ(ζ 1 )dσ(ζ 2 ) Also, h(ζ) = p=0 q=0 h pq ζ p 1 ζq 2,
20 242 MA Dritschel and J Rovnyak where the coefficients are vectors in G and all but finitely many are zero, and h 0 h p0 h = h 1, h p = h p1, p =0, 1, 2, It follows that T acts as a bounded operator on a suitable direct sum of copies of G SinceQ(ζ) 0onT 2, T 0 Fix N m 2 Set T 0 T 1 T 2 T T 1 T 0 T 1 = T 2 T 1 T 0, where T j is the upper (N +1) (N +1)blockofT j with a normalizing factor: Q j0 Q j, 1 Q j, N T j = 1 Q j1 Q j0 Q j, N+1 N +1, j =0, ±1, ±2, Q jn Q j,n 1 Q j0 Then T is the Toeplitz matrix corresponding to the Laurent polynomial m 1 Ψ(w) = T j w j j= m 1 Moreover, T 0 since it is a positive constant multiple of a compression of T Thus Ψ(w) 0for w = 1 By the operator FejérRiesz theorem (Theorem 21), Ψ(w) =Φ(w) Φ(w), w =1, (56) for some analytic polynomial Φ(w) = m 1 j=0 Φ jw j whose coefficients are (N +1) (N + 1) matrices with entries in L(G) Write Φ j = ( ) Φ jn Φ j,n 1 Φ j0, where Φ jk is the kth column in Φ j Set F (z) = m 1 j=0 k=0 N Φ jk z j 1 zk 2 The identity (56) is equivalent to 2m relations for the coefficients of Ψ(w) The coefficients of Ψ(w) are constant on diagonals, there being N + 1 k terms in the kth diagonal above the main diagonal, and similarly below If these terms are summed, the result gives 2m relations equivalent to the identity Q (N) (ζ) = F (ζ) F (ζ), ζ T 2 (57)
21 The Operator FejérRiesz Theorem 243 We omit the calculation, which is straightforward but laborious To convert (57) to the form (55), write F (0) jk F (1) Φ jk = jk, j =0,,m 1 and k =0,,N F (N) jk Then F 0 (z) F 1 (z) F (z) =, F N (z) where F 0 (z),,f N (z) are given by (54), and so (57) takes the form (55) Proof of Theorem 51 for the case d =2 Suppose N m 2,andset Q(z) = m 1 j= m 1 m 2 k= m 2 N +1 N +1 k Q jk z j 1 zk 2 The values of Q(z) are selfadjoint on T 2,and Q(z) =Q(z)+S(z), where S(z) = m 1 m 2 j= m 1 k= m 2 k N +1 k Q jk z j 1 zk 2 Now choose N large enough that S(ζ) <δ, ζ T 2 Then Q(ζ) 0onT 2,and the result follows on applying Lemma 52 to Q(z) Further details can be found in [18], and a variation on this method yielding good numerical results is given in Geronimo and Lai [23] While, as we mentioned, there is in general little hope of finding a factorization of a positive trigonometric polynomial in two or more variables in terms of one or more analytic polynomials of the same degree, it happens that there are situations where the existence of such a factorization is important In particular, Geronimo and Woerdeman consider this question in the context of the autoregressive filter problem [24, 25], with the first paper addressing the scalar case and the second the operatorvalued case, both in two variables They show that for scalarvalued polynomials in this setting there exists a factorization in terms of a single stable (so invertible in the bidisk D 2 ) analytic polynomial of the same degree if and only if a full rank condition holds for certain submatrices of the associated Toeplitz matrix ([24, Theorem 113]) The condition for operatorvalued polynomials is similar, but more complicated to state We refer the reader to the original papers for details
22 244 MA Dritschel and J Rovnyak Stable scalar polynomials in one variable are by definition outer, so the Geronimo and Woerdeman results can be viewed as a statement about outer factorizations in two variables In [19], a different notion of outerness is considered As we saw in 3, in one variable outer factorizations can be extracted using Schur complements The same Schur complement method in two or more variables gives rise to a version of outer factorization which in general does not agree with that coming from stable polynomials In [19], this Schur complement version of outerness is used when considering outer factorizations for polynomials in two or more variables As in the Geronimo and Woerdeman papers, it is required that the factorization be in terms of a single analytic polynomial of the same degree as the polynomial being factored Then necessary and sufficient conditions for such an outer factorization under these constraints are found ([19, Theorem 41]) The problem of spectral factorization can also be considered in the multivariable setting Blower [8] has several results along these lines for bivariate matrixvalued functions, including a matrix analogue of Szegő s theorem similar to Theorem 47 His results are based on a twovariable matrix version of Theorem 51, and the arguments he gives coupled with Theorem 51 can be used to extend these results to polynomials in d>2 variables as well 6 Noncommutative factorization We now present some noncommutative interpretations of the notion of several variables, starting with the one most frequently considered, and for which there is an analogue of the FejérRiesz theorem It is due to Scott McCullough and comes very close to the onevariable result Further generalizations have been obtained by Helton, McCullough and Putinar in [33] For a broad overview of the area, two nice survey articles have recently appeared by Helton and Putinar [34] and Schmüdgen [59] covering noncommutative real algebraic geometry, of which the noncommutative analogues of the FejérRiesz theorem are one aspect In keeping with the assumptions made in [42], all Hilbert spaces in this section are taken to be separable Fix Hilbert spaces G and H, and assume that H is infinite dimensional Let S be the free semigroup with generators a 1,,a d ThusS is the set of words w = a j1 a jk, j 1,,j k {1,,d}, k =0, 1, 2,, (61) with the binary operation concatenation The empty word is denoted e The length of the word (61) is w = k (so e =0)LetS m be the set of all words (61) of length at most m The cardinality of S m is l m =1+d + d d m We extend S to a free group G We can think of the elements of G as words in a 1,,a d,a 1, with two such words identified if one can be obtained 1,,a 1 d from the other by cancelling adjacent terms of the form a j and a 1 j The binary operation in G is also concatenation Words in G of the form h = v 1 w with v, w S play a special role and are called hereditary Notice that a hereditary
23 The Operator FejérRiesz Theorem 245 word h has many representations h = v 1 w with v, w S LetH m be the set of hereditary words h which have at least one representation in the form h = v 1 w with v, w S m We can now introduce the noncommutative analogues of Laurent and analytic polynomials A hereditary polynomial is a formal expression Q = h Q h, (62) h H m where Q h L(G) for all h Analytic polynomials are hereditary polynomials of the special form P = w P w, (63) w S m where P w L(G) for all w Theidentity is defined to mean that Q h = v,w S m h=v 1 w Q = P P P v P w, h H d Next we give meaning to the expressions Q(U) andp (U) for hereditary and analytic polynomials (62) and (63) and any tuple U =(U 1,,U d ) of unitary operators on H First define U w L(H) for any w S by writing w in the form (61) and setting U w = U j1 U jk By convention, U e = I is the identity operator on H Ifh G is a hereditary word, set U h =(U v ) U w for any representation h = v 1 w with v, w S; this definition does not depend on the choice of representation Finally, define Q(U),P(U) L(H G) by Q(U) = U h Q h, P(U) = U w P w h H m w S m The reader is referred to, for example, Murphy [43, 63] for the construction of tensor products of Hilbert spaces and algebras, or Palmer, [45, 110] for a more detailed account Theorem 61 (McCullough [42]) Let Q = h H m h Q h be a hereditary polynomial with coefficients in L(G) such that Q(U) 0 for every tuple U =(U 1,,U d ) of unitary operators on H Then for some l l m,there
24 246 MA Dritschel and J Rovnyak exist analytic polynomials P j = w P j,w, j =1,,l, w S m with coefficients in L(G) such that Q = P1 P Pl P l Moreover, for any tuple U =(U 1,,U d ) of unitary operators on H, Q(U) =P 1 (U) P 1 (U)+ + P l (U) P l (U) In these statements, when G is infinite dimensional, we can choose l =1 As noted by McCullough, when d = 1, Theorem 61 gives a weaker version of Theorem 21 However, Theorem 21 can be deduced from this by a judicious use of Beurling s theorem and an innerouter factorization McCullough s theorem uses one of many possible choices of noncommutative spaces on which some form of trigonometric polynomials can be defined We place this, along with the commutative versions, within a general framework, which we now explain The complex scalarvalued trigonometric polynomials in d variables form a unital algebra P, the involution taking z n to z n,whereforn =(n 1,,n d ), n =( n 1,, n d ) If instead the coefficients are in the algebra C = L(G) for some Hilbert space G, then the unital involutive algebra of trigonometric polynomials with coefficients in C is P C The unit is 1 1 A representation of P C is a unital algebra homomorphism from P C into L(H) for a Hilbert space H Thekey thing here is that z 1,,z d generate C, and so assuming we do not mess with the coefficient space, a representation π is determined by specifying π(z k ), k =1,,d Firstnotethatsincezk z k =1,π(z k ) is isometric, and since zk = z 1 k,we then have that π(z k ) is unitary Assuming the variables commute, the z k s generate a commutative group G which we can identify with Z d under addition, and the irreducible representations of commutative groups are one dimensional This essentially follows from the spectral theory for normal operators (see, for example, Edwards [21, p 718]) However, the onedimensional representations are point evaluations on T d Discrete groups with the discrete topology are examples of locally compact groups Group representations of locally compact groups extend naturally to the algebraic group algebra, which in this case is P, and then on to the algebra P C by tensoring with the identity representation of C Soaseem ingly more complex way of stating that a commutative trigonometric polynomial P in several variables is positive / strictly positive is to say that for each (topologically) irreducible unitary representation π of G, the extension of π to a unital representation of the algebra P C, also called π, has the property that π(p ) 0 / π(p ) > 0 By the way, since T d is compact, π(p ) > 0 implies the existence of some ɛ>0 such that π(p ɛ1 1) = π(p ) ɛ1 0 What is gained through this perspective is that we may now define noncommutative trigonometric polynomials over a finitely generated discrete (so locally
25 The Operator FejérRiesz Theorem 247 compact) group G in precisely the same manner These are the elements of the algebraic group algebra P generated by G; that is, formal complex linear combinations of elements of G endowed with pointwise addition and a convolution product (see Palmer [45, Section 19]) Then a trigonometric polynomial in P C is formally a finite sum over G of the form P = g g P g where P g C for all g We also introduce an involution by setting g = g 1 for g G A trigonometric polynomial P is selfadjoint if for all g, P g = Pg There is an order structure on selfadjoint elements defined by saying that a selfadjoint polynomial P is positive / strictly positive if for every irreducible unital representation π of G, the extension as above of π to the algebra P C (again called π), satisfies π(p ) 0/ π(p ) > 0; where by π(p ) > 0 we mean that there exists some ɛ>0 independent of π such that π(p ɛ(1 1)) 0 Letting Ω represent the set of such irreducible representations, we can in a manner suggestive of the Gel fand transform define ˆP (π) =π(p ), and in this way think of Ω as a sort of noncommutative space on which our polynomial is defined The Gel fandraĭkov theorem (see, for example, Palmer [46, Theorem 1246]) ensures the existence of sufficiently many irreducible representations to separate G, soinparticular,ω For a finitely generated discrete group G with generators {a 1,,a d },let S be a fixed unital subsemigroup of G containing the generators The most interesting case is when S is the subsemigroup generated by e (the group identity) and {a 1,,a d } As an example of this, if G is the noncommutative free group in d generators, then the unital subsemigroup generated by {a 1,,a d } consists of group elements w of the form e (for the empty word) and those which are an arbitrary finite product of positive powers of the generators, as in (61) We also need to address the issue of what should play the role of Laurent and analytic trigonometric polynomials in the noncommutative setting The hereditary trigonometric polynomials are defined as those polynomials of the form P = j w j1 w j2 P j,wherew j1,w j2 S We think of these as the Laurent polynomials Trigonometric polynomials over S are referred to as analytic polynomials The square of an analytic polynomial Q is the hereditary trigonometric polynomial Q Q Squares are easily seen to be positive As a weak analogue of the FejérRiesz theorem, we prove a partial converse below We refer to those hereditary polynomials which are selfadjoint as real hereditary polynomials, and denote the set of such polynomials by H While these polynomials do not form an algebra, they are clearly a vector space Those which are finite sums of squares form a cone C in H (that is, C is closed under sums and positive scalar multiplication) Any real polynomial is the sum of terms of the form 1 A or w2 w 1 B + w1 w 2 B,wherew 1,w 2 S and A is selfadjoint The first of these is obviously the difference of squares Using w w =1foranyw G, we also have w 2 w 1 B + w 1 w 2 B =(w 1 B + w 2 1) (w 1 B + w 2 1) 1 (1 + B B) Hence H = C C
#È²¿ë¼®
http://www.kbc.go.kr/ A B yk u δ = 2u k 1 = yk u = 0. 659 2nu k = 1 k k 1 n yk k Abstract Web Repertoire and Concentration Rate : Analysing Web Traffic Data Yong  Suk Hwang (Research
More informationPage 2 of 5 아니다 means to not be, and is therefore the opposite of 이다. While English simply turns words like to be or to exist negative by adding not,
Page 1 of 5 Learn Korean Ep. 4: To be and To exist Of course to be and to exist are different verbs, but they re often confused by beginning students when learning Korean. In English we sometimes use the
More informationstep 11
Written by Dr. In Ku KimMarshall STEP BY STEP Korean 1 through 15 Action Verbs Table of Contents Unit 1 The Korean Alphabet, hangeul Unit 2 Korean Sentences with 15 Action Verbs Introduction Review Exercises
More informationPage 2 of 6 Here are the rules for conjugating Whether (or not) and If when using a Descriptive Verb. The only difference here from Action Verbs is wh
Page 1 of 6 Learn Korean Ep. 13: Whether (or not) and If Let s go over how to say Whether and If. An example in English would be I don t know whether he ll be there, or I don t know if he ll be there.
More information歯kjmh2004v13n1.PDF
13 1 ( 24 ) 2004 6 Korean J Med Hist 13 1 19 Jun 2004 ISSN 1225 505X 1) * * 1 ( ) 2) 3) 4) * 1) ( ) 3 2) 7 1 3) 2 1 13 1 ( 24 ) 2004 6 5) ( ) ( ) 2 1 ( ) 2 3 2 4) ( ) 6 7 5)  2003 23 144166 2 2 1) 6)
More informationuntitled
Logic and Computer Design Fundamentals Chapter 4 Combinational Functions and Circuits Functions of a single variable Can be used on inputs to functional blocks to implement other than block s intended
More informationOutput file
240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 An Application for Calculation and Visualization of Narrative Relevance of Films Using Keyword Tags Choi JinWon (KAIST) Film making
More information 2 
 1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30 
More informationMicrosoft PowerPoint  ch03ysk2012.ppt [호환 모드]
전자회로 Ch3 iode Models and Circuits 김영석 충북대학교전자정보대학 2012.3.1 Email: kimys@cbu.ac.kr k Ch31 Ch3 iode Models and Circuits 3.1 Ideal iode 3.2 PN Junction as a iode 3.4 Large Signal and SmallSignal Operation
More informationpublic key private key Encryption Algorithm Decryption Algorithm 1
public key private key Encryption Algorithm Decryption Algorithm 1 OneWay Function ( ) A function which is easy to compute in one direction, but difficult to invert  given x, y = f(x) is easy  given
More information歯1.PDF
200176 .,.,.,. 5... 1/2. /. / 2. . 293.33 (54.32%), 65.54(12.13%), / 53.80(9.96%), 25.60(4.74%), 5.22(0.97%). / 3 S (1997)14.59% (1971) 10%, (1977).5%~11.5%, (1986)
More information i   ii   iii   iv   v   vi   1   2   3  1) 통계청고시제 2010150 호 (2010.7.6 개정, 2011.1.1 시행 )  4  요양급여의적용기준및방법에관한세부사항에따른골밀도검사기준 (2007 년 11 월 1 일시행 )  5   6   7   8   9   10 
More informationhttp://www.kbc.go.kr/pds/2.html Abstract Exploring the Relationship Between the Traditional Media Use and the Internet Use MeeEun Kang This study examines the relationship between
More information` Companies need to play various roles as the network of supply chain gradually expands. Companies are required to form a supply chain with outsourcing or partnerships since a company can not
More informationMicrosoft PowerPoint  AC3.pptx
Chapter 3 Block Diagrams and Signal Flow Graphs Automatic Control Systems, 9th Edition Farid Golnaraghi, Simon Fraser University Benjamin C. Kuo, University of Illinois 1 Introduction In this chapter,
More information04다시_고속철도61~80p
Approach for Value Improvement to Increase Highspeed Railway Speed An effective way to develop a highly competitive system is to create a new market place that can create new values. Creating tools and
More information본문01
Ⅱ 논술 지도의 방법과 실제 2. 읽기에서 논술까지 의 개발 배경 읽기에서 논술까지 자료집 개발의 본래 목적은 초 중 고교 학교 평가에서 서술형 평가 비중이 2005 학년도 30%, 2006학년도 40%, 2007학년도 50%로 확대 되고, 2008학년도부터 대학 입시에서 논술 비중이 커지면서 논술 교육은 학교가 책임진다. 는 풍토 조성으로 공교육의 신뢰성과
More information11¹Ú´ö±Ô
A Review on Promotion of Storytelling Local Cultures  265  2266  3267  4268  5269  6 7270  7271  8272  9273  10274  11275  12276  13277  14278  15279  16 7280  17281 
More information300 구보학보 12집. 1),,.,,, TV,,.,,,,,,..,...,....,... (recall). 2) 1) 양웅, 김충현, 김태원, 광고표현 수사법에 따른 이해와 선호 효과: 브랜드 인지도와 의미고정의 영향을 중심으로, 광고학연구 18권 2호, 2007 여름
동화 텍스트를 활용한 패러디 광고 스토리텔링 연구 55) 주 지 영* 차례 1. 서론 2. 인물의 성격 변화에 의한 의미화 전략 3. 시공간 변화에 의한 의미화 전략 4. 서사의 변개에 의한 의미화 전략 5. 창조적인 스토리텔링을 위하여 6. 결론 1. 서론...., * 서울여자대학교 초빙강의교수 300 구보학보 12집. 1),,.,,, TV,,.,,,,,,..,...,....,...
More informationMicrosoft PowerPoint  27.pptx
이산수학 () n항관계 (nary Relations) 2011년봄학기 강원대학교컴퓨터과학전공문양세 nary Relations (n항관계 ) An nary relation R on sets A 1,,A n, written R:A 1,,A n, is a subset R A 1 A n. (A 1,,A n 에대한 n 항관계 R 은 A 1 A n 의부분집합이다.)
More information iii   i   ii   iii  국문요약 종합병원남자간호사가지각하는조직공정성 사회정체성과 조직시민행동과의관계  iv   v   1   2   3   4   5   6   7   8   9   10   11   12   13   14  α α α α  15  α α α α α α
More information우리들이 일반적으로 기호
일본지방자치체( 都 道 府 縣 )의 웹사이트상에서 심벌마크와 캐릭터의 활용에 관한 연구 A Study on the Application of Japanese Local SelfGovernment's Symbol Mark and Character on Web. 나가오카조형대학( 長 岡 造 形 大 學 ) 대학원 조형연구과 김 봉 수 (Kim Bong Su) 193
More information<32382DC3BBB0A2C0E5BED6C0DA2E687770>
논문접수일 : 2014.12.20 심사일 : 2015.01.06 게재확정일 : 2015.01.27 청각 장애자들을 위한 보급형 휴대폰 액세서리 디자인 프로토타입 개발 Development Prototype of Lowend Mobile Phone Accessory Design for Hearingimpaired Person 주저자 : 윤수인 서경대학교 예술대학
More informationHilbert Transform on C1+ Families of Lines
Georgia Institute of Technology June 14, 2004 Outline Background Main Results 1 The Background of the Main Theorem Besicovitch Set Zygmund Conjecture 2 Main Results Main Theorem and Key Proposition Key
More information........pdf 16..
Abstract Prospects of and Tasks Involving the Policy of Revitalization of Traditional Korean Performing Arts YongShik, Lee National Center for Korean Traditional Performing Arts In the 21st century, the
More information퇴좈저널36호4차T.ps, page 2 @ Preflight (2)
Think Big, Act Big! Character People Literature Beautiful Life History Carcere Mamertino World Special Interview Special Writing Math English Quarts I have been driven many times to my knees by the overwhelming
More information장양수
한국문학논총 제70집(2015. 8) 333~360쪽 공선옥 소설 속 장소 의 의미  명랑한 밤길, 영란, 꽃같은 시절 을 중심으로 * 1)이 희 원 ** 1. 들어가며  장소의 인간 차 2. 주거지와 소유지 사이의 집/사람 3. 취약함의 나눔으로서의 장소 증여 례 4. 장소 소속감과 미의식의 가능성 5.
More information<B3EDB9AEC1FD5F3235C1FD2E687770>
오용록의 작품세계 윤 혜 진 1) * 이 논문은 생전( 生 前 )에 학자로 주로 활동하였던 오용록(1955~2012)이 작곡한 작품들을 살펴보고 그의 작품세계를 파악하고자 하는 것이다. 한국음악이론이 원 래 작곡과 이론을 포함하였던 초기 작곡이론전공의 형태를 염두에 둔다면 그의 연 구에서 기존연구의 방법론을 넘어서 창의적인 분석 개념과 체계를 적용하려는
More information歯M991101.PDF
2 0 0 0 2000 12 2 0 0 0 2000 12 ( ) ( ) ( ) < >. 1 1. 1 2. 5. 6 1. 7 1.1. 7 1.2. 9 1.3. 10 2. 17 3. 25 3.1. 25 3.2. 29 3.3. 29. 31 1. 31 1.1. ( ) 32 1.2. ( ) 38 1.3. ( ) 40 1.4. ( ) 42 2. 43 3. 69 4. 74.
More information<B3EDB9AEC1FD5F3235C1FD2E687770>
경상북도 자연태음악의 소박집합, 장단유형, 전단후장 경상북도 자연태음악의 소박집합, 장단유형, 전단후장  전통 동요 및 부녀요를 중심으로  이 보 형 1) * 한국의 자연태 음악 특성 가운데 보편적인 특성은 대충 밝혀졌지만 소박집합에 의한 장단주기 박자유형, 장단유형, 같은 층위 전후 구성성분의 시가( 時 價 )형태 등 은 밝혀지지 않았으므로
More informationBuy one get one with discount promotional strategy
Buy one get one with discount Promotional Strategy KyongKuk Kim, ChiGhun Lee and Sunggyun Park ISysE Department, FEG 002079 Contents Introduction Literature Review Model Solution Further research 2 ISysE
More informationChapter4.hwp
Ch. 4. Spectral Density & Correlation 4.1 Energy Spectral Density 4.2 Power Spectral Density 4.3 TimeAveraged Noise Representation 4.4 Correlation Functions 4.5 Properties of Correlation Functions 4.6
More informationλx.x (λz.λx.x z) (λx.x)(λz.(λx.x)z) (λz.(λx.x) z) Callby Name. Normal Order. (λz.z)
λx.x (λz.λx.x z) (λx.x)(λz.(λx.x)z) (λz.(λx.x) z) Callby Name. Normal Order. (λz.z) Simple Type System   1+malloc(), {x:=1,y:=2}+2,... (stuck) { } { } ADD σ,m e 1 n 1,M σ,m e 1 σ,m e 2 n 2,M + e 2 n
More informationPJTROHMPCJPS.hwp
제 출 문 농림수산식품부장관 귀하 본 보고서를 트위스트 휠 방식 폐비닐 수거기 개발 과제의 최종보고서로 제출 합니다. 2008년 4월 24일 주관연구기관명: 경 북 대 학 교 총괄연구책임자: 김 태 욱 연 구 원: 조 창 래 연 구 원: 배 석 경 연 구 원: 김 승 현 연 구 원: 신 동 호 연 구 원: 유 기 형 위탁연구기관명: 삼 생 공 업 위탁연구책임자:
More information아니라 일본 지리지, 수로지 5, 지도 6 등을 함께 검토해야 하지만 여기서는 근대기 일본이 편찬한 조선 지리지와 부속지도만으로 연구대상을 한정하 기로 한다. Ⅱ. 1876~1905년 울릉도 독도 서술의 추이 1. 울릉도 독도 호칭의 혼란과 지도상의 불일치 일본이 조선
근대기 조선 지리지에 보이는 일본의 울릉도 독도 인식 호칭의 혼란을 중심으로 Ⅰ. 머리말 이 글은 근대기 일본인 편찬 조선 지리지에 나타난 울릉도 독도 관련 인식을 호칭의 변화에 초점을 맞춰 고찰한 것이다. 일본은 메이지유신 이후 부국강병을 기도하는 과정에서 수집된 정보에 의존하여 지리지를 펴냈고, 이를 제국주의 확장에 원용하였다. 특히 일본이 제국주의 확장을
More informationDBPIANURIMEDIA
FPS게임 구성요소의 중요도 분석방법에 관한 연구 2 계층화 의사결정법에 의한 요소별 상관관계측정과 대안의 선정 The Study on the Priority of First Person Shooter game Elements using Analytic Hierarchy Process 주 저 자 : 배혜진 에이디 테크놀로지 대표 Bae, Hyejin AD Technology
More informationMicrosoft PowerPoint  7Work and Energy.ppt
Chapter 7. Work and Energy 일과운동에너지 One of the most important concepts in physics Alternative approach to mechanics Many applications beyond mechanics Thermodynamics (movement of heat) Quantum mechanics...
More information<C7A5C1F620BEE7BDC4>
연세대학교 상경대학 경제연구소 Economic Research Institute Yonsei Universit 서울시 서대문구 연세로 50 50 Yonseiro, SeodaemungS gu, Seoul, Korea TEL: (+822) 21234065 FAX: (+82 2) 3649149 Email: yeri4065@yonsei.ac. kr http://yeri.yonsei.ac.kr/new
More informationProblem New Case RETRIEVE Learned Case Retrieved Cases New Case RETAIN Tested/ Repaired Case CaseBase REVISE Solved Case REUSE Aamodt, A. and Plaza, E. (1994). Casebased reasoning; Foundational
More information<30322D28C6AF29C0CCB1E2B4EB35362D312E687770>
한국학연구 56(2016.3.30), pp.3363. 고려대학교 한국학연구소 세종시의 지역 정체성과 세종의 인문정신 * 1)이기대 ** 국문초록 세종시의 상황은 세종이 왕이 되면서 겪어야 했던 과정과 닮아 있다. 왕이 되리라 예상할 수 없었던 상황에서 세종은 왕이 되었고 어려움을 극복해 갔다. 세종시도 갑작스럽게 행정도시로 계획되었고 준비의 시간 또한 짧았지만,
More information182 동북아역사논총 42호 금융정책이 조선에 어떤 영향을 미쳤는지를 살펴보고자 한다. 일제 대외금융 정책의 기본원칙은 각 식민지와 점령지마다 별도의 발권은행을 수립하여 일본 은행권이 아닌 각 지역 통화를 발행케 한 점에 있다. 이들 통화는 일본은행권 과 等 價 로 연
越 境 하는 화폐, 분열되는 제국  滿 洲 國 幣 의 조선 유입 실태를 중심으로 181 越 境 하는 화폐, 분열되는 제국  滿 洲 國 幣 의 조선 유입 실태를 중심으로  조명근 고려대학교 BK21+ 한국사학 미래인재 양성사업단 연구교수 Ⅰ. 머리말 근대 국민국가는 대내적으로는 특정하게 구획된 영토에 대한 배타적 지배와 대외적 자주성을 본질로 하는데, 그
More informationÀ±½Â¿í Ãâ·Â
Representation, Encoding and Intermediate View Interpolation Methods for Multiview Video Using Layered Depth Images The multiview video is a collection of multiple videos, capturing the same scene at
More informationMicrosoft PowerPoint  CHAP03 [호환 모드]
컴퓨터구성 Lecture Series #4 Chapter 3: Data Representation Spring, 2013 컴퓨터구성 : Spring, 2013: No. 41 Data Types Introduction This chapter presents data types used in computers for representing diverse numbers
More information<5B335DC0B0BBF3C8BF2835B1B35FC0FAC0DAC3D6C1BEBCF6C1A4292E687770>
동남아시아연구 20권 2호(2010) : 73~99 한국 영화와 TV 드라마에 나타난 베트남 여성상 고찰* 1) 육 상 효** 1. 들어가는 말 한국의 영화와 TV 드라마에 아시아 여성으로 가장 많이 등장하는 인물은 베트남 여성이다. 왜 베트남 여성인가? 한국이 참전한 베트 남 전쟁 때문인가? 영화 , , 드라마 을
More information216 동북아역사논총 41호 인과 경계공간은 설 자리를 잃고 배제되고 말았다. 본고에서는 근세 대마도에 대한 한국과 일본의 인식을 주로 영토와 경계인 식을 중심으로 고찰하고자 한다. 이 시기 대마도에 대한 한일 양국의 인식을 살펴볼 때는 근대 국민국가적 관점에서 탈피할
전근대시기 한국과 일본의 대마도 인식 215 전근대시기 한국과 일본의 대마도 인식 하우봉 전북대학교 사학과 교수 Ⅰ. 머리말 브루스 배튼(Bruce Batten)의 정의에 따르면 전근대의 국경에는 국경선으로 이루어진 boundary가 있고, 공간으로 이루어진 frontier란 개념이 있다. 전자 는 구심적이며 내와 외를 격리시키는 기능을 지니고, 후자는 원심적이며
More informationIKC43_06.hwp
2), * 2004 BK21. ** 156,..,. 1) (1909) 57, (1915) 106, ( ) (1931) 213. 1983 2), 1996. 3). 4) 1),. (,,, 1983, 7 12 ). 2),. 3),, 33,, 1999, 185 224. 4), (,, 187 188 ). 157 5) ( ) 59 2 3., 1990. 6) 7),.,.
More information슬라이드 제목 없음
물리화학 1 문제풀이 130403 김대형교수님 Chapter 1 Exercise (#1) A sample of 255 mg of neon occupies 3.00 dm 3 at 122K. Use the perfect gas law to calculate the pressure of the gas. Solution 1) The perfect gas law p
More information09김정식.PDF
0009 2000. 12 ,,,,.,.,.,,,,,,.,,..... . 1 1 7 2 9 1. 9 2. 13 3. 14 3 16 1. 16 2. 21 3. 39 4 43 1. 43 2. 52 3. 56 4. 66 5. 74 5 78 1. 78 2. 80 3. 86 6 88 90 Ex e cu t iv e Su m m a r y 92 < 31> 22 < 32>
More information<BFA9BAD02DB0A1BBF3B1A4B0ED28C0CCBCF6B9FC2920B3BBC1F62E706466>
001 002 003 004 005 006 008 009 010 011 2010 013 I II III 014 IV V 2010 015 016 017 018 I. 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 III. 041 042 III. 043
More information<BCF6BDC3323030392D31385FB0EDBCD3B5B5B7CEC8DEB0D4C5B8BFEEB5B5C0D4B1B8BBF3BFACB1B85FB1C7BFB5C0CE2E687770>
... 수시연구 200918.. 고속도로 휴게타운 도입구상 연구 A Study on the Concept of Service Town at the Expressway Service Area... 권영인 임재경 이창운... 서 문 우리나라는 경제성장과 함께 도시화가 지속적으로 진행되어 지방 지역의 인구감소와 경기의 침체가 계속되고 있습니다. 정부의 다각 적인
More informationOutput file
connect educational content with entertainment content and that production of various contents inducing educational motivation is important. Key words: edutainment, virtual world, fostering simulation
More information204 205
Road Traffic Crime and Emergency Evacuation  202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 Abstract Road Traffic Crime
More information07_Àü¼ºÅÂ_0922
176 177 1) 178 2) 3) 179 4) 180 5) 6) 7) 8) 9) 10) 181 11) 12) 182 13) 14) 15) 183 16) 184 185 186 17) 18) 19) 20) 21) 187 22) 23) 24) 25) 188 26) 27) 189 28) 29) 30)31) 32) 190 33) 34) 35) 36) 191 37)
More informationsnanodeties
Node Centrality in Social Networks Nov. 2015 YounHee Han http://link.koreatech.ac.kr Importance of Nodes ² Question: which nodes are important among a large number of connected nodes? Centrality analysis
More information09È«¼®¿µ5~152s
Korean Journal of Remote Sensing, Vol.23, No.2, 2007, pp.45~52 Measurement of Backscattering Coefficients of Rice Canopy Using a Ground Polarimetric Scatterometer System SukYoung Hong*, JinYoung Hong**,
More information03¹ü¼±±Ô
Relevancy between Aliases of Eight Provinces and Topographical Features during the Chosun Dynasty SeonGyu Beom* Abstract : In Korea, aside from their official administrative names, aliases of each province
More informationJournal of Educational Innovation Research 2016, Vol. 26, No. 3, pp DOI: Awareness, Supports
Journal of Educational Innovation Research 2016, Vol. 26, No. 3, pp.335363 DOI: http://dx.doi.org/10.21024/pnuedi.26.3.201612.335 Awareness, Supports in Need, and Actual Situation on the Curriculum Reconstruction
More informationCrt114( ).hwp
cdna Microarray Experiment: Design Issues in Early Stage and the Need of Normalization Byung Soo Kim, Ph.D. 1, Sunho Lee, Ph.D. 2, Sun Young Rha, M.D., Ph.D. 3,4 and Hyun Cheol Chung, M.D., Ph.D. 3,4 1
More information2009년 국제법평론회 동계학술대회 일정
한국경제연구원 대외세미나 인터넷전문은행 도입과제와 캐시리스사회 전환 전략 일시 2016년 3월 17일 (목) 14:00 ~17:30 장소 전경련회관 컨퍼런스센터 2층 토파즈룸 주최 한국경제연구원 한국금융ICT융합학회 PROGRAM 시 간 내 용 13:30~14:00 등 록 14:00~14:05 개회사 오정근 (한국금융ICT융합학회 회장) 14:05~14:10
More informationJournal of Educational Innovation Research 2018, Vol. 28, No. 4, pp DOI: * A Research Trend
Journal of Educational Innovation Research 2018, Vol. 28, No. 4, pp.295318 DOI: http://dx.doi.org/10.21024/pnuedi.28.4.201812.295 * A Research Trend on the Studies related to Parents of Adults with Disabilities
More information134 25, 135 3, (Aloysius Pieris) ( r e a l i t y ) ( P o v e r t y ) ( r e l i g i o s i t y ) 1 ) 21, 21, 1) Aloysius Pieris, An Asian Theology of Li
25 133162 ( ) I 21 134 25, 135 3, (Aloysius Pieris) ( r e a l i t y ) ( P o v e r t y ) ( r e l i g i o s i t y ) 1 ) 21, 21, 1) Aloysius Pieris, An Asian Theology of Liberation (New York: Orbis Books,
More information04 형사판례연구 1931.hwp
2010년도 형법판례 회고 645 2010년도 형법판례 회고 2)오 영 근* Ⅰ. 서설 2010. 1. 1.에서 2010. 12. 31.까지 대법원 법률종합정보 사이트 1) 에 게재된 형법 및 형사소송법 판례는 모두 286건이다. 이 중에는 2건의 전원합의체 판결 및 2건의 전원합의체 결정이 있다. 2건의 전원합의체 결정은 형사소송법에 관한 것이고, 2건의
More information삼교14.hwp
5 19대 총선 후보 공천의 과정과 결과, 그리고 쟁점: 새누리당과 민주통합당을 중심으로* 윤종빈 명지대학교 논문요약 이 글은 19대 총선의 공천의 제도, 과정, 그리고 결과를 분석한다. 이론적 검증보다는 공천 과정의 설명과 쟁점의 발굴에 중점을 둔다. 4 11 총선에서 새누리당과 민주통합당의 공천은 기대와 달랐고 그 특징은 다음과 같이 요약될 수 있다. 첫째,
More information한국성인에서초기황반변성질환과 연관된위험요인연구
한국성인에서초기황반변성질환과 연관된위험요인연구 한국성인에서초기황반변성질환과 연관된위험요인연구   i   i   ii   iii   iv  χ  v   vi   1   2   3   4  그림 1. 연구대상자선정도표  5   6   7   8  그림 2. 연구의틀 χ  9   10   11 
More information03±èÀçÈÖ¾ÈÁ¤ÅÂ
x x x x Abstract The Advertising Effects of PPL in TV Dramas  Identificaiton by Implicit Memorybased Measures Kim, Jae  hwi(associate professor, Dept. of psychology, ChungAng University) Ahn,
More information현대영화연구
와 에 나타난 섬과 소통의 의미 * 1) 곽수경 국립목포대학교 도서문화연구원 HK연구교수 1. 영화와 섬 2. 물리적인 섬과 상징된 섬 3. 소통수단 4. 결론 목 차 국문초록 최근에는 섬이 해양영토로 인식되고 다양한 방송프로그램을 통해 자주 소 개되면서 다각도로 섬에 대한 관심이 높아지고 있다. 하지만 그에 비해 섬을 배경으로 한 영화는
More informationJournal of Educational Innovation Research 2018, Vol. 28, No. 3, pp DOI: NCS : * A Study on
Journal of Educational Innovation Research 2018, Vol. 28, No. 3, pp.157176 DOI: http://dx.doi.org/10.21024/pnuedi.28.3.201809.157 NCS : * A Study on the NCS Learning Module Problem Analysis and Effective
More information<3130C0E5>
Redundancy Adding extra bits for detecting or correcting errors at the destination Types of Errors SingleBit Error Only one bit of a given data unit is changed Burst Error Two or more bits in the data
More informationePapyrus PDF Document
육아지원연구 2008. 제 3권 1 호, 147170 어린이집에서의 낮잠에 대한 교사와 부모의 인식 및 실제 이 슬 기(동작구 보육정보센터)* 1) 요 약 본 연구의 목적은 어린이집에서의 일과 중 낮잠 시간에 대한 교사와 부모의 인식 및 실제를 알아봄 으로써, 교사와 부모의 협력을 통해 바람직한 낮잠 시간을 모색해 보는 데 있었다. 연구 대상은 서울, 경기지역
More informationVol.259 C O N T E N T S M O N T H L Y P U B L I C F I N A N C E F O R U M
2018.01 Vol.259 C O N T E N T S 02 06 28 61 69 99 104 120 M O N T H L Y P U B L I C F I N A N C E F O R U M 2 2018.1 3 4 2018.1 1) 2) 6 2018.1 3) 4) 7 5) 6) 7) 8) 8 2018.1 9 10 2018.1 11 2003.08 2005.08
More information서강대학원123호
123 2012년 12월 6일 발행인 이종욱 총장 편집인 겸 주간 임종섭 편집장 김아영 (우편번호 121742) 주소 서울시 마포구 신수동1번지 엠마오관 B133호 대학원신문사 전화 7058269 팩스 7131919 제작 일탈기획(07044048447) 웃자고 사는 세상, 정색은 언행 총량의 2%면 족하다는 신념으로 살았습니다. 그 신념 덕분인지 다행히
More information232 도시행정학보 제25집 제4호 I. 서 론 1. 연구의 배경 및 목적 사회가 다원화될수록 다양성과 복합성의 요소는 증가하게 된다. 도시의 발달은 사회의 다원 화와 밀접하게 관련되어 있기 때문에 현대화된 도시는 경제, 사회, 정치 등이 복합적으로 연 계되어 있어 특
한국도시행정학회 도시행정학보 제25집 제4호 2012. 12 : pp.231~251 생활지향형 요소의 근린주거공간 분포특성 연구: 경기도 시 군을 중심으로* Spatial Distribution of Daily LifeOriented Features in the Neighborhood: Focused on Municipalities of Gyeonggi Province
More informationI&IRC5 TG_08권
I N T E R E S T I N G A N D I N F O R M A T I V E R E A D I N G C L U B The Greatest Physicist of Our Time Written by Denny Sargent Michael Wyatt I&I Reading Club 103 본문 해석 설명하기 위해 근래의 어떤 과학자보다도 더 많은 노력을
More information<C1A4BAB8B9FDC7D031362D335F3133303130322E687770>
권리범위확인심판에서는 법원이 진보성 판단을 할 수 없는가? Can a Court Test the Inventive Step in a Trial to Confirm the Scope of a Patent? 구대환(Koo, DaeHwan) * 41) 목 차 Ⅰ. 서론 Ⅱ. 전원합의체판결의 진보성 판단 관련 판시사항 1. 이 사건 특허발명 2. 피고 제품 3.
More information2017.09 Vol.255 C O N T E N T S 02 06 26 58 63 78 99 104 116 120 122 M O N T H L Y P U B L I C F I N A N C E F O R U M 2 2017.9 3 4 2017.9 6 2017.9 7 8 2017.9 13 0 13 1,007 3 1,004 (100.0) (0.0) (100.0)
More information0508 087ÀÌÁÖÈñ.hwp
산별교섭에 대한 평가 및 만족도의 영향요인 분석(이주희) ꌙ 87 노 동 정 책 연 구 2005. 제5권 제2호 pp. 87118 c 한 국 노 동 연 구 원 산별교섭에 대한 평가 및 만족도의 영향요인 분석: 보건의료노조의 사례 이주희 * 2004,,,.. 1990. : 2005 4 7, :4 7, :6 10 * (jlee@ewha.ac.kr) 88 ꌙ 노동정책연구
More informationOR MS와 응용03장
o R M s graphical solution algebraic method ellipsoid algorithm Karmarkar 97 George B Dantzig 979 Khachian Karmarkar 98 Karmarkar interiorpoint algorithm o R 08 gallon 000 000 00 60 g 0g X : : X : : Ms
More information<C7D1B1B9B1A4B0EDC8ABBAB8C7D0BAB85F31302D31C8A35F32C2F75F303132392E687770>
버스 외부 광고의 효과에 관한 탐색적 연구 : 매체 접촉률과 인지적 반응을 중심으로 1) 고한준 국민대학교 언론정보학부 조교수 노봉조 벅스컴애드 대표 이사 최근 몇 년 사이 옥외 광고나 인터넷 광고 등 BTL(Below the Line) 매체가 광고 시장에서 차지하 는 비중이 점점 높아지고 있다. 버스 외부 광고는 2004년 7월 서울시 교통체계개편 이후 이용자
More information<31342D3034C0E5C7FDBFB52E687770>
아카데미 토론 평가에 대한 재고찰  토론승패와 설득은 일치하는가  장혜영 (명지대) 1. 들어가는 말 토론이란 무엇일까? 토론에 대한 정의는 매우 다양하다. 안재현 과 오창훈은 토론에 대한 여러 정의들을 검토한 후 이들을 종합하 여 다음과 같이 설명하고 있다. 토론이란 주어진 주제에 대해 형 식과 절차에 따라 각자 자신의 의견을 합리적으로 주장하여 상대
More informationDBPIANURIMEDIA
The ebusiness Studies Volume 17, Number 4, August, 30, 2016:319~332 Received: 2016/07/28, Accepted: 2016/08/28 Revised: 2016/08/27, Published: 2016/08/30 [ABSTRACT] This paper examined what determina
More informationDBPIANURIMEDIA
The ebusiness Studies Volume 17, Number 6, December, 30, 2016:275~289 Received: 2016/12/02, Accepted: 2016/12/22 Revised: 2016/12/20, Published: 2016/12/30 [ABSTRACT] SNS is used in various fields. Although
More informationJournal of Educational Innovation Research 2017, Vol. 27, No. 3, pp DOI: (NCS) Method of Con
Journal of Educational Innovation Research 2017, Vol. 27, No. 3, pp.181212 DOI: http://dx.doi.org/10.21024/pnuedi.27.3.201709.181 (NCS) Method of Constructing and Using the Differentiated National Competency
More information<B7CEC4C3B8AEC6BCC0CEB9AEC7D0322832303039B3E23130BFF9292E687770>
로컬리티 인문학 2, 2009. 10, 257~285쪽 좌절된 세계화와 로컬리티  1960년대 한국영화와 재외한인 양 인 실* 50) 국문초록 세계화 로컬리티는 특정장소나 경계를 지칭하는 것이 아니라 관계와 시대에 따 라 유동적으로 변화하는 개념이다. 1960년대 한국영화는 유례없는 변화를 맞이하고 있었다. 그 중 가장 특이할 만한 사실은 미국과 일본의 영화계에서
More information1. 서론 11 연구 배경과 목적 12 연구 방법과 범위 2. 클라우드 게임 서비스 21 클라우드 게임 서비스의 정의 22 클라우드 게임 서비스의 특징 23 클라우드 게임 서비스의 시장 현황 24 클라우드 게임 서비스 사례 연구 25 클라우드 게임 서비스에
IPTV 기반의 클라우드 게임 서비스의 사용성 평가  CGames와 Wiz Game 비교 중심으로  Evaluation on the Usability of IPTVBased Cloud Game Service  Focus on the comparison between CGames and Wiz Game  주 저 자 : 이용우 (Lee, Yong Woo)
More informationWRIEHFIDWQWF.hwp
목 차 Abstract 1. 서론 2. 한국영화 흥행의 배경 2.1. 대중문화와 흥행영화 2.2. 흥행변수에 관한 조망 3. 영화 의 내러티브 특징 3.1. 일원적 대칭형 인물 설정 3.2. 에피소드형 이야기 구조 3.3. 감각적인 대사 처리 4. 시청각 미디어 환경의 영향력 4.1. 대중 매체가 생산한 기억 4.2. 기존 히트곡 사용의 위력 5. 결론 참고문헌
More information에너지경제연구 Korean Energy Economic Review Volume 17, Number 2, September 2018 : pp. 1~29 정책 용도별특성을고려한도시가스수요함수의 추정 :, ARDL,,, C4, Q41 
에너지경제연구 Korean Energy Economic Review Volume 17, Number 2, September 2018 : pp. 1~29 정책 용도별특성을고려한도시가스수요함수의 추정 :, ARDL,,, C4, Q41  .  2  . 1.  3  [ 그림 1] 도시가스수요와실질 GDP 추이  4   5   6  < 표 1>
More information i   ii   iii   iv   v   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19  α α  20  α α α α α α  21   22   23 
More information음주측정을 위한 긴급강제채혈의 절차와 법리, A Study on the Urgent Compulsory Blood
음주측정을 위한 긴급강제채혈의 절차와 법리 A Study on the Urgent Compulsory Blood Collecting for Investigation of Driving while Intoxicated 양 동 철 * (Yang, DongChul) < 차 례 > Ⅰ. 서론 Ⅱ. 체내신체검사와 긴급압수ㆍ수색ㆍ검증의 허용범위 Ⅲ. 긴급강제채혈의 허용범위와
More information10송동수.hwp
종량제봉투의 불법유통 방지를 위한 폐기물관리법과 조례의 개선방안* 1) 송 동 수** 차 례 Ⅰ. 머리말 Ⅱ. 종량제봉투의 개요 Ⅲ. 종량제봉투의 불법유통사례 및 방지대책 Ⅳ. 폐기물관리법의 개선방안 Ⅴ. 지방자치단체 조례의 개선방안 Ⅵ. 결론 국문초록 1995년부터 쓰레기 종량제가 시행되면서 각 지방자치단체별로 쓰레기 종량제 봉투가 제작, 판매되기 시작하였는데,
More informationPowerPoint 프레젠테이션
Reasons for Poor Performance Programs 60% Design 20% System 2.5% Database 17.5% Source: ORACLE Performance Tuning 1 SMS TOOL DBA Monitoring TOOL Administration TOOL Performance Insight Backup SQL TUNING
More information16(1)3(국문)(p.4045).fm
w wz 16«1y Kor. J. Clin. Pharm., Vol. 16, No. 1. 2006 x w$btf3fqpsu'psn û w m w Department of Statistics, Chonnam National University Eunsik Park College of Natural Sciences, Chonnam National University
More information<33C2F731323239292DC5D8BDBAC6AEBEF0BEEEC7D02D3339C1FD2E687770>
텍스트언어학 39, 2015, pp. 283~311 한국 대중가요 가사의 문체 분석 장소원(서울대) Chang, Sowon, 2015. The stylistic Analysis of the lyrics of Korean popular song. Textlinguistics 39. The sociological approach, one of the methods
More information영남학17합본.hwp
退 溪 讀 書 詩 에 나타난 樂 의 層 位 와 그 性 格 신 태 수 * 53) Ⅰ. 문제 제기 Ⅱ. 讀 書 詩 의 양상과 樂 의 의미 층위 Ⅲ 敬 의 작용과 樂 개념의 구도 1. 敬 과 靜 味 樂 의 관계 2. 樂 개념의 구도와 敬 의 기능 Ⅳ. 樂 개념이 讀 書 詩 에서 지니는 미학적 성격 1. 樂 의 심상 체계, 그 심미안과 능동성 2. 樂 의 審 美 構
More information<3136C1FD31C8A35FC3D6BCBAC8A3BFDC5F706466BAAFC8AFBFE4C3BB2E687770>
부동산학연구 제16집 제1호, 2010. 3, pp. 117~130 Journal of the Korea Real Estate Analysts Association Vol.16, No.1, 2010. 3, pp. 117~130 비선형 MankiwWeil 주택수요 모형  수도권 지역을 대상으로  NonLinear MankiwWeil Model on Housing
More information00½ÃÀÛ 5š
The Career of Christian Counselor and the Management of Counseling Center Yeo Han Koo 1) 1 17 2011 29 2) 3) 30 4) 5) 6) 7) 2 3 1999 99 2006 23 45 2007 17 19 4 17 2011 95 148 5 100 2014 26 6 29 17 2004
More information2 KHU 글로벌 기업법무 리뷰 제2권 제1호 또 내용적으로 중대한 위기를 맞이하게 되었고, 개인은 흡사 어항 속의 금붕어 와 같은 신세로 전락할 운명에 처해있다. 현대정보화 사회에서 개인의 사적 영역이 얼마나 침해되고 있는지 는 양 비디오 사건 과 같은 연예인들의 사
연구 논문 헌법 제17조 사생활의 비밀과 자유에 대한 소고 연 제 혁* I. II. III. IV. 머리말 사생활의 비밀과 자유의 의의 및 법적 성격 사생활의 비밀과 자유의 내용 맺음말 I. 머리말 사람은 누구나 타인에게 알리고 싶지 않은 나만의 영역(Eigenraum) 을 혼자 소중히 간직하 기를 바랄 뿐만 아니라, 자기 스스로의 뜻에 따라 삶을 영위해 나가면서
More information석사논문.PDF
ABO Rh A study on the importance of ABO and Rh blood groups information in Public Health 2000 2 1 ABO Rh A study on the importance of ABO and Rh blood groups information in Public Health 2000 2 2 ABO Rh
More informationStage 2 First Phonics
ORT Stage 2 First Phonics The Big Egg What could the big egg be? What are the characters doing? What do you think the story will be about? (큰 달걀은 무엇일까요? 등장인물들은 지금 무엇을 하고 있는 걸까요? 책은 어떤 내용일 것 같나요?) 대해 칭찬해
More information대한한의학원전학회지26권4호교정본(1125).hwp
http://www.wonjeon.org http://dx.doi.org/10.14369/skmc.2013.26.4.267 熱入血室證에 대한 小考 1 2 慶熙大學校大學校 韓醫學科大學 原典學敎室 韓醫學古典硏究所 白裕相1, 2 *117) A Study on the Pattern of 'Heat Entering The Blood Chamber' 1, Baik 1
More information2 동북아역사논총 50호 구권협정으로 해결됐다 는 일본 정부의 주장에 대해, 일본군 위안부 문제는 일 본 정부 군 등 국가권력이 관여한 반인도적 불법행위이므로 한일청구권협정 에 의해 해결된 것으로 볼 수 없다 는 공식 입장을 밝혔다. 또한 2011년 8월 헌 법재판소는
일본군 위안부 피해자 구제에 관한 일고( 一 考 ) 1 일본군 위안부 피해자 구제에 관한 일고( 一 考 ) 김관원 / 동북아역사재단 연구위원 Ⅰ. 머리말 일본군 위안부 문제가 한일 간 현안으로 불거지기 시작한 것은 일본군 위안부 피해를 공개 증언한 김학순 할머니 등이 일본에서 희생자 보상청구 소송을 제 기한 1991년부터다. 이때 일본 정부는 일본군이 위안부
More information