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(Non-)Density Properties of Discrete Gabor Multipliers

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(Non-)Density Properties of Discrete Gabor Multipliers Proceedings of the 10th International Conference on Sampling Theory and Applications (Non-)Density Properties of Discrete Gabor Multipliers Dominik Bayer Peter Balazs Acoustics Research Institute Acoustics Research Institute Austrian Academy of Sciences Austrian Academy of Sciences Wien, Austria Wien, Austria Email: [email protected] Email: [email protected] Abstract—This paper is concerned with the possibility of ap- Lemma I.2. Let (fn) be a Bessel sequence with Bessel bound proximating arbitrary operators by multipliers for Gabor frames B. Then, for all n 2 N, or more general Bessel sequences. It addresses the question of p whether sets of multipliers (whose symbols come from prescribed kfnk ≤ B: function classes such as `2) constitute dense subsets of various spaces of operators (such as Hilbert-Schmidt class). We prove a B. Time-frequency analysis number of negative results that show that in the discrete setting 2 subspaces of multipliers are usually not dense and thus too small In the Hilbert space L (R), define the translation operator to guarantee arbitrary good approximation. This is in contrast Txf(t) = f(t − x) and the modulation operator M!f(t) = to the continuous case. e−2πi!tf(t) (for f 2 L2 and x; ! 2 R). These are unitary operators on L2. They combine to form the time-frequency I. PRELIMINARIES shift π(x; !) = M!Tx. The short-time Fourier transform All Hilbert spaces are assumed to be separable and infinite- (STFT) of f with window g is defined as the bilinear time- dimensional. frequency distribution Z −2πi!t A. Bessel sequences Vgf(x; !) = f(t)g(t − x)e dt = hf; π(x; !)gi: R Let H be a Hilbert space with inner product, linear in the If (x ;! ), n 2 , is a discrete subset of 2 and h 2 L2, then first argument, denoted by h·; ·i. A sequence (f ), n 2 , n n N R n N the family of functions (π(x ;! )h) is called a Gabor system. of elements of H is called a Bessel sequence if there exists n n P 2 2 If a Gabor system constitutes a Bessel sequence or a frame a constant B > 0 such that jhh; fnij ≤ Bkhk for n2N for L2, we speak of a Bessel Gabor system or Gabor frame, all h 2 H. Any such number B is called a Bessel bound of respectively. Another important time-frequency distribution is the Bessel sequence, the smallest such constant the optimal the (cross) Wigner distribution of f and g: Bessel bound. If a Bessel sequence satisfies additionally the Z analogous inequality from below, i.e. there exists a constant t t −2πi!t P 2 2 W (f; g)(x; !) = f(x + )g(x − )e dt: A > 0 such that jhh; fnij ≥ Akhk for all h 2 H, then n2N 2 2 the sequence is called a frame for H. Prominent examples of R Bessel sequences are orthonormal systems, which are Bessel It is related to the STFT via the formula W (f; g)(x; !) = 4πi!x sequences with Bessel bound 1. For a Bessel sequence (f ), 2e Vg~f(2x; 2!) with g~(t) = g(−t). Both STFT and n 2 2 2 2 Wigner distribution are in L (R ) if f and g are in L (R). the analysis operator C : H ! ` , h 7! Ch := (hh; fni)n2N, 2 Both can be defined for larger classes of functions or even and the synthesis operator D : ` ! H, c = (cn) 7! Dc := P distributions for f and g. The Wigner distribution is associated cnfn (the series converges in the norm topology of H), n2N are well-defined and adjoint to each other: C = D∗. to the Weyl calculus: every continuous operator T : S! 0 0 A useful characterization of Bessel sequences is the follow- S from Schwartz class S to the tempered distributions S ing (cf. [4]): can be described in the form hT f; gi = hσ; W (g; f)i for f; g 2 S, with a suitable unique (distributional) Weyl symbol 0 2 Lemma I.1. Let (fn) be a sequence in H and (en) be an σ 2 S (R ). If T is a Hilbert-Schmidt operator, then one has 2 2 arbitrary orthonormal basis. Then (fn) is a Bessel sequence σ 2 L (R ). For all of these facts and many more we refer if and only if there exists a bounded operator T 2 B(H) with to [6]. fn = T en for all n 2 N. The optimal Bessel bound B is given 2 C. Compact and Schatten class operators by B = kT kB(H). We will often use the following basic fact about Bessel A bounded operator T : H ! H is compact if the sequences (see e.g. [4]): image of any bounded sequence under T contains a conver- gent subsequence. A compact operator always has a spectral 25 Proceedings of the 10th International Conference on Sampling Theory and Applications P representation T (·) = k sk(T ) h·; φki k with suitably cho- III. BEREZIN TRANSFORM sen orthonormal systems (φk), ( k) and a unique sequence Definition III.1. Let (fn) and (gn) be Bessel sequences in H (sk(T )) with s1(T ) ≥ s2(T ) ≥ ::: ≥ 0, k 2 N. The and T 2 B(H). The Berezin transform of T (associated to sequence (sk(T )) is the sequence of singular values of T . the sequences (f ) and (g )) is defined as the function on p n n N The operator belongs to Schatten p-class S (H), 1 ≤ p < 1, given by P p if k jsk(T )j < 1. These are Banach spaces with norm p 1=p 2 (fn);(gn) p P B (T )(n) := hT fn; gni; n 2 : kT kS = k(sk(T ))kp = ( k jsk(T )j ) . S (H) is also N called Hilbert-Schmidt class, S1(H) trace class. We use the In order to simplify notation we will usually suppress the notation S1(H) to denote the set B(H) of all bounded dependence on the Bessel sequences (f ) and (g ) and simply operators on H, and S0(H) = K(H) to denote the set of n n write B(T ) instead of B(fn);(gn)(T ). all compact operators on H. For more information, refer to e.g. [5] or [7]. Lemma III.2. Let (fn) and (gn) be Bessel sequences with Bessel bounds B and B , respectively. Then the Berezin II. BESSEL MULTIPLIERS F G transform B(T ) is bounded, hence in `1(N), and Definition II.1. Let (fn) and (gn) be Bessel sequences in H 1 p and m = (mn) 2 ` . The Bessel multiplier with symbol m kB(T )k1 ≤ BF BG kT kB(H): (associated to the sequences (fn) and (gn)) is defined as the linear operator on H given by Proof: We have p p (fn);(gn) X jB(T )(n)j ≤ kT k kf kkg k ≤ kT k B B A (m)(h) := mnhh; fnign; h 2 H: B(H) n n B(H) F G n by Lemma I.2, for all n 2 N. In order to simplify notation, we will usually suppress the For later use, we calculate the Berezin transform of a rank- (f ) (g ) dependence on the Bessel sequences n and n and simply one operator. write A(m) instead of A(fn);(gn)(m). We cite without proof several results from [1]. Corollary III.3. Let φ, 2 H and T : H ! H, h 7! hh; φi a rank-one operator. Then Lemma II.2. Let (fn) and (gn) be Bessel sequences with 1 Bessel bounds BF and BG, respectively. If (mn) 2 ` , then B(T )(n) = hfn; φih ; gni: A(m) is a well-defined bounded operator on H with norm p We collect further mapping properties of the Berezin trans- kA(m)kB(H) ≤ BF BG kmk1. form. Lemma II.3. Let (fn) and (gn) be Bessel sequences with 1 Lemma III.4. Suppose T 2 B(H) is a compact op- Bessel bounds BF and BG, respectively. If (mn) 2 ` , then erator and (f ) and (g ) are Bessel sequences. Then A(m) is a trace class operator on H with norm kA(m)kS1 ≤ n n p lim jB(T )(n)j = 0, i.e. B(T ) 2 c ( ). BF BG kmk1. n!1 0 N w Lemma II.4. Let (fn) and (gn) be Bessel sequences with Proof: Since fn −* 0 for n ! 1, we have kT fnk ! 0 Bessel bounds BF and BG, respectively. If limn!1 mn = 0, for n ! 1. Together with Lemma I.2 this yields i.e. m 2 c0(N), then A(m) is a compact operator. jhT fn; gnij ≤ kT fnkkgnk ! 0; for n ! 1: From Lemma II.2 and Lemma II.3, the following is easily proved by interpolation: Lemma III.5. Let (fn) and (gn) be Bessel sequences with Lemma II.5. Let (fn) and (gn) be Bessel sequences with p Bessel bounds BF and BG, respectively. Let T be a Schatten Bessel bounds BF and BG, repsectively. If m 2 ` (N), 1 ≤ p < 1, then A(m) is a Schatten p-class operator, and class operator, with 1 ≤ p < 1. Then the Berezin transform p p B(T ) is in ` (N), and kA(m)kSp ≤ BF BG kmkp. p Table I summarizes these results. kB(T )kp ≤ BF BG kT kSp : Symbol Bessel Multiplier Proof: Let (en) be an arbitrary orthonormal basis for H. 1 1 By Lemma I.1, there are bounded operators R and S in B(H) ` (N) B(H) = S (H) 0 0 such that fn = Ren and gn = Sen for all n and kRkB(H) ≤ c0(N) = ` (N) K(H) = S (H), compact operator p p p p BF and kSkB(H) ≤ BG.
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