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 ∈ N, or more general Bessel sequences. It addresses the question of √ 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 L (R), define the translation operator to guarantee arbitrary good approximation. This is in contrast Txf(t) = f(t − x) and the modulation operator Mωf(t) = to the continuous case. e−2πiωtf(t) (for f ∈ L2 and x, ω ∈ R). These are unitary operators on L2. They combine to form the time-frequency I.PRELIMINARIES shift π(x, ω) = MωTx. 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 −2πiωt A. Bessel sequences Vgf(x, ω) = f(t)g(t − x)e dt = hf, π(x, ω)gi. R Let H be a Hilbert space with inner product, linear in the If (x , ω ), n ∈ , is a discrete subset of 2 and h ∈ L2, then first argument, denoted by h·, ·i. A sequence (f ), n ∈ , n n N R n N the family of functions (π(x , ω )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 |hh, fni| ≤ Bkhk for n∈N for L2, we speak of a Bessel Gabor system or Gabor frame, all h ∈ 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 −2πiωt P 2 2 W (f, g)(x, ω) = f(x + )g(x − )e dt. A > 0 such that |hh, fni| ≥ Akhk for all h ∈ H, then n∈N 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, ω) = 4πiωx sequences with Bessel bound 1. For a Bessel sequence (f ), 2e Vg˜f(2x, 2ω) 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)n∈N, 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), n∈N 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 ∈ 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 ∈ S (R ). If T is a Hilbert-Schmidt operator, then one has 2 2 arbitrary orthonormal basis. Then (fn) is a Bessel sequence σ ∈ L (R ). For all of these facts and many more we refer if and only if there exists a T ∈ B(H) with to [6]. fn = T en for all n ∈ 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 always has a spectral

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P representation T (·) = k sk(T ) h·, φki ψk with suitably cho- III.BEREZIN TRANSFORM sen orthonormal systems (φk), (ψk) 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 ∈ N. The and T ∈ 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 < ∞, given by P p if k |sk(T )| < ∞. These are Banach spaces with norm p 1/p 2 (fn),(gn) p P B (T )(n) := hT fn, gni, n ∈ . kT kS = k(sk(T ))kp = ( k |sk(T )| ) . S (H) is also N called Hilbert-Schmidt class, S1(H) . We use the In order to simplify notation we will usually suppress the notation S∞(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 `∞(N), and Definition II.1. Let (fn) and (gn) be Bessel sequences in H ∞ p and m = (mn) ∈ ` . The Bessel multiplier with symbol m kB(T )k∞ ≤ 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 |B(T )(n)| ≤ kT k kf kkg k ≤ kT k B B A (m)(h) := mnhh, fnign, h ∈ H. B(H) n n B(H) F G n by Lemma I.2, for all n ∈ 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, ψ ∈ H and T : H → H, h 7→ hh, φiψ a rank-one operator. Then Lemma II.2. Let (fn) and (gn) be Bessel sequences with ∞ Bessel bounds BF and BG, respectively. If (mn) ∈ ` , then B(T )(n) = hfn, φihψ, gni. A(m) is a well-defined bounded operator on H with norm √ We collect further mapping properties of the Berezin trans- kA(m)kB(H) ≤ BF BG kmk∞. form. Lemma II.3. Let (fn) and (gn) be Bessel sequences with 1 Lemma III.4. Suppose T ∈ B(H) is a compact op- Bessel bounds BF and BG, respectively. If (mn) ∈ ` , 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 √ lim |B(T )(n)| = 0, i.e. B(T ) ∈ c ( ). BF BG kmk1. n→∞ 0 N w Lemma II.4. Let (fn) and (gn) be Bessel sequences with Proof: Since fn −* 0 for n → ∞, we have kT fnk → 0 Bessel bounds BF and BG, respectively. If limn→∞ mn = 0, for n → ∞. Together with Lemma I.2 this yields i.e. m ∈ c0(N), then A(m) is a compact operator. |hT fn, gni| ≤ kT fnkkgnk → 0, for n → ∞. 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 ∈ ` (N), 1 ≤ p < ∞, then A(m) is a Schatten p-class operator, and class operator, with 1 ≤ p < ∞. Then the Berezin transform √ 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. ∞ ∞ 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 BF and kSkB(H) ≤ BG. Hence ` (N), 1 ≤ p < ∞ S (H), Schatten class operator ∗ hT fn, gni = hT Ren, Seni = hS T Ren, eni TABLE I BESSEL MULTIPLIERS WITH DIFFERENT SYMBOLS for all n. The operator T˜ = S∗TR is again in Sp, so

1 1 ! p ! p X p X p See also the paper [3], which contains somewhat related |B(T )(n)| = |hT˜ en, eni| ≤ kT˜kSp . results for Gabor multipliers. n n

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Since If 1 < p ≤ ∞, then 1 ≤ q < ∞ and T is a compact operator q ˜ ∗ p in S . As such, it has a spectral representation kT kSp ≤ kS kB(H)kT kSp kRkB(H) ≤ kT kSp BF BG, X the proof is finished. T = λkh·, σkiτk Table II summarizes these results. k

with orthonormal bases (σk) and (τk), and λk ≥ 0 with Operator Berezin transform P q q k λk = kT kSq . Choose the particular orthonormal basis B(H) = S∞(H) `∞( ) N (ek) = (σk). Then T ek = T σk = λkτk for all k, and thus 0 0 K(H) = S (H), compact operator c0(N) = ` (N) X p p S (H), 1 ≤ p < ∞, Schatten class ` (N) S = |mn||λk||hσk, fni||hgn, τki| n,k TABLE II 1  X p  p BEREZIN TRANSFORM OF DIFFERENT OPERATORS ≤ |mn| |hσk, fni||hgn, τki| × n,k 1  X q  q Suggested by the results given above, it becomes obvious |λk| |hσk, fni||hgn, τki| . that the concept of Bessel multiplier and the Berezin transform n,k are dual to each other. These two sums can be estimated, the first as The following theorem gives the connection between the X p Berezin transform and Bessel multipliers. |mn| |hσk, fni||hgn, τki| p n,k Theorem III.6. Let m = (mn) ∈ ` (N), 1 ≤ p ≤ ∞, and 1 1 q 1 1 X p X 2 2  X 2 2 T ∈ S , with q the conjugate exponent to p, i.e. p + q = 1. ≤ |mn| |hσk, fni| |hgn, τki| Then n k k hA(m),T iSp,Sq = hm, B(T )i`p,`q . p p ≤ BF BG kmkp Proof: For the moment, let (e ) be an arbitrary orthonor- k and the second similarly as mal basis of H. Then the left hand side can be written as X X |λ |q|hσ , f i||hg , τ i| hA(m),T i = hA(m)(ek), T eki. k k n n k k n,k P  X q p p Inserting A(m) = n mnh·, fnign yields ≤ λk BF kσkk BGkτkk X X k hA(m),T i = mnhek, fnihgn, T eki. (∗) p q k n = BF BG kT kSq . The right hand side gives So, finally, we have for 1 < p ≤ ∞ X hm, B(T )i = mnhgn, T fni p S ≤ BF BG kmkpkT kSq < ∞. n X X ∗ Since in every case S < ∞, Fubini’s theorem yields the = mn hT gn, ekihek, fni n k desired conclusion, the equality of (∗) and (∗∗). by Parseval’s equality. Thus Corollary III.7. (1) Let A : `∞ → B(H) and B : S1 → `1. X X ∗ hm, B(T )i = m he , f ihg , T e i. (∗∗) Then A = B is the adjoint. n k n n k 1 1 ∗ n k (2) Let A : c0 → K(H) and B : S → ` . Then B = A . (3) Let A : `p → Sp, 1 ≤ p < ∞, and B : Sq → `q, with Comparing (∗) and (∗∗), we see that the claimed equality is 1 < q ≤ ∞ the conjugate exponent. Then B = A∗. proved, if we can justify the change of order of summation in the double sum. In order to do so, we examine the Proof: Observe that B(H) = (S1)∗ and `∞ = (`1)∗ in 1 ∗ ∞ 1 ∗ corresponding double sum of the absolute values case (1), S = (K(H)) and c0 ⊆ ` with ` = (c0) in X X case (2), and Sq = (Sp)∗ and `q = (`p)∗ in case (3). The S := |mn||hek, fni||hgn, T eki|. statements then follow immediately from Theorem III.6. n k Consider the case p = 1 (i.e. m ∈ `1 and T ∈ S∞ = B(H)). IV. (NON-)DENSITY RESULTS Then In this section we investigate whether a given operator on 1 1 X  X 2 2  X ∗ 2 2 H can be approximated by a Gabor multiplier with respect S ≤ |mn| |hek, fni| |hT gn, eki| to various norms. In particular, we would like to understand n k k X when the set of Gabor multipliers (associated to a fixed pair ≤ |m |kf kkT ∗g k n n n of Gabor systems) is dense in B(H) or in Sp(H) (if ever). n p In order to examine such density properties, we employ ≤ BF BG kmk1kT kB(H) < ∞. some well known results from functional analysis. Precisely,

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we use the following facts ([5]): B(T ) = 0, but T 6= 0. Hence B : S2 → `2 is not one-to-one. Let X,Y be Banach spaces, T : X → Y be a bounded operator and let T ∗ : Y ∗ → X∗ be the (Banach space) adjoint We can extend this result to the cases 1 ≤ p < 2. operator. Theorem IV.4. Let (f ) and (g ) be Bessel Gabor ∗ ∗ n,m n,m • T is one-to-one on Y , if and only if the range of T is systems and 1 ≤ p < 2. Then the range of A : `p → Sp dense in Y with respect to the norm topology on Y . is not a norm-dense subspace of the Schatten class Sp. ∗ • T is one-to-one on X, if and only if the range of T is dense in X∗ with respect to the weak* topology on X∗. Proof: Let 2 < q ≤ ∞ be the conjugate exponent to p. 2 q ∞ To understand when the mapping a → A(a) has dense Observe that S ⊆ S ⊆ S . By Theorem IV.3, the Berezin transform B : S2 → `2 is not one-to-one, hence, a fortiori, the range, it suffices, in view of Theorem III.6 and its corollary, q q to check when the Berezin transform B is one-to-one. Berezin transform B : S → ` is not one-to-one, either. By Corollary III.7, this is equivalent to the range of A : `p → Sp Lemma IV.1. Let a, b > 0 and assume that (fn,m) = not being norm-dense. (π(an, bm)f) and (gn,m) = (π(an, bm)g) are Gabor systems, For the cases 2 < p < ∞, we conjecture analogous results. f, g ∈ L2(R). Let (z, ν) ∈ R2 and T = π(z, ν) be the For the case p = ∞, we have the following result (whose corresponding time-frequency shift. Then proof we omit for lack of space): 2πi(anν−bmz) B(T )(n, m) = e V (g, f)(z, ν) Theorem IV.5. Let (fn,m) and (gn,m) be Bessel Gabor systems. Then there exists an operator R ∈ B(L2) and a for all n, m ∈ . Z constant δ > 0 such that Corollary IV.2. Let (fn,m) = (π(an, bm)f) and (gn,m) = kR − A(m)k 2 ≥ δ (π(an, bm)g) be Bessel Gabor systems. If there exists a point B(L ) (z, ν) ∈ R2 such that V (g, f)(z, ν) = 0, then the Berezin for all m ∈ `∞. In particular, the range of A : `∞ → B(L2) transform B : B(L2) → `∞ is not one-to-one. is not a norm-dense subspace of B(L2). Proof: We have T = π(z, ν) 6= 0 in B(L2), but B(T ) = 0 One can take for R the Fourier transform, fractional Fourier by the preceding lemma. transforms or any other operator that incorporates time- For the particular case of Hilbert-Schmidt operators, we frequency shifts of arbitrarily large size. have the following negative result: V. CONCLUSION Theorem IV.3. Let (f ) and (g ) be Bessel Gabor n,m n,m Our results show that subsets of Gabor multipliers with systems. Then the range of A : `2 → S2 is not a norm-dense symbols in `p-spaces are not dense in the respective Schatten subspace of Hilbert-Schmidt class. There are thus Hilbert- classes, but span proper subspaces. There exist thus operators Schmidt operators on L2( ) that cannot be approximated in R in these Schatten classes that cannot be approximated arbitrar- Hilbert-Schmidt norm by Gabor multipliers (with a given fixed ily well by multipliers in the respective . This is pair of Gabor systems). in sharp contrast to the case of continuous (STFT) multipliers, Proof: In view of Corollary III.7, it suffices to show that as shown in [2]. For approximation of bounded operators in B : S2 → `2 is not one-to-one. Let T ∈ S2. Now note that operator norm, however, the negative result shown in this paper there is a bijective correspondence between Hilbert-Schmidt also holds analogously in the continuous case. operators and Weyl symbols in L2( 2). Thus there exists a R VI.ACKNOWLEDGEMENT unique Weyl symbol σ ∈ L2(R2) such that The first author would like to thank K. Grochenig¨ for hT φ, ψi = hσ, W (ψ, φ)i his substantial help (in particular with [2]) and continual encouragement. for all φ, ψ ∈ L2(R). Thus REFERENCES B(T )(n, m) = hσ, W (gn,m, fn,m)i [1] P. Balazs. Basic definition and properties of Bessel multipliers. J. Math. = hσ, W (π(an, bm)g, π(an, bm)f)i Anal. Appl., 325(1):571–585, 2007. [2] D. Bayer. Bilinear Time-Frequency Distributions and Pseudodifferential = hσ, T(an,bm)W (g, f)i. Operators. PhD thesis, University of Vienna, 2010. Observe that W (g, f) ∈ L2(R2). As is well-known, a discrete [3] J. Benedetto and G. Pfander. Frame expansions for Gabor multipliers. 2 2 Appl. Comp. Harm. Anal., 20(1):26–40, 2006. countable family of translates of a function F ∈ L (R ) is [4] O. Christensen. An Introduction to Frames and Riesz Bases. Applied and never complete, thus Numerical Harmonic Analysis. Birkhauser,¨ 2003. [5] J.B. Conway. A Course in . Springer, New York, 2nd U := span{T(an,bm)W (g, f) | n, m ∈ Z} edition, 1990. [6] K. Grochenig.¨ Foundations of Time-Frequency Analysis. Appl. Numer. is a proper closed subspace of L2(R2). Choose 0 6= σ ∈ U ⊥. Harmon. Anal. Birkhauser¨ Boston, Boston, MA, 2001. [7] K. Zhu. Operator Theory in Function Spaces. Mathematical Surveys and Then the corresponding Hilbert-Schmidt operator T satisfies Monographs. American Mathematical Society, 2nd edition, 2007. B(T )(n, m) = hσ, W (gn,m, fn,m)i = 0 for all n, m ∈ Z, thus

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