A Beurling-Helson type theorem for modulation spaces
aa r X i v : . [ m a t h . C A ] J a n A BEURLING-HELSON TYPE THEOREM FOR MODULATIONSPACES
KASSO A. OKOUDJOU
Abstract.
We prove a Beurling-Helson type theorem on modulation spaces. Moreprecisely, we show that the only C changes of variables that leave invariant themodulation spaces M p,q ( R d ) are affine functions on R d . A special case of our resultinvolving the Sj¨ostrand algebra was considered earlier by A. Boulkhemair. Introduction
Given a function φ defined from the torus T to itself, let φ ∗ be the change ofvariables defined by(1) φ ∗ ( u ) = u ◦ φ for any function u defined on T .In 1953, A. Beurling and H. Helson proved that if φ is continuous from T intoitself and if φ ∗ is a bounded linear operator on the Fourier algebra A ( T ) = A ( T ) ofabsolutely convergent Fourier series, then necessarily φ ( t ) = kt + φ (0) for some k ∈ Z [1]. The proof of this result involved some nontrivial arithmetical considerations. Adifferent proof was given by J.-P. Kahane [15]. The Beurling-Helson theorem waslater extended to the higher dimensional setting by W. M. Self [17]. More recently,V. Lebedev and A. Olevskiˇı [16] further extended and generalized the Beurling-Helsontheorem. In particular, for d ≥ ≤ p < ∞ let A p ( R d ) = F L p ( R d ) equippedwith the norm k f k A p = k ˆ f k L p where F is the Fourier transform defined by F f ( ω ) =ˆ f ( ω ) = R R d f ( t ) e − πit · ω dt. It was proved in [16] that if φ : R d → R d is C , and if φ ∗ maps A p ( R d ) into itself for some 1 ≤ p < ∞ , p = 2, then φ ( x ) = Ax + φ (0) where A is a real invertible d × d matrix. In this higher dimensional setting, the case p = 1was already proved in [17]. Observe that since A ( R d ) = F L ( R d ) = L ( R d ), theclass of functions φ such that φ ∗ is bounded on A ( R d ) is quite large. For instance,for any homeomorphism φ on T such that φ − satisfies the Lipschitz condition, φ ∗ isbounded on A ( T ), and a transference argument can be used to prove similar resultfor A ( R d ).In this note, we shall characterize the C changes of variables that leave invariantthe modulation spaces (to be defined below). In particular, our result applies to aspecial subspace of the Fourier algebra called Feichtinger algebra. This space denoted Date : October 31, 2018.2000
Mathematics Subject Classification.
Primary 42B15; Secondary 42A45, 42B35.
Key words and phrases.
Beurling-Helson theorem, changes of variables, Feichtinger algebra,Fourier multipliers, modulation spaces, Sj¨ostrand algebra. S was introduced by H. Feichtinger [5] and is the smallest Banach algebra that isinvariant under both the translation and the modulation operators. Moreover, theFeichtinger algebra is an example of a modulation space and plays an important rolein the theory Gabor frames [11]. In fact, the modulation spaces have also been playingan increasing role in the analysis of pseudodifferential operators [12, 13, 20]. Fur-thermore, a Banach algebra of pseudodifferential operators known as the Sj¨ostrandalgebra, denoted S ω , and which contains the H¨ormander class S , , was introducedindependently by Feichtinger [6] and J. Sj¨ostrand [18]. This space is yet anotherexample of a modulation space. We refer to [7] for an updated version of [6] whichcontains some historical perspectives on the modulation spaces. In 1997, A. Boulkhe-mair [4] proved that if φ is a C mapping on R d such that φ ∗ maps S ω into itself,then φ must be an affine function: This is a Beurling-Helson type theorem for theSj¨ostrand algebra. It is therefore natural to seek a characterization of the changesof variables that leave invariant modulation spaces. The goal of this note is to ex-tend and generalize this Beurling-Helson type theorem to all the modulation spaces.The main argument in the proof of our result is the fact that the intersection of amodulation space with the space of functions with compact support coincides withthe subspace of compactly supported functions in A p ( R d ) = F L p ( R d ). The proof ofthis fact as well as the definition of the modulation spaces will be given in Section 2.Finally in Section 3 we shall prove our main result. In the sequel, we shall denote by | A | the Lebesgue measure of a measurable subset A of R d . Preliminaries
Modulation spaces.
The Short-Time Fourier Transform (STFT) of a function f with respect to a window g is V g f ( x, y ) = Z R f ( t ) g ( t − x ) e − πiyt dt, whenever the integral makes sense. This definition can be extended to f ∈ S ′ ( R d )and g ∈ S ( R d ) and yields a continuous function V g f , see [11]. Definition 1.
Given 1 ≤ p, q ≤ ∞ , and given a window function 0 = g ∈ S , themodulation space M p,q = M p,q ( R d ) is the space of all distributions f ∈ S ′ for whichthe following norm is finite:(2) k f k M p,q = (cid:18)Z R d (cid:18)Z R d | V g f ( x, y ) | p dx (cid:19) q/p dy (cid:19) /q , with the usual modifications if p and/or q are infinite. Remark . The definition is independent of the choice of the window g in the senseof equivalent norms.The modulation spaces were originally introduced by Feichtinger [6]. We refer to[11] and the references therein for more details about modulation spaces. EURLING-HELSON TYPE THEOREM 3
The Feichtinger algebra S which coincides with the modulation space M , ( R d ) isa Banach algebra under both pointwise multiplication and convolution. Furthermore, M , ( R d ) like M p,p ( R d ) 1 ≤ p ≤ ∞ is invariant under the Fourier transform [5, 8, 9].While the Beurling-Helson theorem completely classifies the changes of variablesthat operate in A ( T ) (and also on A ( R )) it was still unknown what changes ofvariables operate on the Feichtinger algebra and more generally on the modulationspaces. This question will be completely settled below.2.2. Local modulation spaces.
The theory of modulation can be defined in thegeneral setting of locally compact Abelian groups [7]. In particular, it can be shownthat for G = Z d (or any discrete group), M p,q ( G ) = ℓ p ( G ). Similarly, if G = T d (or any compact group), M p,q ( G ) = F L q ( G ). Here we focus on functions that arelocally in a modulation space.In the sequel we shall denote by M p,qcomp ( R d ) the subspace of M p,q ( R d ) consistingof compactly supported functions, and by M p,qloc ( R d ) the space of functions that arelocally in M p,q ( R d ). In particular, u ∈ M p,qloc ( R d ) if and only if for each g ∈ C ∞ ( R d )with supp ( g ) ⊂ K where K is a compact subset of R d , we have u K = g u ∈ M p,q ( R d ),i.e., u K ∈ M p,qcomp ( R d ) . ( F L q ) comp ( R d ) and ( F L q ) loc ( R d ) are defined similarly.The next result contains the key argument in the proof of our main result. We wishto point out that some special cases of the result are already known. For instance, theresult was proved for M ∞ , ( R d ) in [4, Theorem 5.1], while [9] dealt with M p,p ( R d )1 ≤ p < ∞ . Furthermore, an independent and different proof of part b. of Lemma 1using convolution relations on generalized amalgam spaces was indicated to us byH. Feichtinger [10]. Lemma 1.
Let ≤ p, q ≤ ∞ . Then the following statements hold a. M p,qcomp ( R d ) = ( F L q ) comp ( R d ) . b. M p,qloc ( R d ) = ( F L q ) loc ( R d ) . Proof.
We shall only prove part a. of the result as part b. follows from the definitionof M p,qloc ( R d ). Furthermore, to prove a. it suffices to show that given a compact subset K of R d M p,q ( R d ) | K = F L q ( R d ) | K . Note that this last equation holds not only asset equality, but also as equality of Banach spaces with equivalent norms.Let R > u ∈ F L q ( R d ) such that supp ( u ) ⊂ B R (0). Let g ∈ C ∞ c ( R d ) with supp ( g ) ⊂ B R (0). Then, for each ω ∈ R d , V g u ( · , ω ) is supported in B R (0). Thus, using the fact that | V g u ( x, ω ) | = | V ˆ g ˆ u ( ω, − x ) | = |F − (ˆ u · T ω ˆ g )( x ) | wehave the following estimates k V g u ( · , ω ) k L p ≤ | B R (0) | /p k V g u ( · , ω ) k L ∞ = | B R (0) | /p kF − (ˆ u · T ω ˆ g ) k L ∞ ≤ | B R (0) | /p k ˆ u · T ω ˆ g k L ≤ | B R (0) | /p | ˆ u | ∗ | ˆ g | ( ω ) . K. A. OKOUDJOU
Consequently, k V g u k L p,q ( R d ) ≤ | B R (0) | /p k ˆ u k L q ( R d ) k ˆ g k L ( R d ) , that is k u k M p,q ( R d ) | BR (0) ≤ C ( R, p, q, d ) k u k F L q ( R d ) | BR (0) . Thus, F L q ( R d ) | B R (0) ⊂ M p,q ( R d ) | B R (0) . For the converse, let
R > u ∈ M p,q ( R d ) such that supp ( u ) ⊂ B R (0) . Let g ∈ C ∞ ( R d ) such that g ≡ B R (0). It is trivially seen that for all x ∈ B R (0)and for all t ∈ B R (0), g ( t − x ) = 1. Thus, for all ω ∈ R d and for x ∈ B R (0),ˆ u ( ω ) χ B R (0) ( x ) = χ B R (0) ( x ) V g u ( x, ω ) = χ B R (0) ( x ) Z B R (0) u ( t ) e − πit · ω g ( t − x ) dt. Therefore, | B R (0) | /p | ˆ u ( ω ) | = k χ B R (0) ( · ) V g u ( · , ω ) k L p . Hence, k ˆ u k L q ≤ | B R (0) | − /p k V g u k L p,q , that is k u k F L q ( R d ) | BR (0) ≤ C ( R, p, q, d ) k u k M p,q ( R d ) | BR (0) . Therefore, M p,q ( R d ) | B R (0) ⊂ F L q ( R d ) | B R (0) . We can now conclude that M p,q ( R d ) | B R (0) = F L q ( R d ) | B R (0) . (cid:3) Main results
Before stating our main result, we wish to indicate that it is trivially seen that allthe modulation spaces are invariant under affine changes of variables. That is, let1 ≤ p, q ≤ ∞ and φ : R d → R d be an affine mapping, i.e., φ ( x ) = Ax + b where A isa d × d real invertible matrix and b ∈ R d . Then the linear operator φ ∗ given by (1)maps M p,q ( R d ) into itself, that is φ ∗ ( M p,q ( R d )) ⊂ M p,q ( R d ) . Indeed, let g ∈ S and u ∈ M p,q ( R d ), and ˜ g = g ◦ A − where A − is the inverse of A .The result follows from V g φ ∗ ( u )( x, ω ) = | detA | e − πiω · A − b V ˜ g u ( Ax + b, ( A ∗ ) − ω )where A ∗ denote the conjugate of A .If we restrict our attention to the modulation spaces M p,p ( R d ) 1 < p < ∞ thefollowing stronger result can be proved. For Proposition 1 we assume that R d = ∪ Nk =1 Q k where for each k, Q k is a (possible infinite) “cube” with sides parallel to thecoordinates axis. Moreover, we assume that for k = 1 , . . . , N the Q k s have disjointinteriors. EURLING-HELSON TYPE THEOREM 5
Proposition 1.
Let φ be a continuous on R d such that for k = 1 , . . . , N , the restric-tion φ k of φ to Q k is an affine function given by φ k ( x ) = A k x + b k where A k is a realinvertible d × d matrix and b k ∈ R d . Then the linear operator φ ∗ given by (1) maps M p,p ( R d ) into itself, that is φ ∗ ( M p,p ( R d )) ⊂ M p,p ( R d ) . Proof.
It is evident from the definition of the modulation spaces that M p,p is invariantunder the Fourier transform, see [8, 9]. Let u ∈ M p,p ( R d ), then φ ∗ ( u ) = u ◦ φ = N X k =1 χ Q k · ( u ◦ φ ) = N X k =1 χ Q k · ( u ◦ φ k ) , and so k φ ∗ ( u ) k M p,p ≤ N X k =1 k χ Q k · ( u ◦ φ k ) k M p,p . As indicated above, u ◦ φ k ∈ M p,p . Hence, v k = F − ( u ◦ φ k ) ∈ M p,p ( R d ) as well.Moreover, note that χ Q k is a bounded Fourier multiplier on all M p,p ( R d ): this followsfrom [2, Theorem 1] in the case d = 1, and from [3, Theorem 6] when d > M p,p ( R d ) under the Fourier transform, weconclude that there exists c k > k χ Q k · ( u ◦ φ k ) k M p,p = kF − ( χ Q k · ˆ v k ) k M p,p ≤ c k k u k M p,p , from which the proof follows. (cid:3) Remark . The conclusion of Proposition 1 holds if we used an infinite decompositionof R d , that is if we assume that R d = ∪ ∞ k =1 Q k where the cubes Q k still have sides par-allel to the coordinate axis and disjoint interiors. In this case, the extra assumptionneeded to prove the previous result is that the constants c k appearing in the aboveproof, are uniformly bounded, i.e., sup k c k < ∞ . We are now ready to state and prove our main result.
Theorem 1.
Let φ : R d → R d be a C function. Assume that the operator φ ∗ defined by (1) maps M p,q ( R d ) into itself, i.e., φ ∗ ( M p,q ( R d )) ⊂ M p,q ( R d ) for some ≤ p, q ≤ ∞ , with = q < ∞ . Then φ is an affine mapping, that is φ ( x ) = Ax + φ (0) for some real invertible d × d matrix A .In particular, the Feichtinger algebra M , ( R d ) is preserved by, and only by affinechanges of variables.Proof. Because φ ∗ ( M p,q ( R d )) ⊂ M p,q ( R d ) and φ ∗ ( u ) = u ◦ φ is compactly supportedwhenever u is, Lemma 1 implies that φ ∗ maps M p,qcomp ( R d ) = ( F L q ) comp ( R d ) into itselfas well as M p,qloc ( R d ) = ( F L q ) loc ( R d ) into itself. Therefore,when d = 1 and q = 1, the Beurling-Helson Theorem [1, pp. 84-86], implies that φ ( x ) = ax + φ (0);when d = 1 and 1 < q < ∞ , q = 2, it follows from [16, Theorem 3] that φ ( x ) = ax + φ (0); K. A. OKOUDJOU when d > q = 1, it follows from [17, Corollary 1] that φ ( x ) = Ax + φ (0),where A is a real invertible d × d matrix;when d > < q < ∞ , q = 2, it follows from [16, Theorem 6] that φ ( x ) = Ax + φ (0), where A is a real invertible d × d matrix. (cid:3) Remark . The fact that q = 2 in Theorem 1 was justified in the Introduction.Moreover, we restricted to q < ∞ , because the key ingredients in the proof of ourmain result are [16, Theorem 3, Theorem 6] whose proofs are based on a densityargument. It is not clear to us if Theorem 1 holds for q = ∞ . Remark . Using Lemma 1 and [16, pp. 214], it follows that if φ : R d → R d is nonlinearand C , then φ ∗ is not bounded on M p,q . This fact together with Proposition 1, showthat the C condition in Theorem 1 is the only nontrivial smoothness condition toimpose on φ .For the Sj¨ostrand algebra S ω which coincides with the modulation space M ∞ , ( R d ),Theorem 1 was proved in under a weaker assumption on φ . More specifically, it wasproved in [4, Theorem 5.1] that if φ is a proper mapping, i.e., φ is continuous on R d and φ − ( K ) is a compact set for any compact subset K of R d , and if φ ∗ ( M ∞ , ( R d )) ⊂M ∞ , ( R d ) then φ ( x ) = Ax + φ (0). It is also straightforward to prove Theorem 1 underthis weaker assumption on φ .Finally, we wish to conclude this paper by pointing out the connection of our mainresult to certain Fourier multipliers. More precisely, let σ be a function defined on R d . The Fourier multiplier with symbol σ is the operator H σ initially defined on S by H σ f ( x ) = Z R d σ ( ξ ) ˆ f ( ξ ) e πiξ · x dξ. We refer to [19] for more on Fourier multipliers. As mentioned above, there is a strongconnection between the L p -continuity of the Fourier multipliers and the Beurling-Helson theorem. In particular, the family of homomorphisms e iφ ( ξ ) on the spaceof L p -Fourier multipliers was investigated by H¨ormander in [14, Section 1.3]. It iseasily seen that σ ( ξ ) = e iξ , then H σ is bounded on all L p ( R d ) for 1 ≤ p ≤ ∞ and d ≥
1. H¨ormander proved that if φ : R d → R d is C and if φ ∗ ( σ )( ξ ) = σ ( φ ( ξ )) = e iφ ( ξ ) is a bounded Fourier multiplier on L p ( R d ) for some 1 < p < ∞ and p = 2,then φ is an affine function [14, Theorem 1.15]. It is interesting to note that thereexist nonlinear (non-affine) functions φ on R d such that the Fourier multipliers withsymbols φ ∗ ( σ )( ξ ) = σ ( φ ( ξ )) = e iφ ( ξ ) are bounded on all modulation spaces [3].4. Acknowledgment
The author would like to thank Chris Heil for bringing some of the questionsdiscussed in this work to his attention. He also thanks ´Arp´ad B´enyi, Hans Feichtinger,Karlheinz Gr¨ochenig, Norbert Kaiblinger, and Luke Rogers for helpful discussions.
EURLING-HELSON TYPE THEOREM 7
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Kasso A. Okoudjou, Department of Mathematics, University of Maryland, Col-lege Park, MD, 20742 USA
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