Abstract
Let \(\mathcal H\) be a Hilbert space of distributions on \(\mathbf{R}^{d}\) which contains at least one nonzero element of the Feichtinger algebra \(S_0\) and is continuously embedded in \(\mathscr {D}'\). If \(\mathcal H\) is translation and modulation invariant, also in the sense of its norm, then we prove that \(\mathcal H= L^2\), with the same norm apart from a multiplicative constant.
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1 Introduction
In the paper we show that any Hilbert space of distributions on \(\mathbf{R}^{d}\) which is translation and modulation invariant with respect to the norms, agrees with \(L^2(\mathbf{R}^{d})\). These considerations are strongly linked with Feichtinger’s minimization property, which shows that the Feichtinger algebra \(S_0(\mathbf{R}^{d})\), which is the same as the modulation space \(M^{1,1}(\mathbf{R}^{d})\), is the smallest nontrivial Banach space of tempered distributions which is norm invariant under translations and modulations. Our investigations may therefore be considered as a Hilbert space analogue of those investigations which lead to Feichtinger’s minimization property.
We remark that the search for the smallest Banach space possessing such norm invariance properties, seems to be the main reason that Feichtinger led to introduce and investigate \(S_0(\mathbf{R}^{d})\) and in its prolongation the foundation of classical modulation spaces (see [2]). The space \(S_0(\mathbf{R}^{d})=M^{1,1}(\mathbf{R}^{d})\) is small in the sense that it is contained in every Lebesgue space \(L^p(\mathbf{R}^{d})\) and its Fourier image, \(p\in [1,\infty ]\) (cf. [2, 5, 9] and the references therein).
On the contrary, the modulation space \(M^{\infty ,\infty }(\mathbf{R}^{d})\), which is the dual of \(M^{1,1}(\mathbf{R}^{d})\), contains all these Lebesgue and Fourier Lebesgue spaces. By straightforward arguments (see Proposition A.1) it follows that for a translation and modulation invariant Banach space \(\mathcal B\) of tempered distributions which contains at least one nonzero element in \(S_0(\mathbf{R}^{d})\), we have
Here the first inclusion is a reformulation of the Feichtinger’s minimization property (in the unweighted case). We refer to Sect. 1 and [5, 6] for notations and some facts on distributions.
Feichtinger’s minimization property has been extended in different ways, e. g. to weighted spaces (see e. g. [5, Chapter 12]), and to the quasiBanach situation (see e. g. [9]). At the same time minimization property has been applied e. g. in nonuniform samplings, and for deducing sharp Schattenvon Neumann and nuclear results for operators with kernels in modulation spaces (see e. g. [9]). Note also that translation and modulation invariant Banach spaces are important in Gabor analysis, e. g. when searching for suitable windows for Gabor frames. It also seems necessary for a Banach space to be translation and modulation invariant, if it should be conveniently discretizable by Gabor expansions (see e. g. [5] and the references therein).
In our investigations we do not present any such weighted analogies in the Hilbert space case. On the other hand, we consider translation and modulation invariant Banach spaces, \(\mathcal B\), which are continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\), the set of all distributions on \(\mathbf{R}^{d}\). As a first step we prove in Proposition 1.5 that any such \(\mathcal B\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\), the set of all tempered distributions on \(\mathbf{R}^{d}\). Note that the original approaches in [2, 5] have the more restrictive assumption that these \(\mathcal B\) are continuously embedded in the subspace \(\mathscr {S}'(\mathbf{R}^{d})\) of \(\mathscr {D}'(\mathbf{R}^{d})\).
2 Translation and modulation invariant Hilbert spaces
In this section we first recall the definition of translation and modulation invariant spaces. Thereafter we consider such spaces which at the same time are Hilbert spaces of distributions on \(\mathbf{R}^{d}\). We show some features on how differentiations and multiplications by polynomials of such spaces behave in the inner product of such Hilbert spaces. In the end we show that such Hilbert spaces agree with \(L^2(\mathbf{R}^{d})\).
The definition of translation and modulation invariant Banach spaces is given in the following (cf. [2, 5]).
Definition 1.1
Let \(\mathcal B\) be a Banach space which is continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\). Then \(\mathcal B\) is called translation and modulation invariant, if \(y\mapsto f(yx)e^{i\langle y,\xi \rangle }\) belongs to \(\mathcal B\) and
for every \(f\in \mathcal B\) and \(x,\xi \in \mathbf{R}^{d}\).
Remark 1.2
We identify \(\mathscr {S}'(\mathbf{R}^{d})\) as a subspace of \(\mathscr {D}'(\mathbf{R}^{d})\) in the same way as in [6]. In particular it follows that \(f \in \mathscr {D}'(\mathbf{R}^{d})\) belongs to \(\mathscr {S}'(\mathbf{R}^{d})\), if and only if there is a constant \(C>0\) and seminorm \(\Vert \, \cdot \, \Vert \) on \(\mathscr {S}(\mathbf{R}^{d})\) such that
for every \(\psi \in C_0^\infty (\mathbf{R}^{d})\). For such f, there is a unique \(f_0\in \mathscr {S}'(\mathbf{R}^{d})\) such that \(\langle f,\psi \rangle =\langle f_0,\psi \rangle \) for every \(\psi \in C_0^\infty (\mathbf{R}^{d})\). (See [6][Chapter VII].) Then we identify f with \(f_0\), and thereby consider f as an element in \(\mathscr {S}'(\mathbf{R}^{d})\).
Our main result is the following.
Theorem 1.3
Let \(\mathcal H\) be a translation and modulation invariant Hilbert space, continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\) and contains at least one element of \(M^{1,1}(\mathbf{R}^{d})\setminus \{ 0\}\). Then \(\mathcal H= L^2(\mathbf{R}^{d})\) with
for some constant \(c>0\) which is independent of \(f\in \mathcal H= L^2(\mathbf{R}^{d})\).
We observe that \(L^2(\mathbf{R}^{d})\) is the same as the modulation space \(M^{2,2}(\mathbf{R}^{d})\) (see [2]). Hence the analogy of (0.1) in the Hilbert space case is that for translation and modulation invariant Hilbert spaces \(\mathcal H\) one has
Remark 1.4
It is obvious that the constant c in (1.1) can be evaluated by
for any fixed \(f\in \mathcal H\setminus \{ 0\}\).
We need some preparations for the proof of Theorem 1.3. First we note that the restriction of the \(L^2\) scalar product \((\, \cdot \, ,\, \cdot \, )_{L^2}\) on \(\mathscr {S}(\mathbf{R}^{d})\times \mathscr {S}(\mathbf{R}^{d})\) is uniquely extendable to a continuous sesquilinear form on \(\mathscr {S}(\mathbf{R}^{d})\times \mathscr {S}'(\mathbf{R}^{d})\) and that the dual of \(\mathscr {S}(\mathbf{R}^{d})\) can be identified by \(\mathscr {S}'(\mathbf{R}^{d})\) through this extension.
As a consequence of the following proposition we have that any translation and modulation invariant Banach space on \(\mathbf{R}^{d}\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\).
Proposition 1.5
Let \(\mathcal B\) be a Banach space, continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\) such that \(y\mapsto f(yx)\) belongs to \(\mathcal B\) for every \(x\in \mathbf{R}^{d}\) and
for some constants \(C,r>0\) which are independent of \(f\in \mathcal B\) and \(x \in \mathbf{R}^{d}\). Then \(\mathcal B\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\).
Proposition 1.5 follows by similar arguments as in the proof of BochnerSchwartz theorem for positive definite distributions (see e. g. [7] and the references therein). In order to be self contained we give a proof in Appendix A.
Remark 1.6
By Proposition 1.5 it follows that results on Feichtinger’s minimization property in e. g. [2, 5, 9], still hold true with relaxed assumptions that the involved Banach spaces are embedded in \(\mathscr {D}'(\mathbf{R}^{d})\) instead of its subspace \(\mathscr {S}'(\mathbf{R}^{d})\).
For a translation and modulation invariant Banach space \(\mathcal B\) in Definition 1.1, there is also a maximization property, analogous to Feichtinger’s minimization property (see Proposition A.1 in Appendix A). In the unweighted case, the largest possible such \(\mathcal B\) is given by \(S_0'(\mathbf{R}^{d})=M^{\infty ,\infty }(\mathbf{R}^{d})\) in the BanachGelfand triple \((S_0(\mathbf{R}^{d}), L^2(\mathbf{R}^{d}), S_0'(\mathbf{R}^{d}))\) studied in [4] (see Appendix A for the definition of \(M^{\infty ,\infty }(\mathbf{R}^{d})\)).
As a consequence of Proposition 1.5 we have that \(\mathcal H\) in Theorem 1.3 is contained in \(\mathscr {S}'(\mathbf{R}^{d})\). Since \(\mathcal H\) contains at least one element in \(M^{1,1}(\mathbf{R}^{d})\) we get the more refined
with continuous inclusions, by Proposition A.1 and Feichtinger’s minimization property. In particular, the standard Gaussian \(h_0(x) = \pi ^{\frac{d}{4}}e^{\frac{1}{2}x^2}\) belongs to \(\mathcal H\).
For the proof of Theorem 1.3, we also need some properties on Hermite functions. Recall that the Hermite function \(h_\alpha \) of order \(\alpha \in \mathbf{N}^{d}\) on \(\mathbf{R}^{d}\) is defined by
It is wellknown that \(\{h_\alpha \} _{\alpha \in \mathbf{N}^{d}}\) is an orthonormal basis for \(L^2(\mathbf{R}^{d})\), and a basis for \(\mathscr {S}(\mathbf{R}^{d})\) (see e. g. [7, 8]).
We may pass from one Hermite function to another by successively applying the annihilation and creation operators, which are given by
respectively, \(j=1,\dots ,d\). It is then wellknown that if \(e_j\) is the jth vector in the standard basis in \(\mathbf{R}^{d}\), then
and
(Cf. e. g. [1].) This implies
where
Furthermore,
We have now the following lemma.
Lemma 1.7
Let \(\mathcal H\) be a translation and modulation invariant Hilbert space on \(\mathbf{R}^{d}\). Then the following is true:

(1)
for every \(f,g \in \mathcal H\) and \(x,\xi \in \mathbf{R}^{d}\) it holds
$$\begin{aligned} (f (\, \cdot \, x),g )_{\mathcal H}&= (f ,g (\, \cdot \, +x))_{\mathcal H} \end{aligned}$$(1.9)and
$$\begin{aligned} (f \cdot e^{i\langle \, \cdot \, ,\xi \rangle },g )_{\mathcal H}&= (f ,g \cdot e^{i\langle \, \cdot \, ,\xi \rangle })_{\mathcal H} \text{; } \end{aligned}$$(1.10) 
(2)
\((\partial ^\alpha f ,g )_{\mathcal H} = (f ,(\partial )^\alpha g )_{\mathcal H}\) and \((x^\alpha f ,g )_{\mathcal H} = (f ,x^\alpha g )_{\mathcal H}\) for every \(f ,g \in \mathscr {S}(\mathbf{R}^{d})\) and \(\alpha \in \mathbf{N}^{d}\).
We observe that (2) in the previous lemma gives
Proof
We have by Definition 1.1
which by polarization gives
when \(f ,g \in \mathcal H\) and \(x,\xi \in \mathbf{R}^{d}\). This gives (1).
The assertion (2) follows by applying \(\partial _x^\alpha \) and \(\partial _\xi ^\alpha \) on (1.9) and (1.10) with \(f,g\in \mathscr {S}(\mathbf{R}^{d})\), and then putting \(x=\xi =0\). \(\square \)
Proof of Theorem 1.3
Suppose that \(\alpha ,\beta \in \mathbf{N}^{d}\) are such that \(\beta _j>\alpha _j\) for some \(j\in \{ 1,\dots ,d \}\). By (1.11), (1.6) and (1.8) we get
and
This implies that \( \{ \Vert h_0\Vert _{\mathcal H}^{1}h_\alpha \} _{\alpha \in \mathbf{N}^{d}} \) is an orthonormal system for \(\mathcal H\).
Hence, if \(f\in \mathscr {S}(\mathbf{R}^{d})\), then
Since the inclusions
are continuous and dense (see e. g. [2, 5]), it follows that \(L^2(\mathbf{R}^{d})\) is continuously embedded in \(\mathcal H\), and that (1.1) holds. Furthermore, let from now on the original \(\mathcal H\) norm be replaced by the equivalent Hilbert norm \(f\mapsto \Vert h_0\Vert _{\mathcal H}^{1}\Vert f\Vert _{\mathcal H}\). Then it follows that the inclusion \(i\, :\ L^2(\mathbf{R}^{d})\rightarrow \mathcal H\) is an isometric injection.
We shall use HahnBanach’s theorem to prove that the latter map is in fact bijective. Suppose that \(\ell \) is a linear continuous form on \(\mathcal H\) which is zero on \(L^2(\mathbf{R}^{d})\). Then \(\ell (f)=(f,g_0)_{\mathcal H}\) for some unique \(g_0\in \mathcal H\). We need to prove that \(g_0=0\).
Since the forms \((\, \cdot \, ,\, \cdot \, )_{L^2}\) and \((\, \cdot \, ,\, \cdot \, )_{\mathcal H}\) agree on \(L^2(\mathbf{R}^{d})\), which contains \(\mathscr {S}(\mathbf{R}^{d})\), and since \(\mathcal H\subseteq \mathscr {S}'(\mathbf{R}^{d})\), the same extension and duality properties hold true with \((\, \cdot \, ,\, \cdot \, )_{\mathcal H}\) in place of \((\, \cdot \, ,\, \cdot \, )_{L^2}\) (cf. the paragraph after Remark 1.4). In particular,
We have \(g_0\in \mathcal H\subseteq \mathscr {S}'(\mathbf{R}^{d})\). Recall that any element in \(\mathscr {S}'(\mathbf{R}^{d})\) has a Hermite series expansion converging in \(\mathscr {S}'(\mathbf{R}^{d})\), whose coefficients are polynomially bounded with respect to their orders (see [7]). Consequently,
for some \(\{ c(\alpha )\} _{\alpha \in \mathbf{N}^{d}}\) such that \(c(\alpha )\le C(1+\alpha )^N\) for some constants \(C,N> 0\). Since \((f,g_0)_{\mathcal H}=0\) when \(f\in L^2(\mathbf{R}^{d})\), and that \(h_\alpha \in \mathscr {S}(\mathbf{R}^{d})\) we get
giving that \(g_0=0\).
By HahnBanach’s theorem it follows that \(L^2(\mathbf{R}^{d})\) is dense in \(\mathcal H\). Since \(L^2(\mathbf{R}^{d})\) is also a closed subset of \(\mathcal H\) in view of (1.13), it follows that \(\mathcal H= L^2(\mathbf{R}^{d})\), and the result follows.
\(\square \)
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Acknowledgements
The first author was supported by Vetenskapsrådet (Swedish Science Council) within the project 201904890. The second author is grateful for the support received from NBHM grant (0204/19/2019R&DII/10472). The third author is thankful to Indian Institute of Science (C.V. Raman PDF, file no: R(IA)CVRPDF/2020/224) for the financial support. The fourth author wishes to thank the infosys foundation for their generous financial support to HarishChandra Research Institute (HRI), Prayagraj (Allahabad), for inviting foreign experts and facilitating research collaboration. The authors are also grateful to HRI for the excellent hospitality and research facilities.
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Appendix A. Some properties of translation invariant Banach spaces
Appendix A. Some properties of translation invariant Banach spaces
In this appendix we give a proof of Proposition 1.5 and discuss maximality of translation and modulation invariant Banach spaces of distributions on \(\mathbf{R}^{d}\).
Proof of Proposition 1.5
Let \(\langle x\rangle =1+x\), \(\varepsilon >0\), Q be the cube \([0,1+\varepsilon ]^d\subseteq \mathbf{R}^{d}\) and \(0\le \varphi \in C_0^\infty (\mathbf{R}^{d})\) be such that
Since \(\mathcal B\) is continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\), it follows from Chapter II in [6] that there exists a constant \(C>0\) and integer \(N\ge 0\) such that
for every \(\psi _0\in C_0^\infty (Q)\) and \(f\in \mathcal B\). Here we have also used the fact that a distribution restricted to a compact set has finite order.
Hence, if \(\psi \in C_0^\infty (\mathbf{R}^{d})\), Leibniz’ rule gives
for some seminorm \(\Vert \, \cdot \, \Vert \) in \(\mathscr {S}(\mathbf{R}^{d})\), and some constants \(C_1\), \(C_2\) and \(C_3\) which are independent of \(f\in \mathcal B\) and \(\psi \). Here \(C_\varphi \) depends only on \(\varphi \), and we have used
Hence,
for some constant C which is independent of \(f\in \mathcal B\) and \(\psi \in C_0^\infty (\mathbf{R}^{d})\). By Remark 1.2 and (A.1) it follows that \(\mathcal B\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\).
\(\square \)
Next we prove the maximization property for translation and modulation invariant Banach spaces. Let \(\omega \in L^\infty _{loc}(\mathbf{R}^{2d};\mathbf{R}_+ )\) be such that
for some constants \(C,r>0\) which are independent of \(x,y,\xi ,\eta \in \mathbf{R}^{d}\). Also let \(\phi \in \mathscr {S}(\mathbf{R}^{d})\setminus \{ 0\}\). Then the modulation space \(M_{(1/\omega )}^{\infty ,\infty }(\mathbf{R}^{d})\) is given by
Here \(V_\phi f\) is the shorttime Fourier transform of \(f\in \mathscr {S}'(\mathbf{R}^{d})\), given by
The Banach space topology in \(M_{(1/\omega )}^{\infty ,\infty }(\mathbf{R}^{d})\) is then given by the norm
(Cf. [2, 3, 5].) We set \(M^{\infty ,\infty }=M^{\infty ,\infty }_{(\omega )}\) when \(\omega =1\).
For other choices of \(\omega \) satisfying (A.2), the \(L^2\) scalar product \((\, \cdot \, ,\, \cdot \, )_{L^2}\) on \(\mathscr {S}(\mathbf{R}^{d})\times \mathscr {S}(\mathbf{R}^{d})\) is uniquely extendable to a continuous sesquilinear form on \(M^{1,1}_{(\omega )}(\mathbf{R}^{d})\times M^{\infty ,\infty }_{(1/\omega )}(\mathbf{R}^{d})\), where \(M^{1,1}_{(\omega )}(\mathbf{R}^{d})\) is the weighted version of \(M^{1,1}(\mathbf{R}^{d})\) with respect to \(\omega \). Furthermore, \(M^{\infty ,\infty }_{(1/\omega )}(\mathbf{R}^{d})\) is the dual of \(M^{1,1}_{(\omega )}(\mathbf{R}^{d})\) with respect to this product (cf. [2, 5]).
Proposition A.1
Let \(\omega \in L^\infty _{loc}(\mathbf{R}^{2d};\mathbf{R}_+ )\) satisfies (A.2). Let \(\mathcal B\) be a translation and modulation invariant Banach space which is continuously embedded in \(\mathscr {D}'(\mathbf{R}^{d})\) and contains at least one element of \(M^{1,1}_{(\omega )}(\mathbf{R}^{d})\setminus \{ 0\}\). Suppose that for some constant \(C>0\) it holds
Then \(\mathcal B\) is continuously embedded in \(M_{(1/\omega )}^{\infty ,\infty }(\mathbf{R}^{d})\).
Proof
By Proposition 1.5 it follows that \(\mathcal B\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\). Hence, for any fixed \(\phi \in \mathscr {S}(\mathbf{R}^{d}){\setminus }0\), the map
is linear and continuous from \(\mathcal B\) to \(\mathbf{C}\), because \(\mathcal B\) is continuously embedded in \(\mathscr {S}'(\mathbf{R}^{d})\), giving that
This gives
which shows that
for some constant \(C>0\). Hence
which implies \(f\in M_{(1/\omega )}^{\infty ,\infty }(\mathbf{R}^{d})\). Furthermore, (A.3) shows that
Consequently, \(\mathcal B\) is continuously embedded in \(M_{(1/\omega )}^{\infty ,\infty }(\mathbf{R}^{d})\), and the assertion follows.
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Toft, J., Gumber, A., Manna, R. et al. Translation and modulation invariant Hilbert spaces. Monatsh Math 196, 389–398 (2021). https://doi.org/10.1007/s00605021015897
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DOI: https://doi.org/10.1007/s00605021015897