Translation and modulation invariant Hilbert spaces

We show that for any Hilbert space of distributions on $\textbf{R}^d$ which is translation and modulation invariant, is equal to $L^2(\textbf{R}^d)$, with the same norm apart from a multiplicative constant.


Introduction
In the paper we show that any Hilbert space of distributions on R d which is translation and modulation invariant agrees with L 2 (R d ). These considerations are strongly linked with Feichtinger's minimization property, which shows that the modulation space (also called the Feichtinger algebra) M 1,1 (R d ) is the smallest non-trivial Banach space which is norm invariant under translations and modulations. Our investigations may therefore be considered as a Hilbert space analogy of those investigations which lead to Feichtinger's minimization property.
We remark that the search of the smallest Banach space possessing such norm invariance properties, seems to be the main reason that Feichtinger was led to introduce and investigate M 1,1 (R d ) and in its prolongation the foundation of classical modulation spaces (see [2]). The space M 1,1 (R d ) is small in the sense that it is contained in any Lebesgue space L p (R d ), as well as the Fourier image of these spaces (cf. [2,4,5] and the references therein). This fact is also an immediate consequence of Feichtinger's minimization property. On the contrary, the modulation space M ∞,∞ (R d ), which is the dual of M 1,1 (R d ), contains all these Lebesgue and Fourier Lebesgue spaces. By a straight-forward duality approach it can be proved that for suitable assumptions on a translation and modulation invariant Banach space B, we have (where the first inclusion is a reformulation of the Feichtinger's minimization property).
Feichtinger's minimazation property has been extended in different ways, e. g. to weighted spaces (see e. g. [4,Chapter 12]), and to the quasi-Banach situation (see e. g. [5]). At the same time minimization

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 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 (R d ).
We use the same notations as in [3]. The definition of translation and modulation invariant quasi-Banach spaces is given in the following.
Our main result is as follows.

Theorem 1.2. Let H be a translation and modulation invariant Hilbert
for some constant c > 0 which is independent of f ∈ H = L 2 (R d ).
Remark 1.3. It is obvious that the constant c in (1.1) can be evaluated by We need some preparations for the proof. Since H in Theorem 1.2 is continuously embedded in D ′ (R d ), it follows by some straight-forward arguments that H is continuously embedded in S ′ (R d ). In order to be self-contained we here present some motivations.
In fact, let Q d,r be the cube for some constants C > 0 and N ≥ 0 which are independent of ψ 0 ∈ C ∞ 0 (Q d,1+ε ) and f ∈ H.
for some semi-norm · in S (R d ).
Here C ϕ only depends on ϕ. Hence, for some constant C which is independent of f ∈ H and ψ ∈ C ∞ 0 (R d ).
, it follows that the definition of (f, ψ) extends uniquely to any f ∈ H and ψ ∈ S (R d ), and that (1.2) holds. This shows that H is continuously embedded in S ′ (R d ).
By Feichtinger's minimization property it follows that for H in Theorem 1.2 we have with continuous inclusions. In particular, the normalized standard Gaussian on R d , h 0 (x) = π − d 4 e − 1 2 |x| 2 belongs to H. We have now the following lemma. (1) for every f, g ∈ H and x, ξ ∈ R d it holds (1.6) when f, g ∈ H and x, ξ ∈ R d . This gives (1). The assertion (2) follows by applying ∂ α x and ∂ α ξ on (1.4) and (1.5), and then letting x = ξ = 0.
We shall apply the previous result to deduce essential information of Hermite functions and their role in the Hilbert space H. We recall that the Hermite function h α of order α ∈ N d on R d is defined by It is well-known that {h α } α∈N d is an orthonormal basis for L 2 (R d ), and a basis for S (R d ).
We may pass between different Hermite functions by applying the annihilation and creation operators, which are given by respectively, j = 1, . . . , d. It is then well-known that if e j is the jth vector in the standard basis in R d , then and C j h α = α j + 1 h α+e j , α ∈ N d .
By Hahn-Banach's theorem it follows that L 2 (R d ) is dense in H. Since L 2 (R d ) is also a closed subset of H, it follows that H = L 2 (R d ), and the result follows.