Mapping properties for the Bargmann transform on modulation spaces

Abstract

Abstract. We investigate mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions. IntroductionIn [1], V. Bargmannn introduce a transform V which is bijective and isometric from L 2 (R d ) into the Hilbert space A 2 (C d ) of all entire analytic functions F on C d such that F · e −| · | 2 /2 ∈ L 2 (C d ). (We use the usual notations for the usual function and distribution spaces. See e. g. [21], and refer to Section 1 for specific definitions and other notations.) Furthermore, several important properties for V were established. For example:• the Hermite functions are mapped into the normalized analytical monomials. Furthermore, the latter set forms an orthonormal basis for A 2 (C d ); • the creation and annihilation operators, and harmonic oscillator on appropriate elements in L 2 , are transformed by V into simple operators; • there is a convenient reproducing formula for elements in A 2 . In [2], Bargmann continued his work and discussed mapping properties for V on more general spaces. For example, he proves that V(S ′ ), the image of S ′ under the Bargmann transform is given by the formulaHere A p,q (ω) (C d ) is the set of all entire functions F on C d such that F · e −| · | 2 /2 · ω 0 belongs to the mixed Lebesgue space L p,q (C d ), for some appropriate modification ω 0 of the weight function ω.The Bargmann transform can easily be reformulated in terms of the short-time Fourier transform, with a particular Gauss function as window function. In this context we remark that the (classical) modulation spaces M p,q , p, q ∈ [1, ∞], as introduced by Feichtinger in [5], consist of all tempered distributions whose shorttime Fourier transforms (STFT) have finite mixed L p,q norm. It follows that the parameters p and q to some extent quantify the degrees of asymptotic decay and singularity of the distributions in M p,q . The theory of modulation spaces was developed further and generalized in [7, 9, 10, 14], where Feichtinger and Gröchenig established the theory of coorbit spaces. In particular, the modulation space M p,q (ω) , where ω denotes a weight function on phase (or time-frequency shift) space, appears 1 as the set of tempered (ultra-) distributions whose STFT belong to the weighted and mixed Lebesgue space L p,q (ω) . Here the weight ω quantifies the degree of asymptotic decay and singularity of the distribution in M p,q (ω) . A major idea behind the design of these spaces was to find useful Banach spaces, which are defined in a way similar to Besov spaces, in the sense of replacing the dyadic decomposition on the Fourier transform side, characteristic to Besov spaces, with a uniform decomposition. From the construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense are rather similar, and sharp embeddings between these spaces can be found in [26,27], which are improvements of certain embeddings in [13]. (See also [25] for verification of the sharpne...

The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

“…In comparison to already established theories of such spaces (cf. [22,27,47] and the references therein) the conditions for the involved weight functions are significantly relaxed in the present paper. This leads to that our family of modulation spaces are significantly larger compared to the "usual" families of such spaces.…”

Radial symmetric elements and the Bargmann transform

“…Gelfand-Shilov spaces and their distribution spaces can also, in some sense more convenient ways, be characterized by means of estimates of short-time Fourier transforms, (see e. g. [10,12]). We recall here the details and start by recalling the definition of the short-time Fourier transform.…”

On the Inverse to the Harmonic Oscillator

“…It follows that H is an invertible and globally elliptic operator on S and S s , and their dual spaces, for every s ≥ 1/2 (cf. e. g. [17,18] ′ , and their proofs that these results remain valid after the standard harmonic oscillator has been replaced by the operator in (2.29).…”

“…(See e. g. [9-12, 14, 16], and Proposition 2.2 and Theorem 3.10 in [17].) Furthermore, the operator possesses useful regularity properties.…”

“…One way to obtain the latter property is to use Theorem 3.10 in [17]. The other standard way is to use Theorem 25.4 (with m = m 0 = 2) in [16], which implies that 4) and using appropriate mapping properties of pseudo-differential operators with symbols satisfying (0.4).…”