Characterization of L. Schwartz’ convolutor and multiplier spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\) by the short-time Fourier transform

Original Paper

DOI: 10.1007/s13398-013-0144-4

Cite this article as:
Bargetz, C. & Ortner, N. RACSAM (2014) 108: 833. doi:10.1007/s13398-013-0144-4
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Abstract

A new definition of the short-time Fourier transform for temperate distributions is presented and its mapping properties are investigated. K.-H. Gröchenig and G. Zimmermann characterized the spaces \(\mathcal S \) and \(\mathcal S '\) of rapidly decreasing functions and temperate distributions, respectively, by their short-time Fourier transform. Following an idea of G. Zimmermann, we give analogous characterizations of the spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\). These spaces, being (PLB)-spaces, have a much more complicated structure than \(\mathcal S \) and \(\mathcal S '\), which is the reason why we have to use the technical machinery of L. Schwartz’ theory of vector-valued distributions.

Keywords

Temperate and (very) rapidly decreasing distributions and functions Fourier transform Short-time Fourier transform 

Mathematics Subject Classification (2010)

Primary 42B10 Secondary 46F12 

1 Introduction and Notation

Originally, the Fourier–Bros–Iagolnitzer transform of a temperate distribution was used to characterize real analyticity in a certain point of \(\mathbb R ^{n}\). This characterization appears in the monograph “Harmonic Analysis in Phase Space” of G. B. Folland ([4, p. 159–164]). The corresponding transform
$$\begin{aligned} P^{t}f(\xi ,x) = \int \limits _\mathbb{R ^{1}} \mathrm{e}^{-2\pi \mathrm{i}y\xi -\pi t(y-x)^{2}} f(y) \,\mathrm{d}y \end{aligned}$$
is called therein wave-packet transform. By replacing \(\mathrm{e}^{-tx^{2}}\) with a rapidly decreasing “window function” \(g\in \mathcal S \) the “short-time Fourier transform” is defined by
$$\begin{aligned} V_{g}f(x,\xi ) = \int \limits _\mathbb{R ^{n}} \mathrm{e}^{-\mathrm{i}y\xi } f(y) g(y-x)\,\mathrm{d}y \end{aligned}$$
in [5] and [6] (up to constants and complex conjugation). In [5], it is shown that
$$\begin{aligned} V_{g}:\mathcal S '_{x} \rightarrow \mathcal S '_{x\xi }, \end{aligned}$$
and the characterization of Schwartz functions (which L. Schwartz called “spherical functions”) by the range of \(V_{g}\), i.e.,
$$\begin{aligned} V_{g}f\in \mathcal S _{x,\xi } \Leftrightarrow f\in \mathcal S _{x}. \end{aligned}$$
More generally, we define in Sect. 4 the mapping \((f,g) \mapsto V_{g}f\) on \(\mathcal S '_{x}\times \mathcal S '_{\xi }\) as well as its left inverse \(W_{h}:\mathcal S '_{x\xi } \rightarrow \mathcal S '_{z}\) using the theory of vector-valued distributions. Hereby, we apply three propositions on the vector-valued multiplication of distributions, namely Proposition 21 bis. in [15, p. 70], Proposition 25 in [16, p. 120] and Proposition 1 (the proof of which is given in Sect. 3).

In Sect. 5, we determine the target spaces of \(f\mapsto V_{g}f\) if \(f\in \mathcal S \) or \(\mathcal O _{C}'\) or \(\mathcal O _{M}\) or \(\mathcal S '\) and \(g\) is fixed in one of the spaces \(\mathcal S \), \(\mathcal O _{C}'\) and \(\mathcal O _{M}\). In Proposition 7, also the target spaces for \(W_{h}\), \(h\in \mathcal S \), are determined.

In Sect. 6, we assert that \(W_{h}\) is the left-inverse of \(V_{g}\) on \(\mathcal S \) and \(\mathcal O _{C}'\) if \(h,g\in \mathcal S \) and on \(\mathcal S '\) if \(h\in \mathcal S \), \(g\in \mathcal S '\).

The final Sect. 7 yields the characterizations of the spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\), i.e., the space of rapidly decreasing distributions and the space of slowly increasing functions, respectively, by the ranges
$$\begin{aligned} \mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi }\, \hbox {and} \, \mathcal O _{C,x}\widehat{\otimes }\mathcal S '_{\xi } \end{aligned}$$
of the short-time Fourier transform.
Generally we adopt L. Schwartz’ notations in [11, 13, 15, 16] with the exceptions
$$\begin{aligned} \mathcal F f(\xi ) = \int \limits _\mathbb{R ^{n}} \mathrm{e}^{-\mathrm{i}\xi y} f(y)\,\mathrm{d}y, \qquad \mathcal F ^{-1}f(x) = (2\pi )^{-n}\int \limits _\mathbb{R ^{n}} \mathrm{e}^{\mathrm{i} x\xi } f(\xi )\,\mathrm{d}\xi , \end{aligned}$$
\(f\) being absolutely integrable and denoting by
$$\begin{aligned} \xi y = \xi _{1}y_{1}+\cdots +\xi _{n}y_{n} \end{aligned}$$
the standard inner product on \(\mathbb R ^{n}\), and the notation \(f(\hat{y})\) for which we simply write \(f(y)\).

\(\mathcal F _{y}\) denotes the partial or vector-valued Fourier transform acting on “distributions in \(y\)”. To give an example:

Let \(g\) be a rapidly decreasing window-function, i.e., \(g\in \mathcal S (\mathbb R ^{1})\) and \(\mathrm{vp\,}\frac{1}{x}\in \mathcal S '(\mathbb R ^{1})\) the principal value distribution. Then, the \(\mathcal C ^{\infty }_{x\xi }\)-function
$$\begin{aligned} V_{g}\left( \mathrm{vp\,}\tfrac{1}{x}\right) :(x,\xi )\mapsto \langle 1_{y}, \mathrm{e}^{-\mathrm{i}y \xi } g(y-x)\mathrm{vp\,}\tfrac{1}{y} \rangle , \end{aligned}$$
given pointwise by the expression
$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^{+}} \int \limits _{|y|\ge \varepsilon } \mathrm{e}^{-\mathrm{i}y\xi } \frac{g(y-x)}{y}\, \mathrm{d}y, \end{aligned}$$
defines the short-time Fourier transform \(V_{g}(\mathrm{vp\,}\frac{1}{x})\) of \(\mathrm{vp\,}\frac{1}{x}\)which we define as the vector-valued Fourier transform
$$\begin{aligned} V_{g}\left( \mathrm{vp\,}\tfrac{1}{x}\right) (x,\xi ) = \mathcal F _{y}\left( g(y-x)\mathrm{vp\,}\tfrac{1}{y}\right) (\xi ). \end{aligned}$$
In Proposition 4 1., it is shown that we have, more precisely,
$$\begin{aligned} V_{g}\left( \mathrm{vp\,}\tfrac{1}{x}\right) \in \mathcal O _{C,x}\widehat{\otimes }\mathcal O _{C,\xi } \subset \mathcal C ^{\infty }_{x,\xi } = \mathcal E _{x,\xi }. \end{aligned}$$
The occurring distribution spaces are defined in Sect. 2.

For locally convex spaces \(E\) and \(F\), \(\mathcal L (E,F)\) is the space of linear and continuous maps \(E\rightarrow F\). The indices \(b\) and \(c\) denote the topologies of uniform convergence on all bounded and all compact, absolutely convex subsets, respectively, of \(E\).

Completed tensor products \(E\widehat{\otimes }F\) without an index denote the completions of the tensor product \(E\otimes _{\varepsilon }F = E\otimes _{\pi }F\) if \(E\) or \(F\) is a nuclear space.

2 Recapitulation: spaces, Fourier transforms, vector-valued Fourier transforms

The space \(\mathcal S =\mathcal S (\mathbb R ^{n})\) of rapidly decreasing functions is defined as
$$\begin{aligned} \mathcal S (\mathbb R ^{n}) = \left\{ \varphi \in \mathcal C ^{\infty }(\mathbb R ^{n}); \forall m\in \mathbb N _{0}\,\forall \alpha \in \mathbb N _{0}^{n}:\left( 1+|x|^{2}\right) ^{m/2}\,\partial ^{\alpha }f \in \mathcal C _{0}(\mathbb R ^{n}) \right\} , \end{aligned}$$
the space of very slowly increasing functions
$$\begin{aligned} \mathcal O _{C}(\mathbb R ^{n}) = \left\{ \varphi \in \mathcal C ^{\infty }(\mathbb R ^{n}); \exists m\in \mathbb N _{0}\,\forall \alpha \in \mathbb N _{0}^{n}:\left( 1+|x|^{2}\right) ^{-m/2}\,\partial ^{\alpha }f \in \mathcal C _{0}(\mathbb R ^{n}) \right\} \end{aligned}$$
(“fonctions à croissance très lente”: [8, p. 173], [1, p. 131]), the space of slowly increasing functions
$$\begin{aligned} \mathcal O _{M}(\mathbb R ^{n}) = \left\{ \varphi \in \mathcal C ^{\infty }(\mathbb R ^{n}); \forall \alpha \in \mathbb N _{0}^{n}\,\exists m\in \mathbb N _{0}:\left( 1+|x|^{2}\right) ^{-m/2}\,\partial ^{\alpha }f \in \mathcal C _{0}(\mathbb R ^{n}) \right\} . \end{aligned}$$
\(\mathcal O _{M}\) is the space of multipliers of\(\mathcal S \)and\(\mathcal S '\), i.e.,
$$\begin{aligned} \mathcal O _{M} = \left\{ \varphi \in \mathcal C ^{\infty }; \forall \psi \in \mathcal S :\psi \cdot \varphi \in \mathcal S \right\} = \left\{ \varphi \in \mathcal C ^{\infty }; \forall T\in \mathcal S ':\varphi \cdot T \in \mathcal S '\right\} \end{aligned}$$
(see [13, p. 243]). \(\mathcal O _{M}'\) is the space of very rapidly decreasing distributions (“distributions à decroissance très rapide”: [1, p. 130]), \(\mathcal O _{C}'\) the space of rapidly decreasing distributions ([13, p. 244]).
We have
$$\begin{aligned} \mathrm{e}^{\mathrm{i} |x|^{2}} \in \left( \mathcal O _{C}'\!\setminus \!\mathcal O _{M}'\right) \cap \left( \mathcal O _{M}\!\setminus \!\mathcal O _{C}\right) \end{aligned}$$
(see Exemple in [13, p. 245]).
The multiplication of test functions is a hypocontinuous bilinear mapping
$$\begin{aligned} \mathcal O _{M}\times \mathcal S \rightarrow \mathcal S , (\varphi , \psi ) \mapsto \varphi \cdot \psi \end{aligned}$$
and hence also the multiplication
$$\begin{aligned} \mathcal O _{C}\times \mathcal S \rightarrow \mathcal S , (\varphi , \psi ) \mapsto \varphi \cdot \psi \end{aligned}$$
as \(\mathcal O _{C}\hookrightarrow \mathcal O _{M}\) is contained with a finer topology.
The multiplication of test functions with distributions is a hypocontinuous bilinear mapping
$$\begin{aligned} \begin{aligned} \mathcal O _{M} \times \mathcal S '&\rightarrow \mathcal S ', (\varphi , T) \mapsto \varphi \cdot T, \\ \mathcal O _{M} \times \mathcal O _{C}'&\rightarrow \mathcal S ', (\varphi , T) \mapsto \varphi \cdot T,\\ \mathcal O _{M} \times \mathcal O _{M}'&\rightarrow \mathcal O _{M}', (\varphi , T) \mapsto \varphi \cdot T, \\ \mathcal O _{C} \times \mathcal O _{C}'&\rightarrow \mathcal O _{C}', (\varphi , T) \mapsto \varphi \cdot T. \end{aligned} \end{aligned}$$
By Fourier transform and transposition, we have the regularization property
$$\begin{aligned} \mathcal S '\times \mathcal S \rightarrow \mathcal O _{C}, (T,\varphi ) \mapsto \varphi *T \end{aligned}$$
(Proposition 7 in [8, p. 420]), whereas
$$\begin{aligned} \mathcal O '_{C}\times \mathcal S \rightarrow \mathcal S , (T,\varphi ) \mapsto \varphi *T \end{aligned}$$
(Théorème IX in [13, p. 244]).
The distribution \(\delta (x-y)\in \mathcal S '_{xy} = \mathcal S '(\mathbb R ^{n}_{x}\times \mathbb R ^{n}_{y})\) is the kernel of the identity mapping, i.e., if \(\mathcal H \) is a normal space of distributions then
$$\begin{aligned} \delta (x-y) \in \mathcal L (\mathcal H ,\mathcal H ) \subset \mathcal H '_{c}{{\mathrm{\varepsilon }}}\mathcal H . \end{aligned}$$
Herein, \(E{{\mathrm{\varepsilon }}}F\) means L. Schwartz’ \(\varepsilon \)-product (see Définition in [15, p. 18]) of the spaces \(E\) and \(F\). By Corollaire 1 in [15, p. 47], we have
$$\begin{aligned} \mathcal H '_{c}{{\mathrm{\varepsilon }}}\mathcal H = \mathcal H '_{c}\widehat{\otimes }_{\varepsilon }\mathcal H \end{aligned}$$
if \(\mathcal H '_{c}\) and \(\mathcal H \) are complete and one of them satisfies the approximation property. Furthermore, for nuclear\(\mathcal H \) it holds \(\mathcal H '_{c} = \mathcal H '\) and hence
$$\begin{aligned} \mathcal H '_{c}\widehat{\otimes }_{\varepsilon }\mathcal H = \mathcal H '\widehat{\otimes }\mathcal H = \mathcal L _{b}(\mathcal H , \mathcal H ). \end{aligned}$$
By Théorème 16 in [1, p. 131] and Théorème 10 in [1, p. 55], the spaces
$$\begin{aligned} \begin{array}{*7{c}} \mathcal S &{} \subset &{} &{} \subset &{} \mathcal O _{C} &{} \subset &{} \mathcal O _{M} \\ \bigcap &{} &{} &{} &{} &{} &{} \bigcap \\ \mathcal O _{M}' &{} \subset &{} \mathcal O _{C}' &{} \subset &{} &{} \subset &{} \mathcal S ' \end{array} \end{aligned}$$
are complete, nuclear, normal spaces of distributions. The inclusions are continuous and we have
$$\begin{aligned} \delta (x-y) \in \mathcal S '_{x}\widehat{\otimes }\mathcal S _{y},\,\, \delta (x-y) \in \mathcal O _{C,x}\widehat{\otimes }\mathcal O '_{C,y} \text { and } \delta (x-y) \in \mathcal O _{M,x}'\widehat{\otimes }\mathcal O _{M,y}. \end{aligned}$$
Taking into account the scalar isomorphism properties of the Fourier transform
$$\begin{aligned} \mathcal F :\mathcal S \rightarrow \mathcal S \text { and } \mathcal F :\mathcal O _{C}' \rightarrow \mathcal O _{M}, \end{aligned}$$
we obtain
  1. (i)

    \(\mathrm{e}^{-\mathrm{i} x\xi } \in \mathcal S '_{x}\widehat{\otimes }\mathcal S _{\xi }\),

     
  2. (ii)

    \(\mathrm{e}^{-\mathrm{i} x\xi } \in \mathcal O _{C,x}\widehat{\otimes }\mathcal O _{M,\xi }\),

     
  3. (iii)

    \(\mathrm{e}^{-\mathrm{i} x\xi } \in \mathcal O _{M,x}'\widehat{\otimes }\mathcal O _{C,\xi }'\),

     
(cf. [12, p. 225]) by applying the vector-valued partial Fourier transform (see [15, p. 73]). Let us remark that (i) is the Lemma in [15, p. 133] whereas (ii) means that the vector-valued function
$$\begin{aligned} \mathbb R ^{n}_{x} \rightarrow \mathcal O _{C}(\mathbb R ^{n}_{\xi }), x \mapsto \left[ \xi \mapsto \mathrm{e}^{-\mathrm{i}x\xi }\right] \end{aligned}$$
belongs to the space \(\mathcal O _{M}(\mathbb R ^{n}_{x}; \mathcal O _{C}(\mathbb R ^{n}_{\xi })) = \mathcal O _{M,x} \widehat{\otimes }\mathcal O _{C,\xi }\).
Considering \(\mathrm{e}^{-\mathrm{i}xy}\) as the kernel of the map \(\mathcal F :\mathcal S '\rightarrow \mathcal S '\), we state that (i) means: the Fourier transform is the transpose of the Fourier transform \(\mathcal F :\mathcal S \rightarrow \mathcal S \) defined as the restriction of the Fourier transform of absolutely integrable functions. The assertions (ii) and (iii) can be interpreted in the following way:
$$\begin{aligned} \mathcal F :\mathcal O _{C}' \rightarrow \mathcal O _{M} \text { and } \mathcal F :\mathcal O _{M}' \rightarrow \mathcal O _{C} \end{aligned}$$
are isomorphisms of topological vector spaces (Théorème XV in [13, p. 268].

3 On the existence and uniqueness of vector-valued bilinear maps

In order to establish the mapping properties of the \(W_{h}\)-transform in Proposition 7, we apply a more abstract result on the existence and uniqueness of vector-valued bilinear maps.

Given continuous linear maps \(u:E\rightarrow F\) and \(v:G \rightarrow H\) and tensor product topologies \(\lambda \le \mu \) on \(E\otimes G\) and \(F\otimes H\), there is a unique continuous linear map
$$\begin{aligned} u\otimes v:E\widehat{\otimes }_{\lambda }F\rightarrow G\widehat{\otimes }_{\mu }H \end{aligned}$$
which is the continuation of the continuous mapping \(u\otimes v\) on the tensor products.

For (non-continuous but hypo- or partially continuous) bilinear maps, the situation is more complicated. In the article [16], L. Schwartz considered this problem of the existence of suitable “tensor products” of bilinear maps. Based on the “théorèmes de croisement” he proves that such a vector-valued bilinear map exists if, besides certain assumptions on the spaces, one of the bilinear maps is continuous and the other one is hypocontinuous (Corollaire to Propostion 3 in [16, p. 38]). L. Schwartz also gives a much more complicated result for the multiplication and the convolution of vector valued distributions based on two partially continuous bilinear maps, where the spaces in the pre-image domain are only subspaces of the completed tensor products of the pre-image domains of the scalar bilinear maps (Proposition 25 and Proposition 38 in [16, p. 120 and p. 159]).

Our purpose for this section is to give a result on the “tensor product” of two hypocontinuous bilinear mappings with assumptions on the occurring spaces which are more easy to check than the ones in Proposition 25 and Proposition 38 in [16, p. 120 and p. 159].

In the following, we will use an additional topology on the tensor product of two locally convex spaces \(E\) and \(F\)—the \(\beta \)-topology. It is the finest locally convex topology such that the canonical mapping
$$\begin{aligned} E \times F \rightarrow E\otimes _{\beta } F \end{aligned}$$
is hypocontinuous. If we equip \(E\otimes F\) with the \(\beta \)-topology the canonical isomorphism between bilinear maps on \(E\times F\) and linear maps on \(E\otimes _{\beta } F\) maps the hypocontinuous bilinear maps into the continuous linear maps on the tensor product. For the definition of \(\beta \)-\(\beta \) and \(\beta \)-\(\gamma \)-decomposable set, we refer to [16, p. 15].

Proposition 1

(cf. Proposition 31 in [3, p. 48]) Let \(\mathcal H \), \(\mathcal K \) and \(\mathcal L \) be complete spaces of distributions (or more general complete locally convex topological vector-spaces), where \(\mathcal H \) is nuclear. Let \(E\), \(F\) and \(G\) be three complete locally convex topological vector spaces and
$$\begin{aligned} \cup :\mathcal H \times \mathcal K \rightarrow \mathcal L \text { and } b:E \times F \rightarrow G \end{aligned}$$
be two hypocontinuous bilinear maps.
If one of the assumptions
  1. 1.

    \(\mathcal H \) and \(E\) are Fréchet spaces

     
  2. 2.

    \(\mathcal H \) and \(E\) are (DF)-spaces

     
is satisfied, there is a hypocontinuous bilinear map
$$\begin{aligned} \begin{array}{l}\cup \\ b\end{array}:\mathcal H (E)\times \mathcal K (F)\rightarrow \mathcal L (G) \end{aligned}$$
satisfying the consistency property
$$\begin{aligned} \begin{array}{l}\cup \\ b\end{array} (S\otimes e, T\otimes f) = (S\cup T)\otimes b(e,f). \end{aligned}$$
If \(\mathcal K \) satisfies the approximation property \(\begin{array}{l}\cup \\ b\end{array}\) is the unique partially continuous bilinear map satisfying this property.

We prove this proposition by an adaptation of the proof of Proposition 3 in [16, p. 37].

Proof

In Proposition 2 in [16, p. 18] the bilinear map
$$\begin{aligned} \Gamma _{\beta ,\beta }:(\mathcal H \widehat{\otimes }_{\beta }E) \times (\mathcal K {{\mathrm{\varepsilon }}}F) \rightarrow (\mathcal H \widehat{\otimes }_{\beta }\mathcal K ) {{\mathrm{\varepsilon }}}(E\widehat{\otimes }_{\beta }F) \end{aligned}$$
is defined. It is hypocontinuous with respect to bounded sets of \(\mathcal K (F)\) and \(\beta \)-\(\beta \)-decomposable sets of \(\mathcal H \widehat{\otimes }_{\beta }E\). On \((\mathcal H \otimes E)\times (\mathcal K \otimes F)\rightarrow \mathcal H \otimes \mathcal K \otimes E\otimes F\) it coincides with the canonical mapping.

If \(\mathcal H \) and \(E\) are Fréchet spaces then every partially continuous bilinear map on \(\mathcal H \times E\) is continuous by Theorem 1 in [8, p. 357] hence it holds \(\mathcal H \widehat{\otimes }_{\beta }E = \mathcal H \widehat{\otimes }_{\pi }E\).

If \(\mathcal H \) and \(E\) are (DF)-spaces then every hypocontinuous bilinear map on \(\mathcal H \times E\) is continuous by Théorème 2 in [7, p. 64] hence \(\mathcal H \widehat{\otimes }_{\beta }E = \mathcal H \widehat{\otimes }_{\pi }E\).

As \(\mathcal H \) is nuclear, we have \(\mathcal H \widehat{\otimes }_{\pi }E = \mathcal H \widehat{\otimes }_{\varepsilon }E\). As by Theorem 13.1 in [14, p. 69] nuclear spaces satisfy the approximation property, Corollaire 1 in [15, p. 47] yields \(\mathcal H \widehat{\otimes }_{\varepsilon }E = \mathcal H {{\mathrm{\varepsilon }}}E\). Hence the identity \(\mathcal H \widehat{\otimes }_{\beta }E = \mathcal H (E)\) holds.

If \(\mathcal H \) and \(E\) are (DF)-spaces every bounded subset of \(\mathcal H (E)\) is \(\beta \)-\(\beta \)-decomposable according to Proposition 1 in [16, p. 16].

If  \(\mathcal H \) and \(E\) are Fréchet spaces Proposition 1 in [16, p. 16] yields that every bounded subset of \(\mathcal H \widehat{\otimes }E = \mathcal H \widehat{\otimes }_{\beta }E\) is \(\gamma \)-\(\beta \)-decomposable, i.e., for all bounded sets \(\Xi \subset \mathcal H (E)\) there exists a compact set \(A\subset \mathcal H \) and a bounded set \(B\subset E\) such that \(\Xi \subset \overline{\mathrm{ac\,}(A\otimes B)}\). As \(A\) is compact, \(A\) is bounded and hence \(\Xi \) is \(\beta \)-\(\beta \)-decomposable.

Summing up the above results, we obtain that
$$\begin{aligned} \Gamma _{\beta ,\beta }:\mathcal H (E)\times \mathcal K (F) \rightarrow (\mathcal H \widehat{\otimes }_{\beta }\mathcal K ){{\mathrm{\varepsilon }}}(E\widehat{\otimes }_{\beta }F). \end{aligned}$$
is hypocontinuous with respect to bounded sets of \(\mathcal H (E)\) and \(\mathcal K (F)\) and therefore also
$$\begin{aligned} (\tilde{\cup }{{\mathrm{\varepsilon }}}\mathrm{id\,})\circ \Gamma _{\beta ,\beta }:\mathcal H (E)\times \mathcal K (F) \rightarrow \mathcal L (E\widehat{\otimes }_{\beta }F), \end{aligned}$$
where \(\tilde{\cup }\) denotes the linear map \(\mathcal H \widehat{\otimes }\mathcal K \rightarrow \mathcal L \) associated with \(\cup \). Additionally it holds
$$\begin{aligned} ((\tilde{\cup }{{\mathrm{\varepsilon }}}\mathrm{id\,})\circ \Gamma _{\beta ,\beta })(S\otimes e, T\otimes f) = (S\cup T)\otimes (e\otimes f), \end{aligned}$$
as \(\Gamma _{\beta ,\beta }|_{(\mathcal H \otimes E)\times (\mathcal K \otimes F)} = \mathrm{can}:(\mathcal H \otimes E)\times (\mathcal K \otimes F) \rightarrow (\mathcal H \otimes \mathcal K )\otimes (E\otimes F).\) If \(\mathcal K \) satisfies the approximation property then \(\mathcal K \otimes F\subset \mathcal K (F)\) is a dense subspace and \((\tilde{\cup }{{\mathrm{\varepsilon }}}\mathrm{id\,})\circ \Gamma _{\beta ,\beta }\) is the unique partially continuous bilinear mapping satisfying the above consistency property.

The map \((S,T)\mapsto \begin{array}{l}\cup \\ b\end{array}(S,T) :=((\mathrm{id\,}{{\mathrm{\varepsilon }}}\tilde{b})\circ (\tilde{\cup }{{\mathrm{\varepsilon }}}\mathrm{id\,})\circ \Gamma _{\beta ,\beta }) (S,T)\) satisfies the properties stated above.

Remark 1

By replacing the completed tensor products by quasi-completed ones, the proposition above also holds true if \(\mathcal H \), \(\mathcal K \), \(\mathcal L \) and \(G\) are just quasi-complete and \(\mathcal H \) satisfies the strict approximation property. The completeness assumptions on \(E\) and \(F\) can be dropped completely. For the details, we refer to [3, p. 48].

4 Definition of the short-time Fourier transform \(V_{g}\) and of the \(W_{h}\)-transform for distributions

Analogously to the Fourier-Wigner transform in Chapter 1.4 in [4, p. 30ff], the short-time Fourier transform\(V_{g}f\) is defined in [5, p. 206] and [6, p. 26] (up to complex conjugation and factors in the exponent) by the function
$$\begin{aligned} V_{g}f(x,\xi ) = \int \limits _\mathbb{R ^{n}} f(y)\mathrm{e}^{-\mathrm{i}\xi y} g(y-x)\, \mathrm{d}y \end{aligned}$$
if \(f,g\in L^{2}(\mathbb R ^{n})\). Denoting by \(\mathcal C _{0}\) the space of continuous functions vanishing at infinity, the classical properties
$$\begin{aligned} L^{2}\times L^{2} \rightarrow \mathcal C _{0}, (f,g)\mapsto f*g \end{aligned}$$
([17, p. 115]), and
$$\begin{aligned} \mathcal F :L^{1}\rightarrow \mathcal C _{0}, \end{aligned}$$
(Riemann-Lebesgue) imply that the mapping
$$\begin{aligned} L^{2}\times L^{2} \rightarrow \mathcal BC , (f,g)\mapsto V_{g}f \end{aligned}$$
is well-defined, bilinear and continuous.
By the same reasons the mapping
$$\begin{aligned} L^{p} \times L^{q} \rightarrow \mathcal BC , (f,g)\mapsto V_{g}f, 1<p<\infty , \frac{1}{q} = 1 - \frac{1}{p} \end{aligned}$$
is well-defined, bilinear and continuous.

In [5, 6], the function \(g\) is called “window function”. If \(g=\mathrm{e}^{-t x^{2}}\), \(t>0\), then \(V_{g}\) coincides with the FBI-transform, investigated in (3.30) in [4, p. 159] and in [10, p. 137].

4.1 Definition of the short-time Fourier transform for temperate distributions

For distributions \(f,g\in \mathcal S '\) the expression \(f(y)g(y-x)\) is defined as the image of
$$\begin{aligned} f(\xi )\otimes g(\eta )\in \mathcal S '(\mathbb R ^{2n}_{\xi ,\eta }) \end{aligned}$$
under the linear map
$$\begin{aligned} \mathbb R ^{2n}_{x,y} \rightarrow \mathbb R ^{2n}_{\xi ,\eta }, \xi =y, \eta = x-y \end{aligned}$$
(cf. 2\(^{\circ }\) Convolution in [15, p. 131]).
If \(f,g\in \mathcal S '(\mathbb R ^{n})\) then \(V_{g}f\) is defined by the partial or vector-valued Fourier transform ([15, p. 73]):
$$\begin{aligned} V_{g}f = \mathcal F _{y}(f(y)g(y-x)) \in \mathcal S '_{x,y}. \end{aligned}$$

Proposition 2

The mapping
$$\begin{aligned} \mathcal S _{x}'\times \mathcal S _{y}' \rightarrow \mathcal S '_{x,\xi }, (f,g)\mapsto V_{g}f \end{aligned}$$
is bilinear, continuous and injective if \(g\ne 0\).

Proof

The canonical mapping \(\mathcal S _{\xi }'\times \mathcal S _{\eta }'\rightarrow \mathcal S '_{\xi ,\eta }, (f,g)\mapsto f\otimes g\) is continuous, as the tensor product decomposition \(\mathcal S '_{\xi ,\eta } = \mathcal S '_{\xi }\widehat{\otimes }\mathcal S '_{\eta }\) holds by Proposition 28 in [15, p. 98] and the definition of the \(\pi \)-topology. As the mapping \(\mathcal S _{x,y}\rightarrow \mathcal S _{\xi ,\eta }, \psi \mapsto \psi (\xi +\eta ,\xi )\) is continuous, the same holds true for its transpose \(\mathcal S '_{\xi ,\eta }\rightarrow \mathcal S '_{x,y}\) which is the above coordinate transform. As by [15, p. 73] the vector-valued Fourier transform is continuous, the mapping
$$\begin{aligned} \mathcal S _{x}'\times \mathcal S _{y}' \rightarrow \mathcal S '_{x,\xi }, (f,g)\mapsto V_{g}f \end{aligned}$$
is the composition of a continuous bilinear mapping and continuous linear mappings and hence bilinear and continuous itself.

Remark 2

Note that the bilinear mapping in the proposition above is defined (less explicitly) by continuous extension from \(\mathcal S \times \mathcal S \) into \(\mathcal S \) in (1.42) Proposition in [4, p. 31]. The proof of the continuity is missing there.

4.2 Definition of the \(W_{h}\)-transform

Definition and Proposition 3

If \(h\in \mathcal S \) and \(F\in \mathcal S '_{x,\xi }\) then the \(W_{h}\)-transform
$$\begin{aligned} W_{h}:\mathcal S '_{x,\xi } \rightarrow \mathcal S '_{z}, F \mapsto {_\mathcal{O _{C,x}}\langle 1_{x}, (\mathcal F _{\xi }^{-1}F)(x,z) h(z-x)\rangle _\mathcal{O _{C,x}'(\mathcal S '_{z})}} \end{aligned}$$
is well-defined, linear and continuous.
The bracket \({_\mathcal{O _{C,x}}\langle \cdot , \cdot \rangle _\mathcal{O _{C,x}'(\mathcal S '_{z})}}\) is the \(\mathcal S '\)-valued extension of the evaluation mapping
$$\begin{aligned} \mathcal O _{C} \times \mathcal O _{C}' \rightarrow \mathbb C , (\varphi , T) \mapsto T(\varphi ) \end{aligned}$$
and hence bilinear and hypocontinuous by Theorem 7.1 in [14, p. 30] or Proposition 4 in [16, p. 41].

Proof

\(h\in \mathcal S \) implies \(h(z-x)\in \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}\) and \(F\in \mathcal S '_{x,\xi }\) implies \((\mathcal F ^{-1}_{\xi }F)(x,z)\in \mathcal S _{x}'\widehat{\otimes }\mathcal S '_{z}\). Proposition 1 yields the unique existence of a hypocontinuous bilinear multiplication
$$\begin{aligned} (\mathcal S '_{x}\widehat{\otimes }\mathcal S '_{z}) \times (\mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}) \rightarrow \mathcal O _{C,x}' \widehat{\otimes } \mathcal S '_{z} \end{aligned}$$
and hence
$$\begin{aligned} (\mathcal F _{\xi }^{-1}F)(x,z)h(z-x) \in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S '_{z}. \end{aligned}$$
Proposition 1 can be applied since the mappings
$$\begin{aligned} \begin{aligned} \mathcal S \times \mathcal S '&\rightarrow \mathcal O _{C}', (\varphi , T)\mapsto \varphi \cdot T, \\ \mathcal O _{C} \times \mathcal S '&\rightarrow \mathcal S ', (\varphi , T) \mapsto \varphi \cdot T \end{aligned} \end{aligned}$$
are hypocontinuous and since \(\mathcal S '\) is a (DF)-space. \(\square \)

Remarks

Note that the mapping \(W_{h}\) restricted to \(\mathcal S _{x,\xi }\) is the transpose of the short-time Fourier transform \(V_{g}\).

5 Ranges of the short-time Fourier transform \(V_{g}\) and of the \(W_{h}\)-transform

We fix \(g\in \mathcal S \) in the following Proposition 4 and investigate the mapping \(f\mapsto V_{g}f\).

Proposition 4

For fixed \(g\in \mathcal S \) the short-time Fourier transform \(V_{g}\) maps
  1. 1.

    \(\mathcal S '\) into \(\mathcal O _{C,x}\widehat{\otimes }\mathcal O _{C,\xi }\),

     
  2. 2.

    \(\mathcal O _{M}\) into \(\mathcal O _{C,x}\widehat{\otimes }\mathcal S _{\xi }\),

     
  3. 3.

    \(\mathcal S \) into \(\mathcal S _{x}\widehat{\otimes }\mathcal S _{\xi } = \mathcal S _{x,\xi }\) and

     
  4. 4.

    \(\mathcal O _{C}'\) into \(\mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi }\)

     
and these linear maps are continuous.

Proof

For \(g\in \mathcal S \) it holds \(g(y-x)\in \mathcal O _{C,x}\widehat{\otimes }\mathcal S _{y}\) and \(g(y-x)\in \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,y}\) by Proposition 7 in [8, p. 420].
  1. 1.
    By vector-valued multiplication (Proposition 21 bis. in [15, p. 70]) with \(f(y)\in \mathcal S '_{y}\) and by observing that \(\mathcal S \cdot \mathcal S '\subset \mathcal O _{M}'\), we obtain
    $$\begin{aligned} f(y)g(y-x) \in \mathcal O _{C,x}\widehat{\otimes }\mathcal O _{M,y}'. \end{aligned}$$
    The partial Fourier transform with respect to \(y\) furnishes
    $$\begin{aligned} (V_{g}f)(x,\xi ) = \left( \mathcal F _{y}(f(y)g(y-x))\right) (\xi ) \in \mathcal O _{C,x}\widehat{\otimes }\mathcal O _{C,\xi }. \end{aligned}$$
     
  2. 2.
    For \(f\in \mathcal O _{M}\), we get \(f(y)g(y-x)\in \mathcal O _{C,x}\widehat{\otimes } \mathcal S _{y}\). The partial Fourier transform with respect to \(y\) yields
    $$\begin{aligned} V_{g}f(x,\xi ) = (\mathcal F _{y}\left( f(y)g(y-x)\right) )(\xi ) \in \mathcal O _{C,x} \widehat{\otimes }\mathcal S _{\xi }. \end{aligned}$$
     
  3. 3.
    We now use the fact \(g(y-x)\in \mathcal O _{C,y}\widehat{\otimes } \,\mathcal S _{x}\). Multiplication with \(f(y)\in \mathcal S _{y}\) and the inclusion \(\mathcal O _{C}\cdot \mathcal S \subset \mathcal S \) deliver
    $$\begin{aligned} f(y)g(y-x) \in \mathcal S _{y} \widehat{\otimes } \mathcal S _{x}. \end{aligned}$$
    The partial Fourier transform implies
    $$\begin{aligned} V_{g}f(x,\xi ) = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal S _{x}\widehat{\otimes }\mathcal S _{\xi } =\mathcal S _{x,\xi }, \end{aligned}$$
    wherein the last equality follows from Proposition 28 in [15, p. 98].
     
  4. 4.
    Vector-valued multiplication of
    $$\begin{aligned} g(y-x) \in \mathcal O _{C,y} \widehat{\otimes } \mathcal S _{x} \end{aligned}$$
    with \(f(y)\in \mathcal O _{C,y}'\) furnishes
    $$\begin{aligned} f(y)g(y-x) \in \mathcal O _{C,y}' \widehat{\otimes } \mathcal S _{x}. \end{aligned}$$
    The partial Fourier transform implies
    $$\begin{aligned} V_{g}f(x,\xi ) = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal S _{x}\widehat{\otimes } \mathcal O _{M,\xi }. \end{aligned}$$
     
The occurring linear mappings are continuous as they are the composition of continuous linear mappings. \(\square \)

Remarks 1

  1. 1.

    Corollary 2.5 in [6, p. 32] is a special case of 1.

     
  2. 2.

    The implication \((i)\Rightarrow (ii)\) of Theorem 2.3 in [5, p. 210] is equivalent to 3.

     
  3. 3.
    The function
    $$\begin{aligned} \mathrm{e}^{-(x^{2}-y)^{2}}\in \left( \mathcal O _{C}(\mathbb R _{x})\widehat{\otimes } \mathcal O _{C}(\mathbb R _{y})\right) \!\setminus \! \mathcal O _{C}(\mathbb R ^{2}_{x,y}) \end{aligned}$$
    shows that \(\mathcal O _{C,x,y}\ne \mathcal O _{C,x}\widehat{\otimes }\mathcal O _{C,y}\) (P. Wagner).
     
  4. 4.
    Let us show the representation
    $$\begin{aligned} \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,\xi }&= \Big \{\varphi \in \mathcal C ^{\infty }(\mathbb R ^{2n}_{x,\xi }); \forall k\in \mathbb N _{0}\, \forall \alpha \in \mathbb N _{0}^{n}\, \exists m\in \mathbb N _{0}\, \forall \beta \in \mathbb N _{0}^{n}:\\&\quad (1+|x|^{2})^{k/2}(1+|\xi |^{2})^{-m/2} \partial ^{\alpha }_{x} \partial ^{\beta }_{\xi }\varphi \in \mathcal C _{0}(\mathbb R ^{2n}_{x,\xi })\Big \}. \end{aligned}$$
    We have
    $$\begin{aligned} \mathcal S \widehat{\otimes }\mathcal O _{C} = \lim _{\leftarrow k} \left( [(1+|x|^{2})^{-k/2}\dot{\mathcal{B }}]\widehat{\otimes }\mathcal O _{C}\right) \end{aligned}$$
    as by 2. Theorem in [9, p. 332] reduced projective limits commute with the projective tensor product. It follows from Proposition 17 in [15, p. 59] that \(\varphi \in \mathcal S \widehat{\otimes }\mathcal O _{C}\) if and only if
    $$\begin{aligned} \forall k\in \mathbb N _{0}\,\forall \alpha \in \mathbb N _{0}^{n}:(1+|x|^{2})^{k/2}\partial ^{\alpha }_{x}\varphi (x,\cdot ) \rightarrow 0 \text { in } \mathcal O _{C}. \end{aligned}$$
    Finally, the assertion follows from the fact, that by [2, p. 77] the inductive limit
    $$\begin{aligned} \mathcal O _{C} = \lim _{m\rightarrow } (1+|\xi |^{2})^{m/2}\dot{\mathcal{B }} \end{aligned}$$
    is a regular one.
     

In the next proposition, we fix \(g\in \mathcal O _{C}'\):

Proposition 5

If \(g\in \mathcal O _{C}'\) then the short-time Fourier transform \(V_{g}\) maps
  1. 1.

    \(\mathcal S '\) into \(\mathcal O _{C,x}'\widehat{\otimes }\mathcal S '_{\xi }\),

     
  2. 2.

    \(\mathcal O _{M}\) into \(\mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{C,\xi }' = \mathcal O _{C,x,\xi }'\),

     
  3. 3.

    \(\mathcal S \) into \(\mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{\xi }\) and

     
  4. 4.

    \(\mathcal O _{C}'\) into \(\mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{M,\xi }\).

     
The linear maps are continuous.

Proof

Note that \(g\in \mathcal O _{C}'\) implies \(g(y-x)\in \mathcal O _{C,x}'\widehat{\otimes } \mathcal O _{C,y}\).
  1. 1.

    By multiplying with \(f(y)\in \mathcal S '_{y}\), we obtain \(f(y)g(y-x)\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S '_{y}\). Hence, we get \(V_{g}f \in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S '_{\xi }\).

     
  2. 2.

    However, multiplication with \(f\in \mathcal O _{M}\) yields \(f(y)g(y-x)\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{M,y}\) and, hence, \(V_{g}f\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{C,\xi }' = \mathcal O _{C,\,x,\xi }'\), wherein the last equality follows by Proposition 28 in [15, p. 98].

     
  3. 3.

    Multiplication by \(f(y)\in \mathcal S _{y}\) yields \(f(y)g(y-x)\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{y}\) and therefore finally \(V_{g}f \in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{\xi }\).

     
  4. 4.

    Multiplication by \(f(y)\in \mathcal O _{C,y}'\) yields \(f(y)g(y-x)\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{C,y}'\) and therefore \(V_{g}f\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal O _{M,\xi }\).

     
Again these maps are continuous as they are compositions of continuous maps. \(\square \)

Proposition 6

Let \(g\in \mathcal O _{M}\). Then the short-time Fourier transform \(V_{g}\) maps
  1. 1.

    \(\mathcal S '\) into \(\mathcal O _{M,x}\widehat{\otimes }\mathcal S '_{\xi }\),

     
  2. 2.

    \(\mathcal O _{M}\) into \(\mathcal O _{M,x}\widehat{\otimes }\mathcal O _{C,\xi }'\),

     
  3. 3.

    \(\mathcal S \) into \(\mathcal O _{M,x}\widehat{\otimes }\mathcal S _{\xi }\) and

     
  4. 4.

    \(\mathcal O _{C}'\) into \(\mathcal O _{M,x}\widehat{\otimes }\mathcal S '_{\xi }\).

     
These linear maps are continuous.

Proof

\(g\in \mathcal O _{M}\) implies \(g(y-x)\in \mathcal O _{M,x}\widehat{\otimes }\mathcal O _{M,y}=\mathcal O _{M,x,y}\).
  1. 1.
    Multiplication of \(g(y-x)\) with \(f(y)\in \mathcal S '_{y}\) implies \(f(y)g(y-x)\in \mathcal O _{M,x}\widehat{\otimes }\mathcal S '_{y}\) and hence
    $$\begin{aligned} V_{g}f = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal O _{M,x}\widehat{\otimes }\mathcal S '_{\xi }. \end{aligned}$$
     
  2. 2.
    Multiplication of \(g(y-x)\) with \(f(y)\in \mathcal O _{M,y}\) implies \(f(y)g(y-x)\in \mathcal O _{M,x}\widehat{\otimes }\mathcal O _{M,y}\) and hence
    $$\begin{aligned} V_{g}f = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal O _{M,x}\widehat{\otimes }\mathcal O '_{C,\xi }. \end{aligned}$$
     
  3. 3.
    Multiplication of \(g(y-x)\) with \(f(y)\in \mathcal S _{y}\) implies \(f(y)g(y-x)\in \mathcal O _{M,x}\widehat{\otimes }\mathcal S _{y}\) and hence
    $$\begin{aligned} V_{g}f = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal O _{M,x}\widehat{\otimes }\mathcal S _{\xi }. \end{aligned}$$
     
  4. 4.
    Multiplication of \(g(y-x)\) with \(f(y)\in \mathcal O '_{C,y}\) implies \(f(y)g(y-x)\in \mathcal O _{M,x}\widehat{\otimes }\mathcal S _{y}'\) and hence
    $$\begin{aligned} V_{g}f = \left( \mathcal F _{y}\left( f(y)g(y-x)\right) \right) (\xi ) \in \mathcal O _{M,x}\widehat{\otimes }\mathcal S '_{\xi } \end{aligned}$$
     
Again these maps are continuous as they are compositions of continuous maps. \(\square \)

Remarks

  1. 1.
    Note that \(\mathcal O _{M,x}\widehat{\otimes }\mathcal O _{M,\xi }\) cannot be achieved as the target space in 4.: If we choose \(g(x) = \mathrm{e}^{\mathrm{i}x^2}\in \mathcal O _{M}\) and \(f(y) = \mathrm{e}^{-\mathrm{i}y^2}\in \mathcal O _{C}'\), we obtain
    $$\begin{aligned} f(y)g(y-x) = \mathrm{e}^{-\mathrm{i}y^{2}}\mathrm{e}^{\mathrm{i}y^{2} - 2\mathrm{i} xy + \mathrm{i}x^{2}} = \mathrm{e}^{\mathrm{i}x^{2}}\mathrm{e}^{-2\mathrm{i}xy} \end{aligned}$$
    and hence
    $$\begin{aligned} V_{g}f = \mathcal F _{y}(f(y)g(y-x)) = (2\pi )^{n} \mathrm{e}^{\mathrm{i}x^2} \delta (\xi +2x)\not \in \mathcal O _{M,x}\widehat{\otimes }\mathcal O _{M,\xi }. \end{aligned}$$
     
  2. 2.

    The ranges of the short-time Fourier transform for \(g\in \mathcal O _{C}\) and \(g\in \mathcal O _{M}'\) can be determined analogously to the Propositions 2, 4, 5 and 6.

     

Next, we fix \(h\in \mathcal S \) and investigate the mapping properties of the \(W_{h}\)-transform \(F\mapsto W_{h}F\).

Proposition 7

Given \(h\in \mathcal S \), the \(W_{h}\)-transform maps
  1. 1.

    \(\mathcal S _{x,\xi }\) into \(\mathcal S _{z}\),

     
  2. 2.

    \(\mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi }\) into \(\mathcal O _{C,z}'\) and

     
  3. 3.

    \(\mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{\xi }\) into \(\mathcal S _{z}\).

     
These linear maps are continuous.

Proof

The vector-valued partial Fourier transform yields
  1. 1.

    \(\mathcal F ^{-1}_{\xi }F \in \mathcal S _{x}\widehat{\otimes }\mathcal S _{z}\) if \(F\in \mathcal S _{x,\xi }\),

     
  2. 2.

    \(\mathcal F ^{-1}_{\xi }F \in \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}'\) if \(F\in \mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi }\) and

     
  3. 3.

    \(\mathcal F ^{-1}_{\xi }F\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{z}\) if \(F\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{\xi }\).

     
Furthermore,
$$\begin{aligned} h(x-z)\in \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z} \text { or } h(x-z)\in \mathcal S _{z}\widehat{\otimes }\mathcal O _{C,x}. \end{aligned}$$
The multiplication mappings
  1. 1.

    \((\mathcal S _{x}\widehat{\otimes }\mathcal S _{z})\times (\mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}) \rightarrow \mathcal S _{x}\widehat{\otimes }\mathcal S _{z}\),

     
  2. 2.

    \((\mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}') \times (\mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}) \rightarrow \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}'\) and

     
  3. 3.

    \((\mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{z}) \times (\mathcal O _{C,x}\widehat{\otimes }\mathcal S _{z}) \rightarrow \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{z}\)

     
are well-defined and hypocontinuous by Proposition 25 in [16, p. 120] and hence
  1. 1.

    \((\mathcal F ^{-1}F)(x,z)h(x-z)\in \mathcal S _{x}\widehat{\otimes }\mathcal S _{z}\),

     
  2. 2.

    \((\mathcal F ^{-1}F)(x,z)h(x-z)\in \mathcal S _{x}\widehat{\otimes }\mathcal O _{C,z}'\) and

     
  3. 3.

    \((\mathcal F ^{-1}F)(x,z)h(x-z)\in \mathcal O _{C,x}'\widehat{\otimes }\mathcal S _{z}\).

     
Vector-valued integration with respect to \(x\) (see [15, p. 129]) furnishes
  1. 1.

    \(W_{h}F \in \mathcal S _{z}\) for \(F\in \mathcal S _{x,\xi }\)

     
  2. 2.

    \(W_{h}F \in \mathcal O _{C,z}'\) for \(F\in \mathcal S _{x} \widehat{\otimes }\mathcal O _{M,\xi }\) and

     
  3. 3.

    \(W_{h}F \in \mathcal S _{z}\) for \(F\in \mathcal O _{C,x}' \widehat{\otimes }\mathcal S _{\xi }\).

     

Remark

1. coincides, in essence, with Proposition 2.2 in [5, p. 209].

6 Inversion

Proposition 8

If \(g,h\in \mathcal S \) then it holds
$$\begin{aligned} W_{h}\circ V_{g} = \langle g, h\rangle \,\mathrm{id} = \int \limits _\mathbb{R ^{n}} g(x)h(x)\,\mathrm{d}x \cdot \mathrm{id} \end{aligned}$$
on \(\mathcal S \) and \(\mathcal O _{C}'\).

Proof

We have by Proposition 4
$$\begin{aligned} V_{g}:\mathcal S \rightarrow \mathcal S _{x,\xi } \text { and } V_{g}:\mathcal O _{C}'\rightarrow \mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi } \end{aligned}$$
and by Proposition 7
$$\begin{aligned} W_{h}:\mathcal S _{x,\xi }\rightarrow \mathcal S \text { and } W_{h}:\mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi } \rightarrow \mathcal O _{C}'. \end{aligned}$$
Hence \(W_{h}\circ V_{g}\) is well-defined. The algebraic part follows by observing
$$\begin{aligned} (\mathcal F ^{-1}(V_{g}f))(x,z) = \left( \mathcal F ^{-1}_{\xi }\mathcal F _{y} (f(y)g(y-x))\right) (x,z) = f(x)g(x-z), \end{aligned}$$
and that
$$\begin{aligned} \langle 1_{x}, (\mathcal F ^{-1}_{\xi }(V_{g}f))(x,z)h(z-x) \rangle = \langle 1_{x}, f(z)g(z-x)h(z-x)\rangle = f(z) \langle g,h\rangle . \end{aligned}$$
Since \(g,h\in \mathcal S \) the bracket \(\langle \cdot ,\cdot \rangle \) simply reduces to the integral \(\langle g, h\rangle = \int _\mathbb{R ^{n}} g(x)h(x)\,\mathrm{d}x\).

Remark 5

Lemma 1.4 (ii) in [5, p. 207] is identical with Proposition 8 for \(\mathcal S \).

More generally, we have the following proposition:

Proposition 8

If \(g\in \mathcal S '\) and \(h\in \mathcal S \) then it holds
$$\begin{aligned} W_{h}\circ V_{g} = \langle g, h\rangle \,\mathrm{id} \end{aligned}$$
on \(\mathcal S '\).

Proof

\(W_{h}V_{g}f\) is well defined for \((f,g)\in \mathcal S '\times \mathcal S '\) and for \(h\in \mathcal S \) by the Propositions 2 and 3. The algebraic part follows as in the proof of Proposition 8. \(\square \)

7 Characterizations of \(\mathcal O _{C}'\) and \(\mathcal O _{M}\)

Lemma 1

$$\begin{aligned} \mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi } = \Big \{\varphi \in \mathcal C ^{\infty }(\mathbb R ^{2n}_{x,\xi }); \forall k\in \mathbb N _{0} \forall \alpha \in \mathbb N _{0}^{2n} \exists m\in \mathbb N _{0}\exists C>0:|(\partial ^{\alpha }\varphi )(x,\xi )|\le C \tfrac{(1+|\xi |^{2})^{m/2}}{(1+|x|^{2})^{k/2}}\Big \}. \end{aligned}$$

Proof

We obtain from Proposition 13 in [11, p. 113]
$$\begin{aligned} \mathcal S _{x}\widehat{\otimes }\,\mathcal O _{M,\xi } = \left\{ \varphi \in \mathcal C ^{\infty }(\mathbb R ^{2n}_{x,\xi }); \forall k\in \mathbb N _{0} \forall \alpha \in \mathbb N _{0}^{2n} \forall \psi \in \mathcal S _{\xi }:(1+|x|^{2})^{k/2}\psi (\xi )\partial ^{\alpha }\varphi \in \mathcal C _{0}\right\} . \end{aligned}$$
The description
$$\begin{aligned} \begin{aligned} \mathcal O _{M}&= \left\{ \varphi \in \mathcal C ^{\infty }; \forall \psi \in \mathcal S :\varphi \cdot \psi \in \mathcal S \right\} \\&= \left\{ \varphi \in \mathcal C ^{\infty }; \forall \alpha \in \mathbb N _{0}^{n}\exists m\in \mathbb N _{0}:(1+|\xi |^{2})^{-m/2} \partial ^{\alpha }\varphi \in \mathcal C _{0}\right\} \end{aligned} \end{aligned}$$
finishes the proof. \(\square \)

Proposition 10

(Characterization of \(\mathcal O _{C}'\)) Let \(g\in \mathcal S \), \(g\ne 0\). Then for \(f\in \mathcal S '\) the following assertions are equivalent:
  1. 1.

    \(f\in \mathcal O _{C}'\),

     
  2. 2.

    \(V_{g}f \in \mathcal S _{x}\widehat{\otimes }\mathcal O _{M,\xi }\) and

     
  3. 3.
    \(V_{g}f\) is continuous and
    $$\begin{aligned} \forall k\in \mathbb N _{0} \,\exists C>0 \,\exists m\in \mathbb N _{0}:|(V_{g}f)(x,\xi )| \le C (1+|x|^{2})^{-k/2} (1+|\xi |^{2})^{m/2} \end{aligned}$$
    for all \((x,\xi )\in \mathbb R ^{2n}\).
     

Proof

Implication 1.\(\Rightarrow \) 2. follows from Proposition 4.

Implication 2.\(\Rightarrow \) 3. follows from the Lemma by taking \(\alpha =0\).

Finally, let us show the implication 3.\(\Rightarrow \) 1.:

The inequality in 3. implies
$$\begin{aligned} \forall k\in \mathbb N _{0}:\left( \mathcal F _{\xi }(V_{g}f)\right) (x,z) \in \left( (1+|x|^{2})^{-k/2} \mathcal C _{0,x}\right) \widehat{\otimes }\mathcal S '_{z}. \end{aligned}$$
As \(\lim _{\leftarrow k} \left( (1+|x|^{2})^{-k/2}\mathcal C _{0,x}\right) \subset \mathcal O _{C}'\) by the definition of the space \(\mathcal O _{C}'\), it holds that \(\mathcal F _{\xi }V_{g}f \in \mathcal O _{C}'\widehat{\otimes }\mathcal S '\). Vector-valued multiplication by Proposition 25 in [16, p. 129] yields
$$\begin{aligned} \left( \mathcal F _{\xi }(V_{g}f)\right) (x,z) g(x-z) \in \mathcal O _{C, x}' \widehat{\otimes } \mathcal O _{C, z}'. \end{aligned}$$
Therefore,
$$\begin{aligned} \langle 1_{x}, \left( \mathcal F _{\xi }(V_{g}f)\right) (x,z)g(x-z)\rangle \in \mathcal O _{C,z}', \end{aligned}$$
i.e., \(W_{g}V_{g}f\in \mathcal O _{C}'\). Hence \(f\in \mathcal O _{C}'\) by Proposition 8. \(\square \)

Proposition 11

(Characterization of \(\mathcal O _{M}\)) Let \(g\in \mathcal S \), \(g\ne 0\). Then for \(f\in \mathcal S '\) the following assertions are equivalent.
  1. 1.

    \(f\in \mathcal O _{M}\),

     
  2. 2.

    \(V_{g}f \in \mathcal O _{C,x}\widehat{\otimes }\mathcal S _{\xi }\) and

     
  3. 3.
    \(V_{g}f\) is continuous and
    $$\begin{aligned} \forall k\in \mathbb N _{0}\,\exists m\in \mathbb N _{0}\,\exists C>0:|(V_{g}f)(x,\xi )| \le C (1+|x|^{2})^{m/2} (1+|\xi |^{2})^{-k/2} \end{aligned}$$
    for all \((x,\xi )\in \mathbb R ^{2n}\).
     

Proof

The implication 1.\(\Rightarrow \) 2. follows from Proposition 4.

We now show the implication 2.\(\Rightarrow \) 1.: The identity
$$\begin{aligned} (V_\mathcal{F g}(\mathcal F f))(x,\xi ) = \mathrm{e}^{-\mathrm{i}x\xi } (V_{g}f)(-\xi ,x) \end{aligned}$$
(Lemma 1.1 in [6, p. 2]) implies \(V_\mathcal{F g}(\mathcal F f) \in \mathcal S _{x}\widehat{\otimes } \mathcal O _{M,\xi }\) since
$$\begin{aligned} \begin{aligned} V_{g}f(-\xi ,x)&\in \mathcal O _{C,\xi }\widehat{\otimes }\mathcal S _{x}&\subset \mathcal O _{C,\xi }\widehat{\otimes }\mathcal O _{M,x} \\ \mathrm{e}^{-\mathrm{i}\xi x}&\in \mathcal O _{C,\xi }\widehat{\otimes }\mathcal O _{M,x} \end{aligned} \end{aligned}$$
and the multiplication
$$\begin{aligned} (\mathcal O _{C,\xi }\widehat{\otimes }\mathcal O _{M,x}) \times (\mathcal O _{C,\xi }\widehat{\otimes }\mathcal O _{M,x}) \rightarrow (\mathcal O _{C,\xi }\widehat{\otimes }\mathcal O _{M,x}) \end{aligned}$$
is well-defined by Proposition 25 in [16, p. 120] as the multiplication on \(\mathcal O _{M}\) is continuous. Thus, \(\mathcal F f\in \mathcal O _{C}'\) and \(f\in \mathcal O _{M}\).
The equivalence of 1. and 3. follows from the identity
$$\begin{aligned} V_\mathcal{F g}(\mathcal F f) \in \mathcal S _{x}\widehat{\otimes } \mathcal O _{M,\xi } \end{aligned}$$
referred to above and Proposition 10.

Acknowledgments

We are indebted to Prof. G. Zimmermann who directed the second author’s attention to the mapping properties of the short-time Fourier transform and to their inversion. Preliminary versions of Propositions 4, 4., 5, 2., 7, 2., 10, 1., 2., and 11, 1., 2 are due to him.

Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Institut für MathematikUniversität Innsbruck InnsbruckAustria

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