# A Sequential Approach to Mild Distributions

## Abstract

**:**

**ECmiCS**). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra $\left(\right)$ can be used to establish this natural identification.

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

- First one considers all possible Cauchy-sequences;
- Then one forms equivalence classes of Cauchy-sequences, which are the objects of the new space;
- The distance of the new elements can be defined in a natural way, as the limit of the distances in the all possible Cauchy-Sequences (CS) generating the equivalence classes, and of course this does not depend on the representative of the class;
- Any element of the original space defines a constant sequence which is a CS, for trivial reasons. The claimed natural embedding mapping assigns simply every element in the given space the corresponding constant equivalence class;
- Then one verifies that this natural embedding of the original metric space into the new one is isometric, so that henceforth the copy with the new object arising from the original objects can be identified with the elements of the original space. (For the case of $\mathbb{Q}\hookrightarrow \mathbb{R}$ we recognize the rational numbers among all the infinite decimal expressons as those which are periodic, for a suitably chosen period.)
- If further structure is available, i.e., if we have a normed space or a normed algebra, the resulting complete object is then a Banach space or a Banach algebra.

## 2. The Short-Time Fourier Transform STFT

**Definition**

**3.**

`www.gaborator.com`, where you can even upload your own piece of music (in the standard WAV-format for audio files).

**Lemma**

**1.**

**Proof.**

## 3. The Usual Approach to $\left(\right)$

**Definition**

**4.**

**Theorem**

**1.**

- For $f\in {\mathit{L}}^{2}\left({\mathbb{R}}^{d}\right)$ one has ${V}_{g}\left(f\right)\in {\mathit{L}}^{2}\left({\mathbb{R}}^{2d}\right)$ and$$\parallel {V}_{g}{\left(f\right)\parallel}_{{\mathit{L}}^{2}\left({\mathbb{R}}^{2d}\right)}={\parallel g\parallel}_{2}{\parallel f\parallel}_{2}\phantom{\rule{1.em}{0ex}}f,g\in {\mathit{L}}^{2}\left({\mathbb{R}}^{d}\right).$$
- A function $f\in {\mathit{L}}^{2}\left({\mathbb{R}}^{d}\right)$ belongs to ${\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$ by definition, if ${V}_{g}\left(f\right)\in {\mathit{L}}^{1}\left({\mathbb{R}}^{2d}\right)$, and$${\parallel f\parallel}_{{\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}}:={\int}_{{\mathbb{R}}^{d}\times \widehat{\mathbb{R}}{}^{d}}\left|{V}_{g}\left(f\right)(t,s)\right|dtds.$$Any function $f\in {\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$ is continuous, bounded and absolutely Riemann-integrable;
- A tempered distribution $\sigma \in {\mathcal{S}}^{\prime}\left({\mathbb{R}}^{d}\right)$ belongs to ${\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime}\left({\mathbb{R}}^{d}\right)$ if and only if$$\parallel {V}_{g}{\left(\sigma \right)\parallel}_{\infty}=\underset{(t,s)\in {\mathbb{R}}^{d}\times \widehat{\mathbb{R}}{}^{d}}{sup}\left|{V}_{g}\left(\sigma \right)(t,s)\right|<\infty .$$Moreover, this expression defines an equivalent norm on $({\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime}\left({\mathbb{R}}^{d}\right),\parallel \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\parallel}_{{\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime}})$.

**Definition**

**5.**

- Dirac sequences, obtained by compression of an integrable function;$${{w}^{*}\phantom{\rule{-1.66672pt}{0ex}}-lim\phantom{\rule{0.166667em}{0ex}}}_{\rho \to 0}\phantom{\rule{0.166667em}{0ex}}{St}_{\rho}\left(g\right)=\left(\right)open="("\; close=")">{\int}_{\mathbb{R}}g\left(x\right)dx$$
- Riemannian sums converging to the integral, e.g., for $f\in {\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$:$$\underset{\alpha \to 0}{lim}\langle {\alpha}^{d}\sum _{k\in {\mathbb{Z}}^{d}}{\delta}_{\alpha k},f\rangle =\underset{\alpha \to 0}{lim}\phantom{\rule{0.166667em}{0ex}}{\alpha}^{d}\sum _{k\in {\mathbb{Z}}^{d}}f\left(\alpha k\right)={\int}_{{\mathbb{R}}^{d}}f\left(x\right)dx=\langle \mathbf{1},f\rangle .$$
- For any $f\in {\mathit{L}}^{1}\phantom{\rule{-1.66672pt}{0ex}}\left({\mathbb{R}}^{d}\right)$ the periodic versions of that function converge to the original function (this is often used to motivate the form of the Fourier integral for non-periodic functions):$${{w}^{*}\phantom{\rule{-1.66672pt}{0ex}}-lim\phantom{\rule{0.166667em}{0ex}}}_{p\to \infty}\sum _{k\in {\mathbb{Z}}^{d}}{T}_{pk}f=f.$$

## 4. Mild Cauchy Sequences

**Definition**

**6.**

**Remark**

**1.**

**Definition**

**7.**

**Definition**

**8.**

**ECmiCS**for an

**E**quivalence

**C**lass if

**mi**ld

**C**auchy

**S**equences. These ECmiCS constitute the new (enlarged) vector space of objects, in fact a normed space with respect to the $CS$-norm. For the rest of this note it will be convenient to use the symbol $\mathbf{F}$ for such an equivalence class, hence ${\parallel \mathbf{F}\parallel}_{CS}$ describes the norm of an ECmiCS.

**Lemma**

**2.**

## 5. The Functional Analytic Viewpoint

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

**Proposition**

**1.**

- 1.
- $\mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{1})\left({\mathbb{R}}^{d}\right)\ast \mathit{W}(\mathcal{F}{\mathit{L}}^{\infty},{\ell}^{\infty})\left({\mathbb{R}}^{d}\right)\subset \mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{\infty})\left({\mathbb{R}}^{d}\right)$;
- 2.
- $\mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{\infty})\left({\mathbb{R}}^{d}\right)\xb7\mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{1})\left({\mathbb{R}}^{d}\right)\subset \mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{1})\left({\mathbb{R}}^{d}\right)$;
- 3.
- $\mathit{W}(\mathcal{F}\phantom{\rule{-1.66672pt}{0ex}}{\mathit{L}}^{1},{\ell}^{1})\left({\mathbb{R}}^{d}\right)\xb7\mathit{W}(\mathcal{F}{\mathit{L}}^{\infty},{\ell}^{\infty})\left({\mathbb{R}}^{d}\right)\subset \mathit{W}(\mathcal{F}{\mathit{L}}^{\infty},{\ell}^{1})\left({\mathbb{R}}^{d}\right)$.

#### Connections to Gabor Analysis

**Lemma**

**4.**

- $f\in {\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$ if and only if ${V}_{g}{\left(f\right)|}_{\mathsf{\Lambda}}\in {\ell}^{1}\left(\mathsf{\Lambda}\right)$;
- $f\in {\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$ if and only if ${V}_{\tilde{g}}{\left(f\right)|}_{\mathsf{\Lambda}}\in {\ell}^{1}\left(\mathsf{\Lambda}\right)$;
- $f\in {\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left({\mathbb{R}}^{d}\right)$ if and only if f has a representation$$f=\sum _{\lambda \in \mathsf{\Lambda}}{c}_{\lambda}{g}_{\lambda},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}some\phantom{\rule{0.166667em}{0ex}}sequence{\left({c}_{\lambda}\right)}_{\lambda \in \mathsf{\Lambda}}\in {\ell}^{1}\left(\mathsf{\Lambda}\right).$$

**Proposition**

**2.**

## 6. Natural Extension of Operators

## 7. Alternative Starting Points

**ECmiCS**) one may ask whether, by the same form of “completion”, other starting points could be chosen, in order to get the same space, but derive more easily additional properties (like Fourier invariance). Alternatively, one might ask, whether certain choices which are closer to applications (like the use of periodic, discrete signals as point of departure) yield other objects. Fortunately this will not be the case. The discussion of these two points makes up the current section.

**Proposition**

**3.**

**ECmiCS**arising from $(\mathit{B},\phantom{\rule{0.166667em}{0ex}}\parallel \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\parallel}_{\mathit{B}})$ can be naturally identified with the space of

**ECmiCS**arising from $\left(\right)$.

**Proof.**

**ECmiCS**allowing only the representatives from $\mathit{B}$ describes an equivalent norm, i.e., to verify that the restriction/modification in the set of representatives does not have an effect on the corresponding infimum’s norm. This is quite plausible because the norm estimate only uses ${\parallel {V}_{{g}_{0}}\sigma \parallel}_{\infty}$ resp. ${\mathit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime}$-norms.

**ECmiCS**are equivalent. □

## 8. References and History

**Lemma**

**5.**

- For every mild CS $\left({h}_{n}\right)$ in ${\mathit{C}}_{\phantom{\rule{-1.66672pt}{0ex}}b}\left({\mathbb{R}}^{d}\right)$ there exists an equivalent sequence $\left({f}_{n}\right)$ in $\mathcal{S}\left({\mathbb{R}}^{d}\right)$. In other words, every equivalence class defining a mild distribution can be constituted with the help of mild Cauchy sequences from $\mathcal{S}\left({\mathbb{R}}^{d}\right)$.
- Any mild CS in $\mathcal{S}\left({\mathbb{R}}^{d}\right)$ is also a regular sequence, which implies that any mild distribution (viewed as ECmiCS) also defines a tempered distribution (in the sense of Lighthill).
- A regular sequence of test functions defines a mild distribution if and only if$$\underset{n\ge 1}{sup}{\parallel {V}_{g}\left({f}_{n}\right)\parallel}_{\infty}<\infty .$$

**Proof.**

**Remark**

**3.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lighthill, M.J. Introduction to Fourier Analysis and Generalised Functions. (Students’ Edition); Cambridge University Press: Cambridge, UK, 1962. [Google Scholar]
- Bracewell, R.N. The Fourier Transform and Its Applications, 2nd ed.; McGraw-Hill Book Company: Auckland, New Zealand, 1983. [Google Scholar]
- Gröchenig, K. Foundations of Time-Frequency Analysis; Birkhäuser: Boston, MA, USA, 2001. [Google Scholar]
- Feichtinger, G.H. On a new Segal algebra. Monatsh. Math.
**1981**, 92, 269–289. [Google Scholar] [CrossRef] - Jakobsen, M.S. On a (no longer) New Segal Algebra: A Review of the Feichtinger Algebra. J. Fourier Anal. Appl.
**2018**, 24, 1579–1660. [Google Scholar] [CrossRef][Green Version] - Feichtinger, H.G. A novel mathematical approach to the theory of translation invariant linear systems. In Novel Methods in Harmonic Analysis with Applications to Numerical Analysis and Data Processing; Springer Science + Business Media: Berlin, Germany, 2016; pp. 1–32. [Google Scholar]
- Cordero, E.; Feichtinger, H.G.; Luef, F. Banach Gelfand triples for Gabor analysis. In Pseudo-Differential Operators; Lecture Notes in Mathematics; Springer: Berlin, Germany, 2008. [Google Scholar]
- Feichtinger, H.G. Banach convolution algebras of Wiener type. In Proceedings on Functions, Series, Operators, Budapest 1980; North-Holland, Colloq. Math. Soc. Janos Bolyai: Amsterdam, The Netherlands, 1983; Volume 35, pp. 509–524. [Google Scholar]
- Feichtinger, H.G.; Gröbner, P. Banach spaces of distributions defined by decomposition methods. I. Math. Nachr.
**1985**, 123, 97–120. [Google Scholar] [CrossRef] - Reiter, H. Classical Harmonic Analysis and Locally Compact Groups; Clarendon Press: Oxford, UK, 1968. [Google Scholar]
- Reiter, H. L
^{1}-Algebras and Segal Algebras; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1971. [Google Scholar] - Feichtinger, H.G.; Strohmer, T. Gabor Analysis and Algorithms. Theory and Applications; Birkhäuser: Boston, MA, USA, 1998. [Google Scholar]
- Feichtinger, H.G.; Strohmer, T. Advances in Gabor Analysis; Birkhäuser: Basel, Switzerland, 2003. [Google Scholar]
- Gröchenig, K.; Leinert, M. Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc.
**2004**, 17, 1–18. [Google Scholar] [CrossRef] - Gröchenig, K.; Leinert, M. Symmetry and inverse-closedness of matrix algebras and symbolic calculus for infinite matrices. Trans. Am. Math. Soc.
**2006**, 358, 2695–2711. [Google Scholar] [CrossRef] - Gröchenig, K. An uncertainty principle related to the Poisson summation formula. Studia Math.
**1996**, 121, 87–104. [Google Scholar] [CrossRef] - Feichtinger, H.G.; Weisz, F. The Segal algebra S
_{0}(R^{d}) and norm summability of Fourier series and Fourier transforms. Monatsh. Math.**2006**, 148, 333–349. [Google Scholar] [CrossRef] - Lyubarskii, Y.I. Frames in the Bargmann space of entire functions. In Entire and Subharmonic Functions; American Mathematical Society (AMS): Providence, RI, USA, 1992; pp. 167–180. [Google Scholar]
- Seip, K. Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J. Reine Angew. Math.
**1992**, 429, 91–106. [Google Scholar] [CrossRef] - Gröchenig, K.; Stöckler, J. Gabor frames and totally positive functions. Duke Math. J.
**2013**, 162, 1003–1031. [Google Scholar] [CrossRef][Green Version] - Jones, D.S. Generalised Functions; McGraw-Hill Book Co.: New York, NY, USA, 1966. [Google Scholar]
- Antosik, P.; Mikusinski, J.; Sikorski, R. Theory of Distributions. The Sequential Approach; Elsevier Scientific Publishing Company: Amsterdam, The Netherlands, 1973. [Google Scholar]
- Bargetz, C.; Ortner, N. Characterization of L. Schwartz’ convolutor and multiplier spaces ${\mathcal{O}}_{C}^{\prime}$ and ${\mathcal{O}}_{M}$ by the short-time Fourier transform. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math.
**2014**, 108, 833–847. [Google Scholar] [CrossRef] - Poguntke, D. Gewisse Segalsche Algebren auf lokalkompakten Gruppen. Arch. Math. (Basel)
**1980**, 33, 454–460. [Google Scholar] [CrossRef][Green Version] - Feichtinger, H.G. Classical Fourier Analysis via mild distributions. MESA Non-Linear Studies
**2019**, 26, 783–804. [Google Scholar] - Feichtinger, H.G.; Jakobsen, M.S. Distribution theory by Riemann integrals. In Mathematical Modelling, Optimization, Analytic and Numerical Solutions; Springer: Singapore, 2020; pp. 33–76. [Google Scholar]
- Feichtinger, H.G. Banach Gelfand triples for applications in physics and engineering. Am. Inst. Phys.
**2009**, 1146, 189–228. [Google Scholar] - Feichtinger, H.G. Ingredients for Applied Fourier Analysis “Sharda Conference Feb. 2018”; Taylor and Francis: Abingdon, UK, 2019; pp. 1–22. [Google Scholar]
- Schwartz, L. Théorie des Distributions. (Distribution Theory). Nouveau Tirage; Paris: Hermann, MO, USA, 1957; Volume 1, 420p. [Google Scholar]
- Friedlander, F.G. Introduction to the Theory of Distributions, 2nd ed.; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Debnath, L. Developments of the theory of generalized functions or distributions—A vision of Paul Dirac. Analysis
**2013**, 33, 57–99. [Google Scholar] [CrossRef]

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Feichtinger, H.G.
A Sequential Approach to Mild Distributions. *Axioms* **2020**, *9*, 25.
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Feichtinger HG.
A Sequential Approach to Mild Distributions. *Axioms*. 2020; 9(1):25.
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2020. "A Sequential Approach to Mild Distributions" *Axioms* 9, no. 1: 25.
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