A Sequential Approach to Mild Distributions

Abstract

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short 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 S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

A Characterization of Multipliers of the Herz Algebra

“…One of the basic results of our study is the observation (already formulated as Theorem C2 in [30] and then published, among others, as Theorem 8.4 in [9]), showing that any multiplier from S 0 (R d ) to S 0 (R d ) is a convolution operator by some σ ∈ S 0 (R d ): (22) In compact form, it is given in the pointwise sense by…”

“…Remark 2. In the context of (mild or) tempered distributions (see [9] and or [10] respectively [11]), one can form PM(R d ) := F −1 (L ∞ (R d )) and call this (by transfer of the norm) the space of pseudomeasures. This space plays an important role for spectral analysis (see the book [12] by J. Benedetto).…”

A universal identifier for communication channels

“…Such a setting provides us with a way to work with norms instead of seminorms. See [22][23][24][25][26] for more information on the Feichtinger algebra and the Banach Gelfand triple…”

On the Reversibility of Discretization

“…It explains their extraordinary role as (1) test functions for tempered distributions, (2) test functions in quantum mechanics [63], p. 12 and [164], pp. 317-318, (3) window functions in the Short Time Fourier Transform (STFT) [41,51,56,165,166] and (4) their validity-satisfying role in Poisson's Summation Formula [20,84,142,165].…”