Search Results (6)

The Banach Gelfand Triple $(S_0,L^2,S_0^{\prime })$ consists of the Feichtinger algebra $S_0\left(R^d\right)$ as a space of test functions, the dual space $S_0^{\prime }\left(R^d\right)$, known as the space of mild distributions, and the intermediate Hilbert space $L^2\left(R^d\right)$. This Gelfand Triple is very useful for the description of mathematical problems in the area of time-frequency analysis, but also for classical Fourier analysis and engineering applications. Because the involved spaces are Banach spaces, we speak of a Banach Gelfand Triple, in contrast to the widespread concept of rigged Hilbert spaces, which usually involve nuclear Frechet spaces. Still, both concepts serve very similar purposes. Based on the manifold properties of $S_0\left(R^d\right)$, it has found applications in the derivation of mathematical statements related to Gabor Analysis but also in providing an alternative and more lucid description of classical results, such as the Shannon sampling theory, with a potential to renew the way how Fourier and time-frequency analysis, but also signal processing courses for engineers (or physicists and mathematicians) could be taught in the future. In the present study, we will demonstrate that one could choose a relatively large variety of similar Banach Gelfand Triples, even if one wants to include key properties such as Fourier invariance (an extended version of Plancherel’s Theorem). Some of them appeared naturally in the literature. It turns out, that $S_0\left(R^d\right)$ is the smallest member of this family. Consequently $S_0^{\prime }\left(R^d\right)$ is the largest dual space among all these spaces, which may be one of the reasons for its universal usefulness. This article provides a study of the basic properties following from a short list of relatively simple assumptions and gives a list of non-trivial examples satisfying these basic axioms. Full article

This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner–Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs. Full article

Let N be the number operator in the space $\mathcal{H}$ of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space $\mathcal{G}$, which is also a dense linear subspace of $\mathcal{H}$, and then by taking its dual ${\mathcal{G}}^{*}$, we obtain a real Gel’fand triple $\mathcal{G}\subset \mathcal{H}\subset {\mathcal{G}}^{*}$. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure ${\gamma }_N$ on ${\mathcal{G}}^{*}$ such that its covariance operator coincides with N. We examine the properties of ${\gamma }_N$, and, among others, we show that ${\gamma }_N$ can be represented as a convolution of a sequence of Borel probability measures on ${\mathcal{G}}^{*}$. Some other results are also obtained. Full article

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$, the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over ${\mathbb{R}}^d$. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space$(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ on G, such as $({\mathit{L}}^p\left(G\right),{\parallel ·\parallel }_p)$, for $1\le p<\infty$, or the Fourier–Stieltjes algebra $\mathcal{F}\mathit{M}\left(G\right)$, and in particular any Segal algebra is a Banach convolution module over $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$ in a natural way. Via the Haar measure we can then identify $\left({\mathit{L}}^1\left(G\right),{\parallel ·\parallel }_1\right)$ with the closure (of the embedded version) of ${\mathit{C}}_c\left(G\right)$, the space of continuous functions with compact support, in $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$, and show that these homogeneous Banach spaces are essential${\mathit{L}}^1\left(G\right)$-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple$({\mathit{S}}_0,{\mathit{L}}^2,{\mathit{S}}_0^{\prime })\left(G\right)$, based on the Segal algebra ${\mathit{S}}_0\left(G\right)$. This direction will be pursued in subsequent papers. Full article

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give an estimation for the value of ${\lambda }_{*}$ such that, for the parameter $\lambda \in [{\lambda }_{*},{\lambda }^{*}]$, there exist exactly three positive solutions. Numerical examples are presented to illustrate various cases. The results complement previous work on this problem. Full article

The Banach Gelfand Triple $({\mathit{S}}_0,{\mathit{L}}^2,{\mathit{S}}_0^{\prime })\left({\mathbb{R}}^d\right)$ consists of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ , a very specific Segal algebra as algebra of test functions, the Hilbert space $\left({\mathit{L}}^2\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_2\right)$ and the dual space ${\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$ , 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 $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ and hence $({\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0^{\prime }})$ , 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 $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ can be used to establish this natural identification. Full article