A Hilbert space approach to distributions

1990 ◽  
Vol 115 (3-4) ◽  
pp. 275-288 ◽  
Author(s):  
Rainer H. Picard
Keyword(s):  
Fourier Transform ◽  
Hilbert Spaces ◽  
Sobolev Type ◽  
Sequential Approach ◽  
Tempered Distributions ◽  
Schwartz Distributions ◽  
The Fourier Transform ◽  
Image Position ◽  
Insight Into ◽  
Space Approach

SynopsisA compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant withrespect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.

scholarly journals Hilbert spaces of tempered distributions, Hermite expansions and sequence spaces

1991 ◽  
Vol 34 (2) ◽  
pp. 271-293
Author(s):  
Rainer H. Picard
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Hermite Functions ◽  
Dimensional Case ◽  
Sobolev Type ◽  
Tempered Distributions ◽  
The One ◽  
Image Position ◽  
Hermite Expansions

Although it is well-known that tempered distributions on ℝn can be expanded into series of Herrnite functions, it does not seem to be known, however, that expansions of this type are accessible through the elementary concept of orthonorma! expansions in Hilbert space. This approach is developed here complementing previous work on a Hilbert space approach to distributions. The basis of the development is the observation that the Hermite functions are a complete orthogonal set in each space of a certain scale of Sobolev type Hilbert spaces associated with the family of differential operators defined byHere Ф denotes a smooth function with compact support. The setting is first developed in the one-dimensional case. By use of the usual multi-index notation this can be extended to the higher-dimensional case. As applications various imbedding results are derived. The paper concludes with a characterization of tempered distributions by convergent Hermite expansions.

On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

1988 ◽  
Vol 108 (3-4) ◽  
pp. 201-228 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansions ◽  
Spectral Measure ◽  
Liouville Problem ◽  
Integral Transforms ◽  
Tempered Distributions ◽  
Sturm Liouville ◽  
The Fourier Transform ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known thatwhere the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals On dilation equations and the Hölder continuity of the de Rham functions

1994 ◽  
Vol 36 (3) ◽  
pp. 309-311
Author(s):  
Yibiao Pan
Keyword(s):  
Fourier Transform ◽  
Approximation Method ◽  
Hölder Continuity ◽  
Simple Approximation ◽  
Holder Continuity ◽  
Integrable Solution ◽  
De Rham ◽  
The Fourier Transform ◽  
Image Position ◽  
Dilation Equations

AbstractWe use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.1. Consider the following dilatation equationwhere |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfywhere is the Fourier transform of f, andwhich immediately leads to

scholarly journals Strong boundedness of analytic functions in tubes

1979 ◽  
Vol 2 (1) ◽  
pp. 15-28
Author(s):  
Richard D. Carmichael
Keyword(s):  
Fourier Transform ◽  
Analytic Function ◽  
Analytic Functions ◽  
Weak Topology ◽  
Complex Space ◽  
Direct Proof ◽  
Tempered Distributions ◽  
Boundedness Property ◽  
Tube Domains ◽  
The Fourier Transform

Certain classes of analytic functions in tube domainsTC=ℝn+iCinn-dimensional complex space, whereCis an open connected cone inℝn, are studied. We show that the functions have a boundedness property in the strong topology of the space of tempered distributionsg′. We further give a direct proof that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology ofg′.

Fourier restriction theorems for curves with affine and Euclidean arclengths

1985 ◽  
Vol 97 (1) ◽  
pp. 111-125 ◽  
Author(s):  
S. W. Drury ◽  
B. P. Marshall
Keyword(s):  
Fourier Transform ◽  
Smooth Manifold ◽  
Fourier Restriction ◽  
Survey Article ◽  
Restriction Theorems ◽  
Schwartz Class ◽  
The Fourier Transform ◽  
Image Position

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequalityfor every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

scholarly journals A Sequential Approach to Mild Distributions

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 25 ◽  
Author(s):  
Hans G. Feichtinger
Keyword(s):  
Fourier Transform ◽  
Number System ◽  
Continuous Functions ◽  
Sequential Approach ◽  
Tempered Distributions ◽  
Segal Algebra ◽  
Gabor Analysis ◽  
Plancherel’S Theorem ◽  
Lca Groups ◽  
Gelfand Triple

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.

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