On Radial Functions and Distributions and Their Fourier Transforms

We give formulas relating the Fourier transform of a radial function in $\mathbb{R}^{n}$ and the Fourier transform of the same function in $\mathbb{R}^{n+1}$ , completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of $\mathbb{R}^{n}$ and $\mathbb{R}^{n+2}$ was considered.     

Fourier Transforms of Distributions and Hausdorff Measures

Consider a distribution whose support has Hausdorff \(h\) -measure zero. How fast can its Fourier coefficients decay to zero? If \(h\) is close to linear, we can improve on known results by a factor of about \((\log n)^{-1/2}\) .     

On Namias's Fractional Fourier Transforms

The objective of this paper is to make rigorous a formal study(Namias, 1980) of fractional powers for the Fourier transform.To make the theory unambiguous, it is found necessary to modifyNamias's fractional operators. Some theorems are then provedfor the modified operators and an operational calculus is developed.Finally, an application to an ordinary differential equationis considered.     

On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes - The question of the representability of a continuous function on $$mathbb R^d$$ in the form of the Fourier integral of a finite Borel complex-valued measure on $$mathbb R^d$$...     

On Radial and Conical Fourier Multipliers

We investigate connections between radial Fourier multipliers on ℝ ^d and certain conical Fourier multipliers on ℝ ^d+1. As an application we obtain a new weak type endpoint bound for the Bochner–Riesz multipliers associated with the light cone in ℝ ^d+1, where d≥4, and results on characterizations of L ^p →L ^p,ν inequalities for convolutions with radial kernels.     

On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Journal of Fourier Analysis and Applications - Let f be an element of the subspace $$(L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)$$ ( $$1\le q \le \alpha \le p \le 2 $$ ) of the Wiener amalgam space...     

On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G_α(P±i0, m, n) [Eq. (I.4.1)] and H_α(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H_zk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n-dimensional ultrahyperbolic operator iterated k times.     

On Mutually Inverse Transforms of Functions on a Half-Line

Doklady Mathematics - Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this...     

自反函数与自反变换

这篇论文的主题是研究由自反函数导出自反级数变换与自反积分变换的一般方法。方法的基本思想来源于著者1964年发表的一篇文章(美国数学家9H.W.Gould曾在德国的Zentralblatt杂志上介绍了该文的主要结果,见该刊1973年254卷#44003)。现在这个工作较系统地发展了早先提出的思想方法,建立了若干一般性定理,有些定理可以作为构造各种级数变换互反公式的方法原则。文中除定理1,2,8之外大部份结果是属于初次发表。值得指出的是,本文中探讨的方法与概念,都是根据对某些特例的分析形成的,积分自反变换方面的结果是以级数变换为借镜,通过类比法得到的。(本文基本内容曾在1981年4月20日大连工学院举办的科学报告会上报告过)。     

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form

On the Laplace Transform of Radial Functions

Let ?(t) (t ∈ R ⁿ be a radial function. Let f(z) be the Laplace transform of ?(t). Then a theorem due to A. Gonzá Domínguez shows that f(z) can be expressed as a Hankel transform. I prove two representation formulae which express the Laplace transform of radial functions by means of the mth-order derivative of the Hankel transform of order 0 and ? ½.     

Quasi-exchange Formulas for Wavelet Transforms of Distributions

We investigate the wavelet transforms of tempered distributions in a way that closely links their Fourier transforms and wavelet transforms. Two exchange formulas of the convolution and the multiplication of wavelet transforms of tempered distributions are established. We call these formulas the quasi-exchange formulas for wavelet transforms of distributions, because of the resemblance between these formulas and the well-known exchange formula for Fourier transforms.     

Pseudo-exchange Formulas for Wavelet Transforms of Distributions

We study the wavelet transforms of tempered distributions via unitary irreducible representations, Fourier transforms, and the wavelet transforms of infinitely differentiable functions of rapid descent. We explore the general properties, and derive the pseudo-exchange formulas for wavelet transforms of tempered distributions. While the new formulas are rather different from the exchange formula for Fourier transforms of distributions, yet the pseudo-exchange formulas still exchange the products of convolution and the products of multiplication of tempered distributions.     

Marcinkiewicz-\theta-Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function . Under some conditions on we show that the maximal operator of the Marcinkiewicz- -means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where depends only on . As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz- -means of a function converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz- -means are uniformly bounded on the spaces and so they converge in norm . Some special cases of the Marcinkievicz- -summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Fourier Spectra of Distributions in Clifford Analysis

In recent papers by Brackx, Delanghe and Sommen, some fundamental higher dimensional distributions have been reconsidered in the framework of Clifford analysis, eventually leading to the introduction of four broad classes of new distributions in Euclidean space. In the current paper we continue the in-depth study of these distributions, more specifically the study of their behaviour in frequency space, thus extending classical results of harmonic analysis.     

Tables of Fourier, Laplace and Hankel Transforms of (u,n)-Dimensional Generalized Functions

The present note contains the Tables of Fourier, Laplace and Hankel transforms of several dimensional generalized functions. They are, mainly, based on the Laplace transform of retarded, Lorentz-invariant functions and the Fourier transforms of causal distributions.     

Transforms and Minors for Binary Functions

We introduce a family of transforms that extends graph- and matroid-theoretic duality, and includes trinities and so on. Associated with each such transform are λ -minor operations, which extend deletion and contraction in graphs. We establish how the transforms interact with our generalised minors, extending the classical matroid-theoretic relationship between duality and minors: ${(M/e)^* =M^* \backslash e}$ . Composition of the transforms is shown to correspond to complex multiplication of appropriate parameters. A new generalisation of the MacWilliams identity is given, using these transforms in place of ordinary duality. We also relate the weight enumerator of a binary linear code at a real argument < –1 to the transform, with parameter on the unit circle, of a close relative of the indicator function of the dual code. This result extends to arbitrary binary codes. The results on weight enumerators can also be recast in terms of the partition function of the Ising model from statistical mechanics. Most of our work is done at the level of binary functions ${f : 2^E \rightarrow \mathbb{C}}$ , which include matroids as a special case. The specialisation to graphs is obtained by letting f be the indicator function of the cutset space of a graph.     

FAST FOURIER TRANSFORMS ON BINARY FIELDS

We give a unified treatment of Fast Fourier Transforms for UDMD systems which contains , as special cases, Fast Fourier algorithms for character groups of many subgroups associated with binary fields.