Related papers: Schwartz functions on real algebraic varieties

Let $Z$ be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on $Z$ which parallels the work of Harish-Chandra. The constant terms $f_I$ of an eigenfunction $f$ are parametrized by subsets $I$…

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the…

In this paper we define Schwartz families in tempered distribution spaces and prove many their properties. Schwartz families are the analogous of infinite dimensional matrices of separable Hilbert spaces, but for the Schwartz test function…

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the…

The purpose of this paper is to introduce the notion of Nash functions in the context of slice regular functions of one quaternionic or octonionic variable. We begin with a detailed analysis of the possible definitions of Nash slice regular…

We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilit\'e de la multiplication des…

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of de-Rham theorem for de-Rham complexes with coefficients in…

We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…

Let G be a solvable Lie group endowed with right Haar measure. We define and study a dense Frechet *-subalgebra S of L1(G), consisting of smooth functions rapidly-decreasing at infinity on G. When G is nilpotent, we recover the classical…

Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as…

We prove a Wiener-Tauberian theorem for $L^1$-spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz theorem for complex groups. As a corollary we obtain a Wiener-Tauberian type theorem for for…

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general…

We investigate the space $X$ of unitary hermitian matrices over $\frp$-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic,…

We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the…

We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \mathscr{S}-regular generalized functions of this…

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to…

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $\beta$-restricted Zernike functions. Mainly, we give the…

In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…

This paper continues the analysis of Schr\"odinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are…

These notes briefly consider some aspects of the Schwartz class of rapidly decreasing smooth functions, tempered distributions, and harmonic functions of polynomial growth.