Related papers: Dunkl harmonic analysis and fundamental sets of co…

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials.…

This paper is devoted to the study of the Dirichlet problem associated with the Dunkl Laplacian $\Delta_k$. We establish, under some condition on a bounded domain $D$ of $\R^d$, the existence of a unique continuous function $h$ on $\R^d$…

We construct N-harmonic functions in a domain with one isolated singularity on the boundary of the domain. By using solutions of the spherical p-harmonic spectral problem, we give an inductive method to produce a large variety of separable…

In this paper, we define a subclass of sense-preserving harmonic functions associated with a class of analytic functions satisfying a differential inequality. We then establish a close relation between both subclasses. Further, we obtain…

In this paper, we establish a connection between Dunkl analysis and slice analysis in the setting of Clifford algebras. Specifically, we show that a Clifford algebra-valued function is slice if, and only if, it belongs to the kernel of the…

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in…

Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was…

In this article we consider functions $f$ meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions. This condition simplifies and generalizes known conditions. We…

Near every point of a real-analytic set in $\mathbb R^n$, we make use of Hironaka's resolution of singularity theorem to construct a family of continuous functions in $W^{1, 1}_{loc}$ such that their weak derivatives have (removable)…

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new methodology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent…

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of…

It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation…

Let $m,n\geq 1$ are integers and $D$ be a domain in the $$ $\mathbb C^n$ or in the $m$-dimensional real space $\mathbb R^m$. We build positive subharmonic functions on $D$ vanishing on the boundary $\partial D$ of $D$. We use such (test)…

Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\partial B_1$, of Clark measures on the unit sphere…

We extend the classical theorem of Uchiyama about constructive Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals to the rational Dunkl setting. On $\mathbb{R}^N$ equipped with a root system $R$ and a…

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space…

We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and…

We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…