Related papers: On moduli of smoothness with Jacobi weights
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…
Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.
In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Gr\"unwald derivatives in $L_p$, $0<p<1$. We extend known inequalities to the whole range of…
The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where…
We consider two types of fractional integral moduli of smoothness, which are widely used in theory of functions and approximation theory. In particular, we obtain new equivalences between these moduli of smoothness and the classical moduli…
The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the…
It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $\theta$ of the form $o((1-|…
In this paper we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show,…
In this work we obtain algebraicity results on special $L$-values attached to Siegel-Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a…
Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points were recently found. The purpose of this paper is to re-express these cyclic identities in terms of ratios of Jacobi…
In this paper we study direct and inverse approximation inequalities in $L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish…
We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using…
As is well known the kernel of the orthogonal projector onto the polynomials of degree $n$ in $L^2(w_{\a,\b}, [-1, 1])$ with $w_{\a,\b}(t) = (1-t)^\a(1+t)^\b$ can be written in terms of Jacobi polynomials. It is shown that if the…
Let $\Omega \subset \mathbb{C}^m$ be an open, connected and bounded set and $\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on $\Omega$. In this article we study quotient Hilbert modules obtained from submodules,…
We define the vector-valued, matrix-weighted function spaces $\dot{F}^{\alpha q}_p(W)$ (homogeneous) and $F^{\alpha q}_p(W)$ (inhomogeneous) on $\mathbb{R}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q \leq \infty$, with the matrix…
We construct coarse moduli spaces of semiquasihomogeneous hypersurface singularities with respect to right equivalence and contact equivalence. We have to fix the principal part of the semiquasihomogeneous singularities. For the moduli…
In this paper, we study the Jacobi frame approximation with equispaced samples and derive an error estimation. We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter $\gamma$ increases in…
The definition of an action functional for the Jacobi sigma models, known for Jacobi brackets of functions, is generalized to \emph{Jacobi bundles}, i.e., Lie brackets on sections of (possibly nontrivial) line bundles, with the particular…
We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is…
A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…