Related papers: Uniform convergence for sequences of best L^{p} ap…

In this paper, we determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the generalized difference operator $\Delta_{a,b}$ on the sequence space $\ell_p \ (1< p < \infty)$ where the real sequences…

We investigate the best order of smoothness of $L^p(L^q)$. We prove in particular that there exists a $C^\infty$-smooth bump function on $L^p(L^q)$ if and only if $p$ and $q$ are both even integers and $p$ is a multiple of $q$.

For a separable symmetric sequence space $X$ of fundamental type we identify the set ${\mathcal F}(X)$ of all $p\in [1,\infty]$ such that $\ell^p$ is block finitely represented in the unit vector basis $\{e_k\}_{k=1}^\infty$ of $X$ in such…

It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank}…

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$…

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…

For a class of systems of semi-linear elliptic equations, including \[ -\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$ (symmetric-type interaction), we…

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the…

M\"untz spaces satisfying the M\"untz and gap conditions are considered. A Fourier approximation of functions in the M\"untz spaces $M_{\Lambda ,p}$ of $L_p$ functions is studied, where $1<p<\infty $. It is proved that up to an isomorphism…

We study the regularity of smooth functions $f$ defined on an open set of $\mathbb{R}^n$ and such that, for certain integers $p\geq 2$, the powers $f^p :x\mapsto (f(x))^p$ belong to a Denjoy-Carleman class $\mathcal{C}_M$ associated with a…

In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in…

We study the best approximation problem: \[ \displaystyle \min_{\alpha\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf x}_i) \right|. \] Here: $\Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\}$ is a list of…

We extend the Kahane-Katznelson-de Leeuw theorem to smoothness spaces by showing that for any $g \in W^{l,2}(\mathbb{T}^d)$, there exists a function $f\in C^l(\mathbb{T}^d)$ satisfying $|\widehat{f}(n)|\geq |\widehat{g}(n)|$ and…

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ such that $ \lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with…

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from $L_p$, $1<p<\infty$. In particular, the paper includes monotonicity properties, equivalence and…

We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value…

A sequence $(x_n)$ in a lattice-normed space $(X,p,E)$ is statistical $p$-convergent to $x\in X$ if there exists a statistical $p$-decreasing sequence $q\stpd 0$ with an index set $K$ such that $\delta(K)=1$ and $p(x_{n_k}-x)\leq q_{n_k}$…

We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in…

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated…

Given a regular compact set $E$ in the complex plane, a unit measure $\mu$ supported by $\partial E,$ a triangular point set $\beta := \{\{\beta_{n,k}\}_{k=1}^n\}_{n=1}^{\infty},\beta\subset \partial E$ and a function $f$, holomorphic on…