arXiv subjects

Z. Ditzian

Publications and source records attributed to Z. Ditzian.

3 recordsLinked to original sources

arXiv2016

On Nikol'skii inequalities for domains in $R^d$

Nikol'skii inequalities for various sets of functions, domains and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree $n$ on a bounded convex domain $D$. That is, we study $σ:= σ(D,d)$ for which \[ \|P\|_{L_q(D)}\le c n^{σ(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\infty, \] where $P$ is a polynomial of total degree $n$. We use geometric properties of the boundary of $D$ to determine $σ(D,n)$ with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal $σ(D,n)$ will be computed explicitly.

math.CA↗

arXiv2014

Discrete $d$-dimensional moduli of smoothness

We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important special case our result implies for $f\in C(I^d)$ and given integer $r$ that when $0<\alpha 0$ and $|u|=1$ such that $x,x+rhu\in I^d$.

math.NA↗

arXiv2007

Polynomial Approximation and $ω^r_ϕ(f,t)$ Twenty Years Later

About twenty years ago the measure of smoothness $ω^r_ϕ(f,t)$ was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.

math.CA↗