Z. Ditzian

Moduli of smoothness andK-functionals inLp, 0<p<1

It is shown that forLp, 0<p<1, the PeetreK-functional betweenLp andWpr is identically zero. Useful measures that are equivalent to the moduli of smoothness are found. The equivalence results that are given are valid for 0<p≤∞.

Kantorovich-Bernstein polynomials

A gap between saturation and direct-converse theorems for Kantoro-vich-Bernstein polynomials will be closed for a steady rate of convergence. The present theorems unify the above-mentioned results. Furthermore, it is shown that for steady rates our converse results are an improvement on both weak-type converse theorems and strong-weak-type converse theorems for the Kantorovich-Bernstein polynomials.

Strong converse inequality for Kantorovich polynomials

AbstractFor the Kantorovich polynomial approximationKn(f, t), 1<p≤∞, we prove that, for somem,nThis equivalence includes a strong converse inequality of type B.

Full length article: Relating smoothness to expressions involving Fourier coefficients or to a Fourier transform

Coefficients of expansion of a function by trigonometric, algebraic and spherical harmonic orthogonal polynomials are related to the smoothness of that function. Hausdorff-Young type and Hardy-Littlewood type inequalities will be utilized. Expressions involving the Fourier transform of a function are also related to the measure of smoothness of that function.

Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients

In a recent paper Bray and Pinsky [1] estimated the growth of [emailxa0protected]^(@x), the Fourier transform of f(x) where [emailxa0protected]?R^d, by some moduli of smoothness. We show here that noticeably better results can be derived as an immediate corollary of previous theorems in [2]. The improvements include dealing with higher levels of smoothness and using the fact that for higher dimensions (when d>=2) the description of smoothness requires less information. Using a similar technique, we also deduce relations between the smoothness of f(x) for [emailxa0protected]?S^d^-^1 or [emailxa0protected]?T^d and the growth of the coefficients of the expansion by spherical harmonic polynomials or trigonometric polynomials.

K functionals and best polynomial approximation in weighted L p (R)

On propose un nouveau module de regularite de la meilleure approximation polynomiale dans L p (R) pondere

Exponential formulae for semi-group of operators in terms of the resolvent

Rate of convergence in terms of the modulus of continuity of eitherT(t)f or ofT(t)Af, whereT(t) is a strongly continuous semi-group of operators, is obtained for Phillips’ and for Widder’s exponential formula.

Polynomial approximation inL p(S) forp>0

AbstractFor a simple polytopeS inRd andp>0 we show that the best polynomial approximationEn(f)p≡En(f)Lp(S) satisfiesn

On Hille’s first exponential formula

E_n left( f right)_p leqslant Comega _S^r left( {f,frac{1}{n}} right)p,