On the Evaluation of Certain Two-Dimensional Singular Integrals with Cauchy Kernels.
On the maximum modulus of polynomials. II.
On the proof of Erdős' inequality
Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality for a constrained polynomial of degree at most , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval and establish a new asymptotically sharp inequality.
One-sided approximation by trigonometric polynomials in -norm, 0
Open problems in constructive function theory.
Optimality of the range for which equivalence between certain measures of smoothness holds
Recently it was proved for 1 < p < ∞ that , a modulus of smoothness on the unit sphere, and , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence does not hold either for p = ∞ or for p = 1.
Polynomial approximation and twenty years later.
Polynomial inequalities in Banach spaces
We point out relations between the injective complexification of a real Banach space and polynomial inequalities. In particular we prove a generalization of a classical Szegő inequality to the case of polynomial mappings between Banach spaces. As an application we observe a complex version of known Bernstein-Szegő type inequalities.
Polynomial inequalities on algebraic sets
We give an estimate of Siciak’s extremal function for compact subsets of algebraic varieties in (resp. ). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.
Polynomial inequalities on general subsets of
Problème de Bernstein pour les fonctions différentiables
Properties of some bivariate approximants.
The smoothness and approximation properties of certain discrete operators for bivariate functions are examined.
Properties of the modulus of continuity for monotonous convex functions and applications.
Rate of convergence on Baskakov-Beta-Bézier operators for bounded variation functions.
Remarks on Voronovskaya's theorem.
Representation formulae for (C₀) m-parameter operator semigroups
Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.
Reverse Markov inequality.
S. N. Bernstein type estimations in the mean on the curves in a complex plane.
Sampling of Band-Limited Vectors.
Sampling of Paley-Wiener Functions on Stratified Groups.