On the approximation properties of bi-parametric potential-type integral operators

Sekin Ç., Güloğlu M., Aliev İ.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.50, no.6, pp.1681-1691, 2021 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 6
  • Publication Date: 2021
  • Doi Number: 10.15672/hujms.821159
  • Journal Name: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
  • Page Numbers: pp.1681-1691
  • Keywords: Abel-Poisson semigroup, Gauss-Weierstrass semigroup, Riesz potentials, Bessel potentials, potentials-type operators, RIESZ-POTENTIALS, TEMPERATURES, TRANSFORMS, INVERSION, SPACES
  • Akdeniz University Affiliated: Yes

Abstract

In this work we study the approximation properties of the classical Riesz potentials I alpha f equivalent to (- increment )-alpha/2f and the so-called bi-parametric potential-type operators J beta alpha f equivalent to (E + (- increment )beta /2)-alpha/beta f as alpha -> alpha 0 > 0 where, alpha > 0, 0 > 0, E is the identity operator and increment is the laplacian. These potential-type operators generalize the famous Bessel potentials when 0 = 2 and Flett potentials when 0 = 1. We show that, if A alpha is one of operators J beta alpha or I alpha, then at every Lebesgue point of f is an element of Lp(Rn) the asymptotic equality (A alpha f)(x) - (A alpha 0f)(x) = O(1)(alpha - alpha 0), (alpha -> alpha+0 ) holds. Also the asymptotic equality parallel to A alpha f - A alpha 0f parallel to p = O(1)(alpha - alpha 0), (alpha -> alpha+0) holds when A alpha = J beta alpha.