q-Fractional Askey-Wilson Integrals and Related Semigroups of Operators

TL;DR

This paper introduces new q-fractional integral operators related to Askey-Wilson polynomials, analyzes their spectral properties, and explores their applications in approximation theory and integral transforms.

Contribution

It presents three novel semigroups of operators, including fractional integrals associated with the Askey-Wilson operator, and studies their spectral and approximation properties.

Findings

Determined spectra of the semigroups of operators.

Established connection relations and bilinear formulas for Askey-Wilson polynomials.

Introduced a q-Gauss-Weierstrass transform with representation and inversion theorems.

Abstract

We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey-Wilson polynomials. We also introduce a q-Gauss-Weierstrass transform and prove a representation and inversion theorem for it.