Approximation properties of a modified Gauss–Weierstrass singular integral in a weighted space

Abstract

Abstract Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w $L^{p,w}$ , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w $f\in L^{p,w}$ . We use the modulus of continuity of the functions to measure the rate of convergence.

Keywords

• Modulus of continuity
• Gauss–Weierstrass singular operator
• Weighted Lebesgue spaces
• Weighted Hölder space