Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters

Abstract

In the present article we investigate both the pointwise convergence and the rate of pointwise convergence of the operators

to f(x ₀) as (x,λ)→ (x ₀,λ ₀) in L ₁〈 a,b〉 where 〈 a,b〉 is an arbitrary interval in R. Here, Λ is a nonempty set of indices with any topology and λ ₀ an accumulation point of Λ in this topology.

Keywords:

• Pointwise convergence
• Rate of convergence
• Nonlinear singular integral
• Locally compact Abelian group
• Haar integral

Keywords:

• 2000 Mathematics Subject Classifications: 41A35
• 41A25
• 41-41
• 47G10
• 28C10

Acknowledgments

The author is grateful to Prof. Dr. Paul L. Butzer for the helpful suggestions and comments, which helped in improving the final version of this article. The author would also like to thank the referees for their patience and the careful study of the manuscript.