A Kantorovich-type variant of Gr\"unwald Interpolation Operators

TL;DR

This paper introduces a new class of integral operators based on Gr"unwald interpolation on Chebyshev nodes, extending convergence results to various function spaces including $L^p$, weighted, and Orlicz spaces.

Contribution

It develops a Kantorovich-type variant of Gr"unwald operators, enabling convergence analysis in broader function spaces beyond continuous functions.

Findings

Operators are uniformly bounded on $L^p$ and other Banach spaces.

Convergence is established using modulus of continuity and K-functionals.

Point-wise estimates are derived via Hardy-Littlewood maximal operator.

Abstract

In this paper, we introduce a new sequence of operators based on the Gr\"unwald interpolation operators on Chebyshev nodes on the space $L^p[0,{\pi}]$. The operators we consider are integral variants of the Gr\"unwald interpolation operators, inspired from the classical Kantorovich operators. Unlike the original Gr\"unwald interpolation operators, our construction enables the derivation of convergence results not only on $C[0,{\pi}]$ but also in the space $L^p[0,{\pi}]$. First, we establish the uniform boundedness of this sequence on these spaces and subsequently prove the convergence of the operators. We obtain quantitative estimates using modulus of continuity and a suitable K-functional. Furthermore, we derive a point-wise estimate via the Hardy-Littlewood maximal operator. By invoking a Korovkin-type theorem, we extend the convergence results to several Banach function spaces on a nontrivial subspace. In particular, we establish these results for weighted Lebesgue spaces, Grand Lebesgue spaces, Morrey spaces, Orlicz spaces etc.