Bernstein and Kantorovich polynomials diminish the $\Lambda$-variation

TL;DR

This paper proves that Bernstein and Kantorovich polynomials reduce the $ ext{ extLambda}$-variation of functions, and uses this to characterize certain function spaces, providing new insights into their structure and properties.

Contribution

It establishes the $ extLambda$-variation diminishing property for Bernstein and Kantorovich polynomials and characterizes the space $C extLambda BV_c$ as a polynomial closure in the $ extLambda BV$ norm.

Findings

Bernstein and Kantorovich polynomials diminish $ extLambda$-variation.

Characterization of $C extLambda BV_c$ as polynomial closure.

Proof of separability of $C extLambda BV_c$.

Abstract

We prove the $\Lambda$-variation diminishing property of the Bernstein and Kantorovich polynomials. Next we apply this result to characterize the space $C\Lambda BV_c$ as the closure of the space of polynomials in the $\Lambda BV$ norm. A new proof of the separability of $C\Lambda BV_c$ is given.