On non-local approximation properties of the binomial power functions $(1+x^q)^r$

TL;DR

This paper investigates the approximation capabilities of binomial power functions $(1+x^q)^r$, establishing their density in continuous and $L^p$ spaces through polynomial systems related to non-local approximation.

Contribution

It introduces polynomial systems associated with binomial power functions to prove their density in various function spaces, advancing approximation theory.

Findings

Density of scattered translated power functions in $L^p$ spaces.

Construction of polynomial systems for non-local approximation.

Establishment of density results in $C[a,b]$.

Abstract

This note mainly concerns the binomial power function, defined as $(1+x^q)^{r}$. We construct systems of polynomials related to non-local approximation, which allows us to establish the density results on $C[a,b]$, where $a,b\in\mathbb{R}$. As a corollary, we show that scattered translated of power functions and certain related functions are dense in the function spaces $L^p([a,b])$, for $1\leq p <\infty$.