Asymptotic Behaviour of the Error of Polynomial Approximation of Functions Like $\vert x\vert^{\alpha+i\beta}$

TL;DR

This paper establishes new asymptotic relations for the approximation errors of functions like |x|^{α+iβ} using polynomials and entire functions, advancing understanding of their approximation behavior.

Contribution

It introduces novel asymptotic relations between approximation errors for specific complex power functions and extends these to related functions involving logarithmic oscillations.

Findings

Derived asymptotic formulas for approximation errors

Applied results to functions with complex powers and logarithmic oscillations

Enhanced understanding of polynomial approximation limits for these functions

Abstract

New asymptotic relations between the $L_p$-errors of best approximation of univariate functions by algebraic polynomials and entire functions of exponential type are obtained for $p\in (0,\iy]$. General asymptotic relations are applied to functions $\vert x\vert^{\al+i\be},\,\vert x\vert^{\al}\cos(\be\log\vert x\vert)$, and $\vert x\vert^{\al}\sin(\be\log\vert x\vert)$.