Entire approximations for a class of truncated and odd functions

TL;DR

This paper develops optimal entire function approximations of prescribed exponential type for a new class of truncated and odd functions, including truncated logarithms and shifted powers, minimizing the L^1 error.

Contribution

It extends previous work on even functions to odd functions, providing explicit solutions for a broader class of truncated and shifted functions.

Findings

Explicit extremal functions for truncated and odd functions derived

Minimized L^1-error achieved for the approximations

Includes new examples like truncated logarithm functions

Abstract

We solve the problem of finding optimal entire approximations of prescribed exponential type (unrestricted, majorant and minorant) for a class of truncated and odd functions with a shifted exponential subordination, minimizing the $L^1(\R)$-error. The class considered here includes new examples such as the truncated logarithm and truncated shifted power functions. This paper is the counterpart of the works of Carneiro and Vaaler (Some extremal functions in Fourier analysis, Part II in Trans. Amer. Math. Soc. 362 (2010), 5803-5843; Part III in Constr. Approx. 31, No. 2 (2010), 259--288), where the analogous problem for even functions was treated.