Some extremal functions in Fourier analysis, III

TL;DR

This paper determines the optimal entire functions of exponential type that best approximate certain even functions in L^1(R), including exponential decay, logarithmic, and fractional power functions, with periodic analogs using trigonometric polynomials.

Contribution

It provides the best approximation results in L^1(R) for a broad class of even functions and extends these results to periodic functions with bounded-degree trigonometric polynomials.

Findings

Optimal approximation functions identified for specific classes of even functions.

Periodic approximation results established using trigonometric polynomials.

Extends classical Fourier analysis approximation theory to new function classes.

Abstract

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.