Duality results for a general trigonometric approximation problem

TL;DR

This paper explores a general trigonometric approximation problem in $L^eta( u)$ spaces, using a duality approach without extra assumptions on the measure, and extends results to multivariate cases for $eta=2$.

Contribution

It introduces a duality method for a broad class of trigonometric approximation problems without additional measure assumptions and extends to multivariate scenarios for $eta=2$.

Findings

Duality method effectively solves the approximation problem.

No extra assumptions on the measure are needed.

Multivariate extensions are achieved for $eta=2$.

Abstract

Let $\alpha\in(1,\infty)$ and $\mu$ be a regular finite Borel measure on a locally compact abelian group. The paper deals with a general trigonometric approximation problem in $L^\alpha(\mu)$, which arises in prediction theory of harmonizable symmetric $\alpha$-stable processes. To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors. The novelty of the present paper is that we do not make any additional assumption on $\mu$. Moreover, for $\alpha=2$, multivariate extensions are obtained.