Stability of trigonometric approximation in $L^p$ and applications to prediction theory

TL;DR

This paper investigates the stability of trigonometric approximation in $L^p$ spaces over LCA groups, analyzing the behavior of metric projections and errors, with applications to prediction theory for stochastic processes.

Contribution

It provides new stability results for trigonometric approximation in $L^p$ spaces on LCA groups and applies these to prediction problems for stochastic processes.

Findings

Stability theorems for prediction of $p$-stable processes.

Analysis of approximation errors in various settings.

Comparison of linear interpolation and extrapolation methods.

Abstract

Let $\Gamma$ be an LCA group and $(\mu_n)$ be a sequence of bounded regular Borel measures on $\Gamma$ tending to a measure $\mu_0$. Let $G$ be the dual group of $\Gamma$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{\mu_n,p}$ the subspace of $L^p(\mu_n)$, $p \in (0,\infty)$, spanned by the characters of $\Gamma$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{\mu_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other.