Fourier analysis in spaces of shifts

TL;DR

This paper introduces a Fourier-based method for approximating functions in shift spaces, providing explicit formulas for best approximation in $L_2( )$, with potential applications in spline and shift-based approximation.

Contribution

It develops a continuous analog of orthogonal decomposition in shift spaces, deriving explicit formulas for best approximation using Fourier transform techniques.

Findings

Explicit expression for best approximation in shift spaces

Connection between Fourier transform and shift-based approximation

Potential applications in spline approximation

Abstract

In this paper, we develop a continual analog of decomposition over orthogonal bases in spaces generated by equidistant shifts of a single function. By doing so, we obtain an explicit expression for best approximation by spaces of shifts in $L_2(\mathbb{R})$. The result is formulated in terms of classical Fourier transform and tends to have various applications in approximation by spaces of shifts and, in particular, in spline approximation.