Approximation, regularization and smoothing of trigonometric splines

TL;DR

This paper explores methods for approximating, regularizing, and smoothing trigonometric interpolation splines, emphasizing their dual interpretation as Fourier series and discrete systems with stored differential properties.

Contribution

It introduces a dual perspective on trigonometric splines and advocates for discrete approaches in approximation and smoothing to leverage stored differential properties.

Findings

Trigonometric splines can be viewed as Fourier series or discrete systems.

Discrete approaches are effective for approximation and smoothing.

Differential properties are preserved in discrete representations.

Abstract

The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier series and as discrete trigonometric Fourier series according to certain systems of functions that are smoothness carriers. It is argued that with approximation and smoothing of trigonometric splines it is expedient to consider as discrete rows, since their differential properties are stored.