About one method of constructing Hermite trigonometric splines

TL;DR

This paper presents a practical method for constructing Hermite trigonometric splines that interpolate periodic functions and their derivatives using linear algebra and Fourier transforms, suitable for real-world applications.

Contribution

It introduces a new approach based on periodicity and linear algebra, simplifying the construction of Hermite trigonometric splines with precomputable solutions.

Findings

Method reduces to solving second-order linear systems

Fast Fourier transform algorithms facilitate coefficient calculation

Examples demonstrate first and second order spline construction

Abstract

The method of constructing trigonometric Hermite splines, which interpolate the values of some periodic function and its derivatives in the nodes of a uniform grid, is considered. The proposed method is based on the periodicity properties of trigonometric functions and is reduced to solving only systems of linear algebraic equations of the second order; solutions of these systems can be obtained in advance. When implementing this method, it is necessary to calculate the coefficients of interpolation trigonometric polynomials that interpolate the values of the function itself and the values of its derivatives at the nodes of the uniform grid; known fast Fourier transform algorithms can be used for this purpose. Examples of construction of trigonometric Hermite splines of the first and second orders are given. The proposed method can be recommended for practical use.