# About one method for constructing Hermite trigonometric polynomials

## TL;DR

This paper presents a new method for constructing Hermite trigonometric polynomials that interpolate a periodic function and its derivatives at uniform grid points, utilizing properties of trigonometric functions and linear algebra solutions.

## Contribution

The paper introduces a novel approach reducing the construction of Hermite trigonometric polynomials to solving two linear systems, enabling efficient computation with FFT algorithms.

## Key findings

- Method reduces interpolation to two linear systems.
- FFT algorithms can be used for coefficient calculation.
- Method is suitable for practical applications.

## Abstract

The method of constructing Hermite trigonometric polynomials, which interpolate the values of a certain periodic function and its derivatives up to (including ) the -th ( ) order in nodes of a uniform grid, is considered. The proposed method is based on the properties of the periodicity of trigonometric functions and is reduced to the solving of only two systems of the linear algebraic equations of the --th order; solutions to these systems may 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 in the nodes of a uniform grid; well-known algorithms of fast Fourier transform can be applied for this purpose. The examples of construction of Hermite trigonometric polynomials for .. are given. The proposed method may be recommended for wide practical application.

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Source: https://tomesphere.com/paper/1902.04327