# About one method of construction of interpolation trigonometric splines

## TL;DR

This paper introduces a method for constructing trigonometric interpolation splines using Fourier series with decreasing coefficients, enabling the creation of various spline classes with specific smoothness and interpolation properties.

## Contribution

It presents a novel approach to building trigonometric splines with controlled smoothness and interpolation features, including classes without polynomial analogues.

## Key findings

- Different classes of trigonometric splines are obtained based on convergence factors.
- Some trigonometric splines do not have polynomial counterparts.
- An example of constructing such splines is provided.

## Abstract

The method of constructing spline classes in the form of trigonometric Fourier series whose coefficients have a certain decreasing order are considered. in turn, this decrement determines the number of continuous derivatives of sum of this series. By grouping members of this series according with the effect of overlaying and introducing a multiplier that provides the interpolation properties of the sum of these series on even grids, we obtain classes of trigonometric interpolation splines. Depending on the types of convergence factors with a certain decreasing order, different classes of such splines are obtained. The classes of trigonometric splines include classes of periodic polynomial splines of even and odd power; At the same time, there exist trigonometric splines that do not have polynomial analogues. An example of the construction of trigonometric splines is given.

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Source: https://tomesphere.com/paper/1902.07970