An extension of the orthogonal derivative with adjustable precision

TL;DR

This paper introduces an extended orthogonal derivative with a customizable accuracy depending on higher derivatives, surpassing traditional methods, and discusses its properties and transfer function.

Contribution

It proposes a new kernel for the orthogonal derivative that allows adjustable accuracy based on higher derivatives, extending previous approaches.

Findings

The new kernel improves derivative accuracy beyond traditional orthogonal derivatives.

The transfer function of the new derivative is explicitly derived.

The kernel is not orthogonal for orders greater than one.

Abstract

The orthogonal derivative is defined as a limit of an integral whose kernel contains an orthogonal polynomial with its measure. When in practice no limit is taken, it means that the accuracy of the derivative depends on the second derivative of the given function. Liptaj shows that it is possible to define a kernel in such a way that the accuracy depends on a higher derivative at your own choice. The accuracy is therefore much greater than with the orthogonal derivative. However Diekema and Koornwinder find a similar extension starting directly from of the orthogonal derivative. The new kernel is not orthogonal for order greater then one. The transfer function for this new derivative is given.