An extension of the orthogonal derivative with adjustable precision

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.