Generalized Discrete Fourier Transform on the Base of Lagrange and Hermite Interpolation Formulas

Abstract

The article offers several generalizations of Discrete Fourier Transform. The theoretical basis for the generalizations is the interpolation formulas of Lagrange and Hermite. It has been established that each polynomial generates its own corresponding Discrete Fourier Transform. The paper proposes an algorithm for constructing new DFT generalizations. It is possible to use the Fast Fourier Transform (FFT) to build new generalizations. The technology for using the FFT-application in the MatLab system is described. We have revealed that the introduction and application of the Discrete Fourier Transform on the base of the interpolation formulas of Lagrange and Hermite allows us to build new generalizations with the necessary properties in practical applications. The authors' approach, as opposed to traditional methods of presentation, has a number of advantages: first, the simplicity and naturalness of the introduction of the Discrete Fourier Transform are achieved, and secondly, it is possible to construct the Generalized Discrete Fourier Transform with specified properties, which is not obvious under the standard approach.