The XFT Quadrature in Discrete Fourier Analysis

This book has two main objectives, the first of which is to extend the power of numerical Fourier analysis and to show by means of theoretical examples and numerous concrete applications that when computing discrete Fourier transforms of periodic and non periodic functions, the usual kernel matrix of the Fourier transform, the discrete Fourier transform (DFT), should be replaced by another kernel matrix, the eXtended Fourier transform (XFT), since the XFT matrix appears as a convergent quadrature of a more general transform, the fractional Fourier transform.

In turn, the book's second goal is to present the XFT matrix as a finite-dimensional transformation that links certain discrete operators in the same way that the corresponding continuous operators are related by the Fourier transform, and to show that the XFT matrix accordingly generates sequences of matrix operators that represent continuum operators, and which allow these operators to be studied from another perspective.

Presents a new quadrature formula for the fractional Fourier transform Many examples are addressed to illustrate the power of the new formula Most of the algorithms presented, are implemented in standard packages as MATLAB or MATHEMATICA Present the XFT matrix as a finite-dimensional transformation

Résumé
"The book gives a concise survey of problems with classical DFT and introduces the XFT as an alternative. The performance is clearly illustrated with many applications ... . Recommended for everyone who uses Fourier transforms in a computational context and wants to learn about its extended XFT alternative and the theory behind it." (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, July 01, 2019)

Contenu
Introduction.- The ordinary discrete Fourier transform.- XFT: A discrete Fourier transform.- Applications of the XFT.- A discrete fractional Fourier transform.