In this thesis, we treat the computation of transforms with asymptotically smooth and oscillatory kernels. We introduce the discrete Laplace transform in a modern form including a generalization to more general kernel functions. These more general kernels lead to specific function transforms. Moreover, we treat the butterfly fast Fourier transform. Based on a local error analysis, we develop a rigorous error analysis for the whole butterfly scheme. In the final part of the thesis, the Laplace and Fourier transform are combined to a fast Fourier transform for nonequispaced complex evaluation nodes. All theoretical results on accuracy and computational complexity are illustrated by numerical experiments.

Three classes of Fourier transforms are presented: Fourier (Laplace) transforms on the halfline, Fourier transforms of measures with compact support and Fourier transforms of rapidly decreasing functions (on whole line). The focus is on the behaviour of Fourier transforms in the region of analyticity and the distribution of their zeros. Applications of results are presented: approximation by exponentials on the finite interval; behavior of the nonharmonic Fourier series; Müntz-Szasz's problem of approximation by powers on unit interval; approximation by weighted exponentials on whole line.

This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science.

Fast Fourier transform (FFT) methods are well established for solving certain types of partial differential equations (PDE). This book is written at an introductory level with the non-specialist user in mind. It first deals with basic ideas and algorithms which may be used to solve problems using simple geometries--the fast Fourier transform is employed and thorough details of the computations are given for a number of illustrative problems. The text proceeds to problems with irregular boundaries, using the capacity matrix approach, and also to more advanced PDE, for which fast solvers may be used as the basis for iterative methods. The use of a numerical Laplace transform technique for certain time-dependent problems is also covered. Throughout the book, the approach is designed to illustrate the essential ideas of the methods employed. References are given for further reading of more advanced or specialized topics.

This monograph reviews the use of the Laplace transform as implemented using the fast Fourier transform. This method has been described earlier by investigators in the electrical power community, but it does not seem to be widely used in the electromagnetic compatibility area. The goal in developing this monograph is to bring this computational method to the attention of the workers in this community by providing several examples and comments on its use for practical problems.

In this book, distributions are introduced via sequences of functions. This approach due to Temple has two virtues:The Fourier transform is defined for functions and generalized to distributions, while the Green function is defined as the outstanding application of distributions. Using Fourier transforms, the Green functions of the important linear differential equations in physics are computed. Linear algebra is reviewed with emphasis on Hilbert spaces. The author explains how linear differential operators and Fourier transforms naturally fit into this frame, a point of view that leads straight to generalized fourier transforms and systems of special functions like spherical harmonics, Hermite, Laguerre, and Bessel functions.

A textbook covering data-science and machine learning methods for modelling and control in engineering and science, with Python and MATLAB®.

This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material.

Presenting an introduction to all Fourier-related transforms, this work includes a number of applications in the different markets. The accompanying disk provides C and Fortran routines that can be implemented.

The fourier transform; Fourier transform properties; Convolution and correlation; Fourier series and sampled waveforms; The discrete fourier transform; Discrete convolutiion and correlation; Applying the discrete fourier transform.