Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

An introduction to Laplace transforms and Fourier series

Abstract: 1. The Laplace Transform.- 1.1 Introduction.- 1.2 The Laplace Transform.- 1.3 Elementary Properties.- 1.4 Exercises.- 2. Further Properties of the Laplace Transform.- 2.1 Real Functions.- 2.2 Derivative Property of the Laplace Transform.- 2.3 Heaviside's Unit Step Function.- 2.4 Inverse Laplace Transform.- 2.5 Limiting Theorems.- 2.6 The Impulse Function.- 2.7 Periodic Functions.- 2.8 Exercises.- 3. Convolution and the Solution of Ordinary Differential Equations.- 3.1 Introduction.- 3.2 Convolution.- 3.3 Ordinary Differential Equations.- 3.3.1 Second Order Differential Equations.- 3.3.2 Simultaneous Differential Equations.- 3.4 Using Step and Impulse Functions.- 3.5 Integral Equations.- 3.6 Exercises.- 4. Fourier Series.- 4.1 Introduction.- 4.2 Definition of a Fourier Series.- 4.3 Odd and Even Functions.- 4.4 Complex Fourier Series.- 4.5 Half Range Series.- 4.6 Properties of Fourier Series.- 4.7 Exercises.- 5. Partial Differential Equations.- 5.1 Introduction.- 5.2 Classification of Partial Differential Equations.- 5.3 Separation of Variables.- 5.4 Using Laplace Transforms to Solve PDEs.- 5.5 Boundary Conditions and Asymptotics.- 5.6 Exercises.- 6. Fourier Transforms.- 6.1 Introduction.- 6.2 Deriving the Fourier Transform.- 6.3 Basic Properties of the Fourier Transform.- 6.4 Fourier Transforms and PDEs.- 6.5 Signal Processing.- 6.6 Exercises.- 7. Complex Variables and Laplace Transforms.- 7.1 Introduction.- 7.2 Rudiments of Complex Analysis.- 7.3 Complex Integration.- 7.4 Branch Points.- 7.5 The Inverse Laplace Transform.- 7.6 Using the Inversion Formula in Asymptotics.- 7.7 Exercises.- A. Solutions to Exercises.- B. Table of Laplace Transforms.- C. Linear Spaces.- C.1 Linear Algebra.- C.2 Gramm-Schmidt Orthonormalisation Process.

Fourier and Laplace transforms

Abstract: 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

A variation norm Carleson theorem

Abstract: By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type: