Books by Rami Shakarchi (Author of Complex Analysis)
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This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.
Introduction: Monday, Jan 19, Fourier series were first used by the French Mathematician Joseph Fourier around in his study of the heat flow. They are one of the most powerful mathematical concepts for applications, ubiquitous in Physics, Electrical Engineering, and many branches of Computing. However, also modern-day Pure Mathematics, for example Number Theory, Geometry and Representation Theory, is permeated by the study of the Fourier transform and its generalizations. This course will introduce some of the tools of Fourier Analysis and present some applications. Our main concern will be the question as to in which sense a function may be represented by its Fourier series or transform.