Fourier Series

Chaurasiya, Vijay Kumar and D, Venku Naidu (2017) Fourier Series. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

Chapter 1 A BRIEF HISTORY OF FOURIER SERIES Fourier series were invented by Fourier who was studying a physical problem. It is no won- der then that they have applications. Of course not all creatures of ”applied .. mathematics have applications in as wide an area as Fourier series do. As this account shows. attempts to understand the behaviour of these series also laid down the foundations of rigorous analysis. Questions like uniform convergence. Cesaro summability and subjects like transfinite cardinals and Lebesgue measure are thought of as ”pure” mathematics. Their history too is related to Fourier series. Among the purest branches of mathematics is number theory-and surely it is a subject quite independent of Fourier series. Yet Heimann Weyl used Fejer’s convergence theorem for Fourier series to prove a beautiful theorem in number theory. called the Weyl Equidistribution Theorem. Every real number x can be written as x = [ x ] + { x } , where [ x ] is the integral part of x and is an integer and { x } is the fractional part of x and is a real number lying in the interval [0 , 1) . One of the areas where Fourier series and transfonns have major applications. is crystal- lography. In 1985 the Nobel Prize in Chemistry was given to H. A. Hauptman and J. Karle who developed a new method for calculating some crystallographic constants from their Fourier coefficients, which can be inferred from measurements. Two crucial ingredients of their analy- sis are Weyl’s equidistribution theorem and theorems of Toeplitz on Fourier series of nonneg... ative functions

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