Fourier Transforms: Mathematical Methods for Physics and Engineering - Volume 2

There is a longstanding conflict between extension and depth in the teaching of mathematics to physics students. This text intends to present an approach that tries to track what could be called the ``middle way'' in this conflict. It is the result of several years of experience of the author teaching the mathematical physics courses at the Physics Institute of the University of São Paulo. The text is organized in the form of relatively short chapters, each appropriate for exposition in one lecture. Each chapter includes a list of proposed problems, which have varied levels of difficulty, including practice problems, problems that complete and extend the material presented in the text, and some longer and more difficult problems, which are presented as challenges to the students. There are complete solutions available, detailed and commented, to all the problems proposed, which are presented in separate volumes. This volume is dedicated to Fourier transforms. This term is used here in a wider sense, including finite Fourier transforms, defined on a finite and discrete lattice, Fourier series, defined on a finite domain within the continuum, and the usual Fourier transforms, defined on the infinite continuum. This constitutes an elementary introduction to what is called, in its more abstract form, harmonic analysis. By means of the device of starting from the finite and discrete version of the formalism, which is done in the spirit of the definition of the Riemann integral, we are able to present in a clear way the basic structure of this whole formalism, while avoiding any need to face on this first moment the difficult convergence questions that arise when one takes the continuum limit. Once in the continuum, the convergence issues are addressed and put in proper perspective through the use of a low-pass filter, which is defined and developed in a fairly precise way. In the last two chapters the whole structure of the Fourier theory of real functions is derived ``ab initio'' once again, this time directly in the continuum, starting from the theory of analytic functions. There we present something that works like a universal summation rule, which applies to all Fourier series, and which allows us to recover any integrable real function from the set of its Fourier coefficients, even when the Fourier series itself diverges.