Harmonic analysis, plays a great role in the development of the many areas of sciences, and in the fact that its ideas permeate much of the present day analysis. In this program we will devote time to an exposition of some basic facts about Fourier series, taken together with a study of elements of Fourier transforms, the first step towards the introduction of Harmonic Analysis. Starting this way allows one to see rather easily certain applications to other sciences, together with the link to such topics as partial differential equations and number theory.
Using our understanding of Fourier analysis we will aim to learn measure theory and Lebesgue integration. For this reason our treatment of Fourier series will be carried out in the context of Riemann integrable functions.
We will learn about Fourier transform and understand its connections with wave equation and the Radon transform. We will conclude with an understanding of various functions spaces, Hilbert spaces to be precise.
It is recommended that you have access to Real analysis; Measure theory, integration, and Hilbert spaces by E. M. Stein and Rami Shakarch