Textbook: An Introduction to Fourier Series and Integrals, by Robert T. Seeley.
The topics we will cover include Fourier series, convolutions, kernels, summation methods, and Fourier transforms, with applications to the wave, heat, and Laplace equations.
Grading is based on
Homework is due at the beginning of class on Thursdays. It may be handed in up to one week late for half credit. Here are the assignments and associated handouts:
Homework 1, due January 18th. See also this handout on complex exponentials and differential equations, and this one from Spivak's Calculus on uniform convergence.
Homework 2, due January 25th.
Fourier Analysis and Boundary Value Problems, by Enrique A. González-Velasco, has interesting historical background and physical applications. It is available free online from the Purdue library.
Fourier Analysis: An Introduction, by Elias M. Stein and Rami Shakarchi, includes applications to other areas of math, including geometry and number theory.
Fourier Analysis, by T. W. Körner, is 'a series of interlinked essays' which goes much further than the above, with many many examples and applications.
A First Course in Harmonic Analysis, Second Edition, by Anton Deitmar, includes a short introduction to distribution theory, which powerfully extends the scope of the Fourier transform. It is also available free online from the Purdue library.
The above all avoid the more advanced machinery of Lebesgue measure and integration. Some basics of this machinery can be found in section 2.9 of González-Velasco's book. A more thorough introduction, with an emphasis on applications to Fourier analysis, is given in Sz.-Nagy's Introduction to real functions and orthogonal expansions, and also in Stein and Shakarchi's Real Analysis. For more on Fourier series, see Hardy and Rogosinski's Fourier Series. For more on distribution theory, see Friedlander and Joshi's Introduction to the Theory of Distributions.
Finally, for more general background and inspiration, I recommend Riemann's habilitation thesis (which is where, incidentally, the Riemann integral was first defined), and which can be found in this beautiful edition of his Collected Papers; see also this handout.