Fourier Series

Gibbs

Overview: The theory of Fourier series has roots in problems of mathematical physics such as wave propagation, heat conduction, and the like. The Fourier expansion of a function, named after Joseph Fourier (1768–1830), is a way of representing a function as a sum of sines and cosines and it is fundamentally different from a Taylor series expansion that you have studied in calculus in that the coefficients are determined not by differentiation but by integration, and the function to be represented need not be differentiable and might even have points of discontinuity.

MAT 329 is an introduction to Fourier Analysis geared towards undergraduate students from both theoretical and applied areas. The course will start with motivating the study of Fourier series by placing emphasis on the history of the subject and some of its applications. Then we shall take a look at Fourier expansions in inner product spaces where there are no issues of convergence, and finally we shall discuss different modes of convergence for Fourier series.

Topics will include Dirichlet’s theorem for pointwise convergence of Fourier series, inner product spaces, orthogonal projections and the Gram–Schmidt process, Fourier expansions and Bessel’s inequality, Parseval’s identity, Mean-square convergence of Fourier series, uniqueness of Fourier coefficients, the Dirichlet and Fejer kernels, convolutions, approximations by trigonometric polynomials, the Gibbs phenomenon, the Discrete Fourier Transform, absolutely convergent Fourier series, and applications of Fourier series to solving boundary value problems such as the heat and wave equations, sampling and signal processing.

Textbook: 

EarlyFourierAnalysisFor some topics, such as windows and kernels, we follow John H. Hubbard and Barbara Burke Hubbard, Topics in Advanced Calculus, (unpublished)

Project: Convolving again and again

Selected Worksheets and Handouts:

Homework:

Skip to toolbar