Short Courses in Discrete and Continuous Fourier Analysis

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An innovative short course on Fourier analysis is being given to groups of 18 college faculty at SIU's Touch of Nature Conference Center during the summers of 1989 and 1990. Lectures are discussing Fourier's representation of discrete and continuous functions; the operator identities that facilitate a deep understanding of the rules and symmetries of this discipline; the fast Fourier transform from Gauss` interpolation bases on derivation to modern variants such as Bracewell's recently patented fast Hartley transform; an elementary new theory of generalized functions, i.e., distributions, that are widely used in mathematics and its applications; and selected applications of these ideas from the mathematical, physical, and engineering sciences. Participants are using personal computers (with especially designed user friendly software) to deepen their understanding of the rules and operator identities developed in the lectures, to experience fast, i.e., 0(N log N), as opposed to slow, i.e., O(N2), N-digit arithmetic, and to solve practical problems using fast transform methods. The lecture notes, exercise sets, and computer discs from the short courses are preparing the participants to introduce corresponding interdisciplinary undergraduate Fourier analysis courses in their home institutions or to enrich and modernize existing courses in numerical analysis, Fourier series, ordinary and partial differentials equations, etc., already in place in the curriculum. The Division of Mathematical Sciences is also providing support for this project. The University is contributing to the cost of the program, with the individual participant's home institutions also bearing a cost of the participation.