Instructor's guide

Orthogonal Series and Boundary Value Problems

Evans M. Harrell II*

*(c) Copyright 1994,1995 by Evans M. Harrell, II. All rights reserved.

This chapter is concerned with the question of how Fourier series converge, and the amount of attention you put into it will depend on the student's orientation to the subject. For undergraduate engineers, it may be too theoretical, and they may be content to see visual evidence of the convergence of Fourier series. Students at this level may be distracted by the example of L2 convergence which is not pointwise, especially as the point is that this phenomenon does not occur for Fourier series. For these students, you could simply point out by example that the Fourier series will not converge at jump points and potentially at the end of the interval of definition. They should be aware that theorems are stated here in case they need to refer to them at some future date.

Different advice applies to mathematics majors, for whom the discussion of convergence of Fourier series in different senses illuminates some of the concepts they encounter in analysis.

I have broken off the theorems for the sine series and cosine series into further links, since they should probably be skipped on a first reading. When these series are introduced, they can be explained as arising from the odd and even periodic extensions of a function, in which case they are special cases of the full Fourier series It is then obvious what happens at jumps and end points. (And for the sine series for 0 ≤ x ≤ L it is obvious anyway that the sum at the endpoints is 0.)

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