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 L^{2} 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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