Convergence and Fourier’s Conjecture
Our initial derivation of the formulas for the Fourier series was based on the conjecture (“Fourier’s bold conjecture”) that any “reasonable” periodic function can be represented by an infinite linear combination of sines and cosines. However, the only evidence thus far given of this conjecture’s validity has been a few pictures and the fact that, if it weren’t true, then this book probably would not have been written.
It is time we look at our conjecture more carefully. After all, if we plan to use Fourier series in real applications, we really should know how well any “reasonable” function of interest can be represented by its Fourier series. How accurately, for example, will any particular partial ...
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