Fourier Series and Approximation
A natural problem is to take a wave output and decompose it into its harmonic parts. Engineers are able to do this with an oscilloscope. A real difficultly occurs when we try to put the parts back together. Mathematically, this amounts to summing up the series obtained from decomposing the original wave. In this chapter, we examine this delicate question: Under what conditions does a Fourier series converge?
We start by looking at the behaviour of the Fourier coefficients. It turns out that they go to zero; and the smoother the function, the faster they go. Then we turn to the more subtle questions of pointwise and uniform convergence. The idea of kernel functions, analogous to the Poisson kernel from the previous chapter, provides an elegant method for understanding these notions of convergence. Then we turn to the L2 norm, where there is a very clean answer. Nice applications of this include the isoperimetric inequality and sums of various interesting series. Finally, we consider applications to polynomial approximation.
KeywordsFourier Series Trigonometric Polynomial Poisson Kernel Piecewise Continuous Function Dirichlet Kernel
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