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Unformatted text preview: Mathematics 421 Essay 2 An operational view of Fourier coefficients Spring 2008 0. Introduction Although Laplace transforms were defined using an integral containing a param- eter, the computation of transforms and inverse transforms used a special list of properties. This approach is generally known as an “operational” approach. Techniques of integration appear only where necessary to derive the properties that are taken as the basic properties of the transform. Although the properties of the transform (except for linearity) are as exotic as the rules of calculus, they become easy to use with a little practice. Unfortunately, the treatment of Fourier coefficients in the text doesn’t develop a similar list of prop- erties, relying instead on classical techniques of integration. Each exercise becomes a separate calculation and results that provide structural hints to the form of the answer are hidden. It is true that Fourier series don’t always converge, in the usual pointwise sense, to the function that they represent, but the coefficients do tend to zero and the rate at which they decrease is related to the smoothness of the function. This provides a visual clue to what to expect from a computation of Fourier coefficients. It is also true, to some extent, that term-by-term differentiation of the series corresponds to differentia- tion of the function. This provides an interpretation of some rules that is easier to remember than what was needed for the proof. Part of the difficulty is that real Fourier series use trigonometric functions, so there are two functions for each positive index, and the term with index zero needs to be treated differently. The use of complex exponentials gives a single list of functions indexed by all integers, and the partial sums of the series are taken to be the sum of terms with index between n and C n for integers n . 1. Real Fourier series Given a function f.x/ on the interval OE p;pŁ , its Fourier series is given by a 2 C 1 X n D 1 a n cos n x p C b n sin n x p where the Fourier coefficients a n and b n are defined by a D 1 p Z p p f.x/dx a n D 1 p Z p p f.x/ cos n x p dx b n D 1 p Z p p f.x/ sin n x p dx The orthogonality if the functions 1; cos n x p ; sin n x p assures us that the Fourier series of one of these functions is just the single term that is the function itself. In particular, the Fourier series of a constant function is itself....
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This note was uploaded on 09/29/2009 for the course 650 421 taught by Professor Bumby during the Spring '08 term at Rutgers.
- Spring '08