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Unformatted text preview: Fourier series Generalizations Applications Chapter 11: Fourier Series Sections 1  5 Chapter 11: Fourier Series Fourier series Generalizations Applications Definition Convergence 1. Fourier series We saw before that the set of functions { 1 , cos( x ) , sin( x ), cos(2 x ) , sin(2 x ) , , cos( mx ) , sin( mx ) , } , where m is a nonnegative integer, forms a complete orthogonal basis of the space of square integrable functions on [ , ]. This means that we can define the Fourier series of any square integrable function on [ , ] as f ( x ) = a + n =1 [ a n cos( nx ) + b n sin( nx )] , where a = 1 2  f ( x ) dx and, for n 1, a n = 1  f ( x ) cos( nx ) dx and b n = 1  f ( x ) sin( nx ) dx . Chapter 11: Fourier Series Fourier series Generalizations Applications Definition Convergence Convergence of Fourier series If f is continuously differentiable on [ , ] except at possibly a finite number of points where it has a lefthand and a righthand derivative, then the partial sum f N ( x ) = a + N n =1 [ a n cos( nx ) + b n sin( nx )] with the a i defined above, converges to f ( x ) as N if f is continuous at x . At a point of discontinuity , the Fourier series converges towards...
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This note was uploaded on 12/01/2009 for the course MATH 322 taught by Professor Staff during the Spring '08 term at University of Arizona Tucson.
 Spring '08
 STAFF
 Math, Fourier Series

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