sines - Fourier Sine Series Examples 16th November 2007 The...

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Unformatted text preview: Fourier Sine Series Examples 16th November 2007 The Fourier sine series for a function f ( x ) defined on x [0 , 1] writes f ( x ) as f ( x ) = X n =1 b n sin( nx ) for some coefficients b n . Because of orthogonality, we can compute the b n very simply: for any given m , we integrate both sides against sin( mx ). In the summation, this gives zero for n 6 = m , and R 1 sin 2 ( mx ) = 1 / 2 for n = m , resulting in the equation b m = 2 Z 1 f ( x ) sin( mx ) dx. Fourier claimed (without proof) in 1822 that any function f ( x ) can be expanded in terms of sines in this way, even discontinuous function. This turned out to be false for various badly behaved f ( x ) , and controversy over the exact conditions for convergence of the Fourier series lasted for well over a century, until the question was finally settled by Carleson (1966) and Hunt (1968): any function f ( x ) where R | f ( x ) | p dx is finite for some p > 1 has a Fourier series that converges almost everywhere to...
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This note was uploaded on 08/10/2008 for the course MATH 151 taught by Professor Sc during the Spring '08 term at Rutgers.

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sines - Fourier Sine Series Examples 16th November 2007 The...

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