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

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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 ) = n =1 b n sin( nπx ) 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( mπx ). In the summation, this gives zero for n = m , and 1 0 sin 2 ( mπx ) = 1 / 2 for n = m , resulting in the equation b m = 2 1 0 f ( x ) sin( mπx ) 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 | f ( x ) | p dx is finite for some p > 1 has a Fourier series that converges almost everywhere to f ( x ) [except at isolated points]. At points where f ( x ) has a jump discontinuity, the Fourier series converges to the midpoint of the jump. So, as long as one does not care about crazy di-

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