hw3 - Mathematics Department Stanford University Math 175...

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Unformatted text preview: Mathematics Department Stanford University Math 175 Homework 3 1. Let L 2 ([- , ]) denote the real L 2 functions f : [- , ] R , equipped with the inner product ( f,g ) = 1 R - fg. (i) Prove that 1 2 , cos( x ) , sin( x ) , cos(2 x ) , sin(2 x ) ,..., cos( nx ) , sin( nx ) ,... is a complete orthonormal sequence for L 2 ([- , ]) , and show that, with respect to this orthonormal sequence, the Fourier series of a function f L 2 ([- , ]) can be written a / 2 + k =1 ( a k cos( kx ) + b k sin( kx )) , where a k = ( f, cos( kx )) , k = 0 , 1 , 2 ,... and b k = ( f, sin( kx )) , k = 1 , 2 ,... . Hint: You can assume the completeness of e inx for the complex L 2 space L 2 C ([- , ]) . (ii) Write down Parsevals identity in this case for a given function f L 2 ([- , ]) . 2. Give an example of a sequence of non-negative C functions on [0 , 1] such that R [0 , 1] f k , yet such that f k ( x ) converges to zero at no point of [0 , 1] ....
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