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HW7_2012

HW7_2012 - Math 602 Homework 7 The problems here are well...

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Math 602 Homework 7 The problems here are well known results. You need to show all your work and explanations. 1. (i) Check that the functions 1 , sin( nx ) , cos( nx ) , ( n = 1 , 2 , 3 . . . ) form an orthogonal system in L 2 [ - π, π ]. (ii) Normalize them to obtain an orthonormal system. (iii) Assuming that f L 2 [ - π, π ] has its Fourier series expansion f = a 0 2 + X n =1 [ a n cos( nx ) + b n sin( nx )] (1) verify that the Fourier coefficients a n and b n are given by the formulas a n = 1 π Z π - π f ( x ) cos( nx ) dx , b n = 1 π Z π - π f ( x ) sin( nx ) dx, 2. (i) Check that the functions e inx , n Z form an orthogonal system in the complex valued functions L 2 [ - π, π ]. (ii) Normalize them to obtain an orthonormal system. (iii) Assuming that f L 2 [ - π, π ] has its Fourier series f = X n = -∞ ˆ f n e inx (2) verify that the Fourier coefficients are given by ˆ f n = 1 2 π Z π - π f ( x ) e - inx dx (iv) Verify that the Fourier coefficients a n , b n , ˆ f n of a function f ( x ) are related by the formulas a n = ˆ f n + ˆ f - n for n = 0 , 1 , 2 , . . . , b n = i ( ˆ f n - ˆ f - n ) for n = 1 , 2 , . . .

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