10_1_2 - ORD & PRTL DIFF EQUATIONS-MATH 108-Spring...

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Name No Calculators. Closed book and notes. Write your final answers in the box if provided. You may use the back of the pages. Each question is worth 10 points. Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 1
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< x < 1 is defined by ( f, g ) = Z 1 0 f ( x ) g ( x ) dx, where bar stands for the complex conjugate. Let the functions φ 1 ( x ), φ 2 ( x ), φ 3 ( x ) , ··· be an orthonormal basis of this set and let f ( x ) be a known function in this set. (a) Derive a formula (justify what you are doing, no credit for writing the formula from memory) for the coefficient a k of the expansion f ( x ) = a 1 φ 1 ( x ) + a 2 φ 2 ( x ) + a 3 φ 3 ( x ) + ··· , 0 < x < 1 , (b) Prove that Z 1 0 | f ( x ) | 2 dx = | a 1 | 2 + | a 2 | 2 + | a 3 | 2 + ··· . HINT: recall
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10_1_2 - ORD &amp; PRTL DIFF EQUATIONS-MATH 108-Spring...

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