10_1_2

# 10_1_2 - ORD PRTL DIFF EQUATIONS-MATH 108-Spring 2010-EXAM...

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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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## This note was uploaded on 09/29/2010 for the course MATH 108 taught by Professor Trangenstein during the Spring '07 term at Duke.

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10_1_2 - ORD PRTL DIFF EQUATIONS-MATH 108-Spring 2010-EXAM...

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