Explanation Given: { ϕ } n = 1 ∞ be an orthonormal system with respect to a positive weight function w ( x ) on [ a , b ] and let f be a continuous function on [ a , b ] . Calculation: Consider the function. f ( x ) ∼ ∑ n = 1 ∞ c n ϕ n ( x ) ... ( 1 ) Since, f ( x ) is square-integral function on [ a , b ] . ∫ a b [ f ( x ) ] 2 d x < ∞ … ( 2 ) The Fourier series associated is, f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) … ( 3 ) Consider another function S N , S N ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) … ( 4 ) Subtract Equation (3) from Equation (4) and square, [ f ( x ) − S N ( x ) ] 2 = f 2 ( x ) − 2 f ( x ) S N ( x ) + S N 2 ( x ) … ( 5 ) Integrate Equation (5) with respect to x . ∫ a b [ f ( x ) − S N ( x ) ] 2 d x = ∫ a b [ f 2 ( x ) − 2 f ( x ) S N ( x ) + S N 2 ( x ) ] d x = ∫ a b [ f 2 ( x ) ] d x − 2 ∫ a b [ f ( x ) S N ( x ) ] d x + ∫ a b [ S N 2 ( x ) ] d x … ( 6 ) Consider the integral, ∫ a b [ S N 2 ( x ) ] d x = ∫ a b [ a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) ] 2 d x = a 0 2 4 ∫ a b d x + ∑ n = 1 ∞ a n 2 ∫ a b cos 2 ( n x ) d x + b n ∫ a b sin 2 ( n x ) d x = a 0 2 4 ∫ a b d x + ∑ n = 1 ∞ 1 2 a n 2 [ 1 + cos ( 2 n x ) ] a b + 1 2 b n [ 1 − cos ( 2 n x ) ] a b = ( b − a ) 2 [ a 0 2 4 + ∑ n = 1 ∞ a n 2 + b n ] Consider the integral,

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ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Chapter 11.3, Problem 25E

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The equation ∑n=1∞cn2≤∫f2(x)w(x)dx is true if f(x)∼∑n=1∞cnϕn(x).

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Chapter 11.3, Problem 24E

Chapter 11.3, Problem 26E

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The equation ∑ n = 1 ∞ c n 2 ≤ ∫ f 2 ( x ) w ( x ) d x is true if f ( x ) ∼ ∑ n = 1 ∞ c n ϕ n ( x ) .

Solution Summary: The author analyzes the Fourier series associated with f(x)=a_02+

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The equation ∑n=1∞cn2≤∫f2(x)w(x)dx is true if f(x)∼∑n=1∞cnϕn(x).

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Chapter 11 Solutions