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Proof of Bessel’s inequality

Proof of Bessel’s inequality

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.4: Values Of The Trigonometric Functions
Problem 23E
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Proof of Bessel’s inequality

and show that aó + > (až + b?). k=1 (b) By considering all possible products in the multiplication of s,(x) by itself, show that n 1 - [s, (x)]*dx = a+ > (až + b?). k-1 (c) By writing 1 [f(x)- s„(x)]*dx 1 Pdx- f(x)s,(x)dx + - |[s,(x)]*dx = 1 1 *dx-ab - > (až + b?), 2 k=1 conclude that 1 + dx, 2 k=1 and from this complete the proof. Observe that the convergence of the series on the left side of (*) implies the following corollary of Bessel's inequality: If a, and b, are coefficients of f(x), then a, → 0 and b, → 0 as n → ∞.
Transcribed Image Text:and show that aó + > (až + b?). k=1 (b) By considering all possible products in the multiplication of s,(x) by itself, show that n 1 - [s, (x)]*dx = a+ > (až + b?). k-1 (c) By writing 1 [f(x)- s„(x)]*dx 1 Pdx- f(x)s,(x)dx + - |[s,(x)]*dx = 1 1 *dx-ab - > (až + b?), 2 k=1 conclude that 1 + dx, 2 k=1 and from this complete the proof. Observe that the convergence of the series on the left side of (*) implies the following corollary of Bessel's inequality: If a, and b, are coefficients of f(x), then a, → 0 and b, → 0 as n → ∞.
1 af + > (a? + b? ) (*) k=1 -T Prove this by the following steps: (a) For any n > 1, define S„(x) = 1 ao + > (az coskx+b; sin kx) k=1
Transcribed Image Text:1 af + > (a? + b? ) (*) k=1 -T Prove this by the following steps: (a) For any n > 1, define S„(x) = 1 ao + > (az coskx+b; sin kx) k=1
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