Proof of Bessel’s inequality and show thataó +> (až + b?).k=1(b) By considering all possible products in the multiplication of s,(x) byitself, show thatn1- [s, (x)]*dx = a+ > (až + b?).k-1(c) By writing1[f(x)- s„(x)]*dx1Pdx-f(x)s,(x)dx + - |[s,(x)]*dx=11*dx-ab - > (až + b?),2k=1conclude that1+dx,2k=1and 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, arecoefficients of f(x), then a, → 0 and b,→ 0 as n → ∞. 1af + > (a? + b? )(*)k=1-TProve this by the following steps:(a) For any n > 1, defineS„(x) =1ao + > (az coskx+b; sin kx)k=1

We need to prove Bessel’s inequality: 12a02+∑k=1∞ak2+bk2≤1π∫-ππf(x)2dx. For (a), For any n≥1, define…

Answered: Proof of Bessel’s inequality

Math

Advanced Math

Proof of Bessel’s inequality

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 → ∞.

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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