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
Related questions
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 → ∞.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc6748d3-4cd8-403e-abcc-dc83fd70c599%2Fd152bcd5-7136-4fd5-a80a-4e6428f0b7a2%2Fj4e549i9_processed.png&w=3840&q=75)
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 → ∞.
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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