Bessels Inequality and Parsevals Equality

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Bessels Inequality and Parsevals Equality
- Mean Convergence of Fourier Series
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BESSELS INEQUALITY
Let f(x) be piecewise continuous on the interval
If the Fourier series of f(x) is given by
then
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The above inequality is called Bessels
inequality.
Proof Let n ? 1.
Define
(sn(x) is often referred to as the sum to n
terms of the Fourier series of f(x).)
Multiplying both sides by
and
and
integrating between
we get
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(using the definitions of ans and bns) that
We note that
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Multiplying both sides by and integrating between
and we get
(using the orthogonality of trigonometric
functions).

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Now
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Since
we get
Letting
we get the Bessels inequality.
Corollary
The Fourier coefficients
as
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Mean convergence of Fourier series PARSEVALS
EQUALITY
it can be
If f(x)2 is integrable on
shown that
We thus say
in the mean
and write
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Hence the previous calculations show that,
Bessels inequality becomes an equality
called PARSEVALS EQUALITY.
Application 1
The Fourier series of f(x) x is given by
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Thus
(and an 0 for all n )
and
Hence Parsevals equality gives
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Application 2 The Fourier series of f(x) x2 is
given by
Thus
for all n 1,2,3,..
Hence, Parsevals equality gives
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or
Application 3 Let f(x) x2. The Fourier Sine
series for f(x) x2
is
in
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where
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Also
Hence we get
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Thus
END OF FOURIER SERIES