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Working with Parseval's Theorem

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A general form of Parseval's Theorem says that if two functions are expanded in a Fourier Series

f(x) =1/2 ao + Sigma [(an cos(nx)) + bn sin(nx)]

g(x) 1/2 ao' + Sigma [(an' cos(nx)) + bn' (sin(nx)]

Then the average value, < f(x)g(x)>, is:
1/4 ao = sigma[an an' + bn bn'] prove this and using any two functions

Please give an example.

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The Fourier eigenfunctions and are orthonormal for any integers n, m in the interval , which means that:
Of course, for any integers n, m
So assume we can write the function as a series expansion of the eigenfnctions:
To find the coefficients we simply multiply both sides of the equation by an eigenfunction.
For we get:
Since the summation is over n and not m we can bring the eigenfunction into the sum:

Solution Summary

The Parseval's Theorem is analyzed. The average value is provided. Two functions are analyzed.