Verifying the Fourier Series is in Hilbert Space

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1. Nov 27, 2015

kostoglotov

The text does it thusly:

But, before I got to here, I attempted it in a different way and want to know if it is still valid.

Check that $f^{*}f$ is finite, by checking that it converges.

$$f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x + b_2^2sin^22x + ...$$

should converge provided the a's and b's are both less than 1 in absolute value.

Is this also a valid way to show that our Fourier Series is an okay vector to work with?

2. Nov 27, 2015

BvU

What is it you want to show ? Orthogonality is OK, and normality is OK (provided you use the correct normalization constants). Completeness ? For all functions you can write the recipe to get the fourier transform.

So yes, the Fourier basis is OK.

3. Nov 29, 2015