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Is Fourier series also defined firstly as $L^1([a,b])→l^∞$, and can be extended to $L^p([a,b])→l^q$ where $p∈(1,2],1/p+1/q=1$ in some way similar to Fourier transform?

I didn't find the answer in the Wikipedia article for Fourier series, or some books I have. So references are also appreciated.

What can we say about generalized Fourier series?

Thanks and regards!

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up vote 2 down vote accepted

Since $L^p[a,b] \subset L^1[a,b]$, there's no need for an "extension". The fact that the Fourier series maps $L^p$ into $\ell^q$ for $p \in [1,2]$ is the Hausdorff-Young theorem.

EDIT: The same proof, using the Riesz-Thorin theorem, shows that this works for generalized Fourier series with respect to any orthonormal basis of functions $u_n$ with $|u_n|$ uniformly bounded.

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