This question states that if we have some function $f \in L^{1}([a,b])$ and its Fourier coefficients $\hat{f}$ satisfy $\hat{f} \in l^{2}(\mathbb{Z})$, then this implies that $f \in L^{2}([a,b])$. Can this idea be generalised? That is, if we have a function $g \in L^{1}([a,b])$ whose Fourier coefficients satisfy $\hat{g} \in l^{p}(\mathbb{Z})$ for some (or perhaps every value of) $p \geqslant 2$, can we deduce that $f \in L^{q}([a,b])$ for some values of $q \geqslant 2$?
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I have worked on a similar question, but I don't think the generalization is straightforward.
note that if $\hat f$ $ \in$ $\ell^2(\mathbb{Z})$, and by hypothesis we have $f\in L^1([a,b])$, we have that actually $f\in L^1([a,b])\cap L^2([a,b])$.
However, the correspondence between $\ell^2(\mathbb{Z})$ and $L^2([a,b])$ doesn't hold for $p>2$ because the norms on the space do not correspond. So just because we have that the Fourier coefficients
$\hat f$ $\in$ $\ell^p(\mathbb{Z})$, doesn't mean $f\in L^p([a,b])$. Further arguments must be made.
Kernel_Dirichlet
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