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Consider $L^2([0,1],\mathbb{R})$.

Then, $$1, \sqrt{2} \cos(2 \pi j x), \sqrt{2} \sin(2 \pi j x ), \quad j =1,2,\ldots$$

is a Schauder basis on $L^2([0,1], \mathbb{R})$.

I am curious, how does this generalize for the Sobolev space $W^{2,2}([0,1]^2, \mathbb{R}^2)$ ?

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  • $\begingroup$ The second answer here seems relevant: mathoverflow.net/questions/261941/… $\endgroup$
    – Christian Remling
    Commented Aug 16, 2021 at 0:35
  • $\begingroup$ See the article:"Développements des distributions en séries de fonctions orthogonales. Séries de Legendre et de Lagurre" by Marianne Guillemot-Teissier which can easily be found online. $\endgroup$
    – bathalf15320
    Commented Aug 16, 2021 at 4:31
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    $\begingroup$ Starting from the trigonometric system $e_j(x)$, sure you get a Hilbert basis of $W^{2,2}([0,1]^2)$ of the form $c_{jk}e_j(x)e_k(y)$ $\endgroup$
    – Pietro Majer
    Commented Aug 17, 2021 at 10:29

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