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I just want to find some standard reference to the following result: let $(a_k)_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L_1(T)$ function in the sense of tempered distributions; then $(a_k)_k \in \ell_2$.

Can someone help on this? And make the statement more precise, if it needs to?

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This seems to be stated, with copious references, and plenty of related results in Jean Pierre Kahane's survey Lacunary Taylor and Fourier series, Bulletin AMS 1964

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  • $\begingroup$ Thanks! It seems to state more than what is actually proved in some of the references, but I have yet to check other references made there, in particular what seems to be the main one: the classic book on trigonometric series by Zygmund. $\endgroup$ Commented May 26, 2012 at 14:10
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This particular result can also be found as Theorem 1.4b in Chapter V of Katznelson's Introduction to Harmonic Analysis (2nd ed., Dover).

Of course much more is known, which I guess would be found or referenced in the survey article that Igor Rivin has already mentioned.

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