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?



convergence of the coefficients of lacunary series
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


