# Convergence of squares of the moduli of partial sums of Fourier series

Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel means of $p_n$ have limits at almost all points of the unit circle. My question is: Are there any facts about the averaged convergence of $|p_n|^2$?

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If $f$ is square-summable wrt the Lebesgue measure $l$ and $d\mu=fdl$, then you easily get the $L^1$-convergence.
It seems that the limit must exist almost everywhere and if $\mu$ has no atoms then it coincides with $|averaged \ lim \ p_n|^2$. For point masses some extra summands appear. If so, I am sure this must be known. –  user14971 May 10 '11 at 19:57
Let my answer be as above, namely, if $\mu$ has no point masses, then we get squared moduli of the limits of the boundary values of the Cauchy tramsforms of $\mu$. For the purely point part of $\mu$ one has to add $\sum\left|\frac{\mu(\{z_n\})}{z-z_n}\right|^2$.