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 $p_n^2$?

If $f$ is squaresummable wrt the Lebesgue measure $l$ and $d\mu=fdl$, then you easily get the $L^1$convergence. 


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 $\sum\left\frac{\mu(\{z_n\})}{zz_n}\right^2$. 

