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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2 Answers
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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.
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$\begingroup$ Yes, and another simple example is the Dirac measure at a point of the circle: then the averaged limit exists, say, for the Cesaro summation method. There are some reasons to think that the fact can be true for every complex measure and that it can be already known. I would very much appreciate any references/arguments. $\endgroup$ May 9, 2011 at 19:04
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$\begingroup$ 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. $\endgroup$ May 10, 2011 at 19:57
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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 $\sum\left|\frac{\mu(\{z_n\})}{z-z_n}\right|^2$.