Is there an explicit example of a one-sided trigonometric series $\sum_{n=0}^\infty a_ne^{int}$ converging everywhere but for which the sequence of the coefficients is not in $\ell^2$?
everywhere convergent trigonometric series
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power-series
trigonometric-series
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0The coeffients are real numbers? Maybe you refer to the *sequence* of coefficients converging in $l^2$.. – 2017-02-20
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0By looking up $l^2$ it seems you may be seeking such an everywhere convergent trig series for which the sum of the squares of the $a_n$ diverges. Please confirm this in the question, or at least in a comment. Another issue: By saying "one sided" trig series, do you just refer to the index $n$ ranging over non-negative integers, as opposed to over all integers? – 2017-02-20
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0coffeemath your second comment is not correct. $\sum_{n=1}^\infty (-1)^n\frac{1}{\sqrt n}$ converges, but not $\sum \frac{1}{n}$. – 2017-02-21
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0You're right, will delete my previous comment. – 2017-02-22