Geodesic
A remark on the partial sums in Hardy spaces
Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 149 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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We prove that a function $f$, analytic in the unit disc, belongs to the Hardy space $H^1$ if and only if $ \sum^n_{j=0} \frac1{+1} \|s_j f\| = O (łog n) \quad (n\to\infty), $ where $s_jf$ are the partial sums of the Taylor series of $f$. As a corollary we have that, for $f\in H^1$, $ \sum^n_{j=0} \frac1{j+1} \|f-s_jf\| = o(łog n), $ The analogous facts for $L^1$ do not hold.
Classification : 30D55
Keywords: Hardy space 
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     author = {M. Pavlovi\'c},
     title = {A remark on the partial sums in {Hardy} spaces},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {149 },
     year = {1995},
     volume = {_N_S_58},
     number = {72},
     zbl = {0945.30029},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1995_N_S_58_72_a16/}
}
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M. Pavlović. A remark on the partial sums in Hardy spaces. Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 149 . http://geodesic.mathdoc.fr/item/PIM_1995_N_S_58_72_a16/