A remark on the partial sums in Hardy spaces
Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 149
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Zbl
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.
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/
@article{PIM_1995_N_S_58_72_a16,
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/}
}