We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on N0∞∪(−N0∞). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H 1
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We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on N0∞∪(−N0∞). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H 1
function almost everywhere to T ∞
. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.
@article{1d4e6a02-9afd-47b0-953e-1f7f7ab23e4c,
abstract = {{<p><br>
We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on N0∞∪(−N0∞). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H <br>
<sup>1</sup><br>
function almost everywhere to T <br>
<sup>∞</sup><br>
. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions. <br>
</p>}},
author = {{Aleman, Alexandru and Olsen, Jan Fredrik and Saksman, Eero}},
issn = {{0021-7670}},
language = {{eng}},
number = {{1}},
pages = {{429--447}},
publisher = {{Magnes Press}},
series = {{Journal d'Analyse Mathematique}},
title = {{Fatou and brothers Riesz theorems in the infinite-dimensional polydisc}},
url = {{http://dx.doi.org/10.1007/s11854-019-0006-x}},
doi = {{10.1007/s11854-019-0006-x}},
volume = {{137}},
year = {{2019}},
}