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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 3(7), Pages 324–340 (Mi msb1745)

Boundary uniqueness theorems for almost analytic functions, and asymmetric algebras of sequences

A. A. Borichev

Abstract: This article concerns algebras of $C^1$-functions in the disk $|z|<1$ such that $|\overline\partial f(z)|<w(1-|z|)$, where $w\uparrow$, and $\int_0\log\log w^{-1}(x) dx=+\infty$. For these functions a factorization theorem (on representation of each such function as the product of an analytic function and an antianalytic function, to within a function tending to zero as the boundary is approached) and a number of boundary uniqueness theorems are proved. One of these theorems is equivalent to a result generalizing the classical Levinson–Cartwright and Beurling theorems and consisting in the following. If $f(z)=\sum_{n<0}a_nz^n$, $|z|>1$, $|a_n|<e^{-p_n}$, $\sum_{n>0}p_n/n^2=\infty$, $F$ is analytic in the disk $|z|<1$, and $|F(z)|=o(w^{-1}(c(1-|z|)))$ as $|z|\to1$ for all $c<\infty$, where $w(x)=\exp(-\sup_n(p_n-nx))$, then $f=0$ and $F=0$ if $F$ has nontangential boundary values equal to the values of $f$ on some subset of the circle $|z|=1$ of positive Lebesgue measure. Here certain regularity conditions are imposed on $p$ and $w$. Uniqueness and factorization theorems for almost analytic functions are applied to the description of translation-invariant subspaces in the asymmetric algebras of sequences
$$\mathfrak A=\{\{a_n\};\forall c\enskip\exists c_1:|a_n|<c_1e^{-cp_n}, n<0, \exists c, \exists c_1:|a_n|<c_1e^{cp_n}, n\geqslant0\}.$$

Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:2, 323–338

Bibliographic databases:

UDC: 517.5
MSC: Primary 30E25; Secondary 30H05

Citation: A. A. Borichev, “Boundary uniqueness theorems for almost analytic functions, and asymmetric algebras of sequences”, Mat. Sb. (N.S.), 136(178):3(7) (1988), 324–340; Math. USSR-Sb., 64:2 (1989), 323–338

Citation in format AMSBIB
\Bibitem{Bor88} \by A.~A.~Borichev \paper Boundary uniqueness theorems for almost analytic functions, and asymmetric algebras of sequences \jour Mat. Sb. (N.S.) \yr 1988 \vol 136(178) \issue 3(7) \pages 324--340 \mathnet{http://mi.mathnet.ru/msb1745} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=959485} \zmath{https://zbmath.org/?q=an:0677.30003|0663.30002} \transl \jour Math. USSR-Sb. \yr 1989 \vol 64 \issue 2 \pages 323--338 \crossref{https://doi.org/10.1070/SM1989v064n02ABEH003311} 

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This publication is cited in the following articles:
1. Dynkin E., “The Pseudoanalytic Extension”, J. Anal. Math., 60 (1993), 45–70
2. E. M. Dyn’kin, “The Pseudoanalytic Extension”, J. Anal. Math, 60:1 (1993), 45
3. Borichev A., “Beurling Algebras and the Generalized Fourier Transform”, Proc. London Math. Soc., 73:Part 2 (1996), 431–480
4. Jean Esterle, “Countable inductive limits of frechet algebras”, J Anal Math, 71:1 (1997), 195
5. J ESTERLE, A VOLBERG, “Sous-espaces invariants par translations bilatérales de certains espaces de Hilbert de suites quasi-analytiquement pondérées”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326:3 (1998), 295
6. Harlouchet I., “Closed Ideals of Certain Quasi-Analytic Beurling Algebras on the Unit Circle”, J. Math. Pures Appl., 79:9 (2000), 863–899
7. Gady Kozma, Alexander Olevskiı̆, “Maximal smoothness of the anti-analytic part of a trigonometric null series”, Comptes Rendus Mathematique, 338:7 (2004), 515
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