BOUNDARY UNIQUENESS THEOREMS FOR ALMOST ANALYTIC FUNCTIONS, AND ASYMMETRIC ALGEBRAS OF SEQUENCES

Abstract

This article concerns algebras of -functions in the disk such that , where and . 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 , 1$ SRC=http://ej.iop.org/images/0025-5734/64/2/A03/tex_sm_3311_img7.gif/>, , 0}p_n/n^2=\infty$ SRC=http://ej.iop.org/images/0025-5734/64/2/A03/tex_sm_3311_img9.gif/>, is analytic in the disk , and as for all , where , then and if has nontangential boundary values equal to the values of on some subset of the circle of positive Lebesgue measure. Here certain regularity conditions are imposed on and . Uniqueness and factorization theorems for almost analytic functions are applied to the description of translation-invariant subspaces in the asymmetric algebras of sequences Bibliography: 15 titles.