Interpolation sets for the algebra of generalized analytic functions

Abstract

According to the well-known Rudin–Carleson theorem (see, e. g., [1], p. 125, 6.3.1), compact subsets of a circumference which are interpolation sets for the disk-algebra are the sets with the null Lebesgue measure. However, even for a polydisk the situation is more sophisticated. If a compact subset of an n-dimensional torus (n > 1) is an interpolation set, then it has the null Lebesgue measure (the Haar measure), but the contrary is not necessarily true (ibid., theorem 6.3.4). Several sufficient conditions for polydisks are described in [1] (see §§ 6.2, 6.3 there) and [2]. They imply the smallness (in a sense) of the compact under consideration. One studies the case of a ball in [3] (see also references therein). An interesting generalization of the concept of analyticity connected with the abstract harmonic analysis is obtained by R. Arens and I. M. Singer in [4]; many papers have originated from this work. In these papers the main subject is the case of the Archimedean ordering (see [5], Chap. VII; [6], Chap. VIII). The contemporary state of this research is described in the monograph [7] and the review [8] (see also [9]). In this paper we study subsets of compact Abelian groups which are interpolation sets for the algebra of generalized analytic functions in the Arens–Singer sense. The concept of the generalized analyticity used in this paper, to put it more precisely, is slightly less restrictive than that in [4]; it follows the approach proposed in [10]. At the same time, we establish certain properties of generalized analytic functions; they, probably, will represent an independent interest. As a rule, we do not assume the Archimedean ordering. We prove the mean value theorem for generalized analytic functions, consider the polynomial approximatibility for the algebra of these functions, and obtain simple conditions, ensuring the interpolation property of compact sets. Some results of the paper were announced in [11].