Algebras of functions on the unit circle
Algebras of functions on the unit circle
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DOI：
10.1090/s0002-9904-1973-13144-1
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发表时间：
1973-03
影响因子：
1.3
通讯作者：
D. Sarason
中科院分区：
文献类型：
--
作者：
D. Sarason
1. Introduction. I should like to discuss in this article a few recent developments in function theory on the unit circle. Rather than attempt a broad survey, I have chosen to concentrate on a fairly narrow circle of ideas which I find especially interesting and which seems suitable for presentation to a general audience. This area, while motivated in large part by functional analytic considerations, has a distinctly classical flavor, and some of the questions I shall mention here might have aroused interest forty or fifty years ago if anyone had thought to raise them at the time.The ideas we shall be concerned with fit into the general contexts of the theory of function algebras and the theory of Hardy spaces and related classes of analytic functions. I have tried to make the bulk of the article intelligible to anyone with a basic knowledge of functional analysis and function theory. Familiarity with a few technical-although by no means esoteric-notions, such as those of a Blaschke product and an inner function, will be helpful to the reader but not indispensable. I have included a few simple proofs here and there where I could do so without being led too far astray. Two minor results below, Theorems 2 and 5, have not to my knowledge been published before. Notations. We denote the open unit disk by D and its boundary, the unit circle, by dD. The independent variable on dD will be denoted either by z or by ew9 according to convenience. The basic algebras we shall be concerned with are C, the algebra of continuous complex valued functions on dD, and L00, the algebra of (classes of) essentially bounded, measurable, complex valued functions with respect to Lebesgue measure on dD. These are Banach algebras under the supremum and essential supremum norms, respectively. We recall that, by the Gelfand-Naimark theorem, L00 is isometrically isomorphic to C (X) for a certain compact Hausdorff space X (the" maximal ideal space" of L00). We denote by A and H00 the algebras of functions in C and L00, respectively, whose Fourier coefficients with negative indices vanish. The functions in A are the boundary functions for the functions that are analytic and uniformly continuous in D; those in Hœ are the boundary
1. Introduction. I should like to discuss in this article a few recent developments in function theory on the unit circle. Rather than attempt a broad survey, I have chosen to concentrate on a fairly narrow circle of ideas which I find especially interesting and which seems suitable for presentation to a general audience. This area, while motivated in large part by functional analytic considerations, has a distinctly classical flavor, and some of the questions I shall mention here might have aroused interest forty or fifty years ago if anyone had thought to raise them at the time.The ideas we shall be concerned with fit into the general contexts of the theory of function algebras and the theory of Hardy spaces and related classes of analytic functions. I have tried to make the bulk of the article intelligible to anyone with a basic knowledge of functional analysis and function theory. Familiarity with a few technical—although by no means esoteric—notions, such as those of a Blaschke product and an inner function, will be helpful to the reader but not indispensable. I have included a few simple proofs here and there where I could do so without being led too far astray. Two minor results below, Theorems 2 and 5, have not to my knowledge been published before. Notations. We denote the open unit disk by D and its boundary, the unit circle, by dD. The independent variable on dD will be denoted either by z or by ew9 according to convenience. The basic algebras we shall be concerned with are C, the algebra of continuous complex valued functions on dD, and L00, the algebra of (classes of) essentially bounded, measurable, complex valued functions with respect to Lebesgue measure on dD. These are Banach algebras under the supremum and essential supremum norms, respectively. We recall that, by the Gelfand-Naimark theorem, L00 is isometrically isomorphic to C (X) for a certain compact Hausdorff space X (the" maximal ideal space" of L00). We denote by A and H00 the algebras of functions in C and L00, respectively, whose Fourier coefficients with negative indices vanish. The functions in A are the boundary functions for the functions that are analytic and uniformly continuous in D; those in Hœ are the boundary