Rudin-Carleson Theorem Research Articles

Pick and peak interpolation

We show how Pick interpolation and interpolation on peak interpolation sets can be combined in an abstract uniform algebra setting. In particular, as a special case, the Rudin-Carleson theorem can be combined with the classical Pick interpolation theorem on the disc.

Fatou’s Interpolation Theorem Implies the Rudin–Carleson Theorem

The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).

A Rudin–Carleson theorem for planar vector fields

This paper generalizes the Rudin–Carleson theorem for homogeneous solutions of locally solvable real analytic vector fields.

Interpolation sets for the algebra of generalized analytic functions

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].

Certain Characterizations of Carathéodory Domains

Let G be a simply connected, bounded domain on the plane with the boundary Γ and let P(Γ) be a uniform closure of polynomials on Γ. It is shown that the Rudin-Carleson Theorem about analytic extensions from zero measure boundary sets is valid for P(Γ) if and only if G is a Caratheodory domain and \({\bar G}\) does not separate the plane. These conditions are also equivalent to maximality of P(Γ) in C(Γ).

Two analogs of the Rudin-Carleson theorem

In this article, a new asymptotic estimate for embeddings of singular measures into H1,∞ is proved and an application to multidimensional analogs of the theorem of Rudin-Carleson is given. Bibliography: 5 titles.

Recent developments on the F. and M. Riesz theorem

Then μ is absolutely continuous with respect to Lebesgue measure on T. From this theorem, we can deduce the famous Rudin–Carleson theorem: Let F be a closed subset of T. Every continuous function on F can be extended to an analytic function inside the unit disk with boundary value on F equal to the given continuous function on F if and only if F is a Lebesgue measure zero set. Later, Helson and Lowdenslager generalized the F. and M. Riesz theorem to compact Abelian groups with ordered duals. De Leeuw and Glicksberg, Doss and Yamaguchi shortly afterwards obtained a number of related results.

A generalized rudin-carleson theorem

The classical Rudin-Carleson theorem has been generlized in several ways in the Literature. In this paper we would like to extend this theorem to Frechet space valued functions More precisely, we answer affirmatively the following question. Given a Frechet space XL:, and a continuous function of f: F, X where F is a closed subset of Lebesgue measure zero on the unit circle in the finite complex plane, does there exist a continuous fuction g from the closed unit disc into X which is also holomorphic in the open unit disc such g|F = f? For the proof, we use an extended version of Mergeleyan's theorem for functions with values in a locally convex topological vector space.

Free interpolation in H∞ and in some other classes of functions. II

The second part of the survey (the first part has been published in Vol. 47 of the same publication) is basically devoted to the description of the set of all restrictions of analytic functions from the class X onto a fixed subset E of their domain of definition. Contents: the proof of Earl's theorem on interpolation by Blaschke products and some of their applications to interpolation on subclasses of the space H∞; interpolation in the spaces of all functions analytic in the unit circle with a sequence of coefficients from ℓp(1⩽p⩽∞); interpolation in some Banach spaces consisting of functions which are analytic in the open unit circle and continuous in its closure (in this case the results are similar to the well-known Rudin-Carleson theorem). In conclusion, one considers problems of interpolation in spaces of analytic functions which are smooth up to the boundary.

Fourier coefficients of the Rudin-Carleson extensions

If F is a closed subset of the unit circle in C of Lebesgue measure 0 then by the Rudin-Carleson theorem every continuous function /: F -> C has a norm-preserving extension belonging to the disc algebra. If we prescribe some Fourier coefficients of the extension g then in general the norm of g will exceed the norm of /. In the paper we relate this problem to an extremal problem and give optimal estimations of the norms of extensions with prescribed Fourier coefficients. In particular, we give a precise description of those functions / which have normpreserving extensions g with finitely many prescribed Fourier coefficients.

A Rudin-Carleson theorem for uniformly convergent Taylor series.

A Rudin-Carleson theorem for uniformly convergent Taylor series.

Erratum to "A Generalization of the Rudin-Carleson Theorem"

p(x) < +(x) + c' for all x E X. is not valid in full generality. However, the statement itself holds in the general form. A proof is in [2]. Since that proof is quite cumbersome, it should be of interest to show how the proof in [1] can easily be repaired. This modified proof is valid also in the case when F is not a G0-set. Replace the paragraph on p. 342 of [1] from line 14 from bottom to line 5 from bottom by the following: We define an upper semicontinuous function 4 on X by

Shorter Notes: The Rudin-Carleson Theorem for Vector-Valued Functions

Shorter Notes: The Rudin-Carleson Theorem for Vector-Valued Functions

The Rudin-Carleson theorem for vector-valued functions

The following generalization of the Rudin-Carleson theorem is proved. Let X X be a complex Banach space and let f : F → X f:F \to X be a continuous function, where F F is a closed subset of the unit circle in C C of Lebesgue measure zero. There exists a continuous function g g from the closed unit disc to X X which is analytic on the open unit disc and satisfies (i) g | F = f g|F = f , (ii) max | z | ⩽ 1 | | g ( z ) | | = max z ϵ F | | f ( z ) | | {\max _{|z| \leqslant 1}}||g(z)|| = {\max _{z\epsilon F}}||f(z)|| .

A Generalization of the Rudin-Carleson Theorem

A Generalization of the Rudin-Carleson Theorem

A generalization of the Rudin-Carleson theorem

The purpose of this paper is to prove a common generalization of a theorem due to T. W. Gamelin [3] and a theorem due to Z. Semadeni [5]. Both these results are generalizations of E. Bishop’s abstract version of the well-known RudinCarleson theorem [2].

A general Rudin-Carleson theorem

which is analytic on D -L such that F(z) =f(z) for all z in S. It is the purpose of this paper to generalize this theoren'r- Before stating the generalization, we remark that the Rudin-Carleson theorem is closely related to a theorem of F. and M. Riesz, which states that any (finite, complex-valued, Baire) measure on L which is orthogonal to all continuous functions F on D which are analytic on D-L is absolutely continuous with respect to Lebesgue measure dO on L. The proofs of the two theorems show that the results of RudinCarleson and of F. and M. Riesz are closely related. We shall state an abstract theorem which shows that the Rudin-Carleson theorem is a direct consequence of the F. and M. Riesz theorem. This abstract theoremi will permit an automatic generalization of the RudinCarleson result to any situation to which the F. and M. Riesz result can be generalized. The theorem to be proved reads as follows: THEOREM 1. Let C(X) be the uniform.ly-normed Bw-nach space of all continuous complex-valued functions on a compact Hausdorff space X. Let B be a closed subspace of C(X). Let B' consist of all (finite, complexvalued, Baire) measures ,u on X such that ffd,u =0 for all f in B. Let ,u be the regular Borel extension of the Baire measure ,u. Let S be a closed subset of X with the property that ,u(T) = 0 for every Borel subset T of S and every L,u in B'. Let f be a continuous complex-valued function on S and A a positive function on X such that jf(x) I <A(x) for all x in S. Then there exists F in B with I F(x) I <A(x) for all x in X and F(x) =f(x) for all x in S. If X is taken to be the set L defined above and B is taken to be those functions in C(L) which are restrictions to L of functions in