An Extremal Problem for Characteristic Functions
نویسندگان
چکیده
Suppose E is a subset of the unit circle T and H∞ ⊂ L∞ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znH∞. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.
منابع مشابه
Gaussian Subordination for the Beurling-selberg Extremal Problem
We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e−πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance [3], [4], [...
متن کاملExtremal Solutions of the Two-dimensional L-problem of Moments, Ii
All extremal solutions of the truncated L-problem of moments in two real variables , with support contained in a given compact set, are described as characteristic functions of semi-algebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal function of a naturally associated hyponormal operator with rank-one self-commutator, provides a natural de...
متن کاملKeywords. Davenport{schinzel Sequence; Tree; Extremal Problem 0 Extremal Problems for Colored Trees and Davenport{schinzel Sequences
In the theory of generalized Davenport{Schinzel sequences one estimates the maximum lengths of nite sequences containing no subsequence of a given pattern. Here we investigate a further generalization, in which the class of sequences is extended to the class of colored trees. We determine exactly the extremal functions associated with the properly 2-colored path of four vertices and with the mo...
متن کاملA Variation on Selberg’s Approximation Problem
Let α ∈ C in the upper half-plane and let I be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of I that vanishes at the point α. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishi...
متن کاملGoldberg’s constants
0 < A0 = A1 = A3 < A2 = A4 < 0.0319, and extremal functions for A0 and A2 exist, but extremal functions for A1, A3 and A4 they do not exist. This is a simple normal families argument; it also shows that extremal functions for A0 have the boundary of the ring {z : A0 < |z| < 1} as the natural boundary, and extremal functions for A2 have the unit circle as the natural boundary. The problem is to ...
متن کامل