Siciak–Zahariuta extremal functions, analytic discs and polynomial hulls

Analytic Discs, Global Extremal Functions and Projective Hulls in Projective Space

Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in P. As a corollary we get a characterization of the projective hull for connected compact sets in P by the existence of analytic discs.

5 Extremal Discs and Analytic Continuation of Product Cr Maps

Analytic Approximation of Matrix Functions and Dual Extremal Functions

We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain spec...

Some Extremal Problems for Analytic Functions

The paper mainly concerns with functions f , analytic in S : |Imz| < 1 and bounded by a constant M > 1. We state sharp estimates for supR |f ′| under the additional condition supR |f | ≤ 1. Using these estimates we deduce well-known Bernstein’s inequality and some its generalizations for entire functions of a finite type with respect to an arbitrary proximate order. Parallely we investigate als...

Extremal properties of polynomial threshold functions

In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: • Almost every Boolean function has PTF degree at most n2 + O( √ n log n). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [32] and Aspnes, Beigel, Furst, and Rudich [4] up to lower order terms. • ...