Harold S. Shapiro

Generalized analytic continuation

One of the most important notions in connection with the study of analytic functions is that of analytic continuation. Several recent investigations have encountered situations where there is associated with an analytic function in a domain, another function analytic in a contiguous domain which can lay claim to being a “continuation” of the original function, even though the original function is nowhere continuable in the classical sense. Such a situation arises, for instance, if the first function has nontangential limiting values almost everywhere on some smooth arc of the boundary and the second function has identical nontangential limiting values almost everywhere on that arc. As follows from a theorem of Lusin and Privalov, the two functions then uniquely determine one another.

What is a Quadrature Domain

We give an overview of the theory of quadrature domains with indications of some if its ramifications.

Certain Hilbert spaces of entire functions

1. Introduction. The research reported on in the present note was motivated by the following Proposition (F), due to Ernest Fischer ([5], see also [4] for an earlier version; actually Fischer proved a more general result, but the special case suffices as a point of departure for our discussion) : (F) Let P denote a homogeneous polynomial in si, • • • , z k with complex coefficients. Then every polynomial in Si, • • • , z k has a unique representation QP+R where

An elementary proof that the biharmonic green function of an eccentric ellipse changes sign

P R. Garabedian showed in 1951 that the Green function for the biharmonic boundary value problem with vanishing Dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse. This refuted a conjecture made by J. Hadamard in 1908. The proof of Garabedian was based on kernel functions; the present note gives an elementary proof.

The Friedrichs operator of a planar domain. II

We consider relations between the Friedrichs operator and constructive aspects of the Dirichlet problems for the Laplace and \({\overline \partial ^{2}}\)-operator. Then we investigate the Fourier expansions in the eigenfunctions of the Friedrichs operator. A link between a generalized Friedrichs operator and minimal nodes quadratures for complex polynomials of a fixed degree is explained. We initiate a discussion of the boundary Friedrichs operator on the Hardy space of a domain. The transformation law of the Friedrichs operator under conformal mappings leads to a modified version of it, based on a symbol function; this object will turn out to be closely related to Hankel operators. We obtain some results concerning which symbols correspond to compact operators.

On Zero and One Points of Analytic Functions

Let f be an analytic function in the open unit disk that assumes the values 0 and 1 at least once and an unequal number of rimes in a disk of radius 1 10-5 entered at the origin. We prove that fmust assume the value 0 or 1 elsewhere in the unit disk. This result is motivated by a control engineering design problem which is also presented.

A minimal area problem in conformal mapping

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU such thatf(0)=f′(0)−1=0. The main result of the paper is that area (f(U) ≥27π/7)(2-α)−2 for allf∈S such that |f″(0)|=2α, 1/2<α<2. This solves a long-standing extremal problem for the class of functions considered.

On the classical Dirichlet problem in the plane with rational data

We consider the Dirichlet problem for the Laplace operator with rational data on the boundary of a planar domain. Our main results include a characterization of the disk as the only domain for which all solutions are rational and a characterization of the simply connected quadrature domains as the only ones for which all solutions are algebraic of a certain type.

The cauchy-kowalevskaya theorem and generalizations

We consider various generalizations of the classical holomorphic Cauchy problem. One of our main results asserts existence and uniqueness of holomorphic solutions to problems of the following type: Where ƒ is a holomorphic functionh is a homogeneous polynomial of degree, say m, and P(z,D) is a holomorphic partial differential operator of the same degree m subjected to certain conditions that depend on h. This result contains as a special case the Cauchy-Kowalevskaya theorem.

Functional equations and harmonic extensions

In this paper, we first consider the posibility of extending to the exterior of a region the solution to a Dirichlet problem. We limit ourselves to the situation where the boundary curve is analytic and the boundary data is polynomial or real-entire. We then turn to the problem of hte existence of solutions to the functional equation f(P(z))-g(Q(z)) = k(z),, where P and Q are given polynomials and k is a given meromorphic function. Central to our results is a generalization of a theorem of A. and C. Renyiconcerning periodic functions of the form f(P). We conclude by showing how the two problems ae related.