Alec Matheson

The Cauchy Transform

Overview Preliminaries The Cauchy transform as a function The Cauchy transform as an operator Topologies on the space of Cauchy transforms Which functions are Cauchy integrals? Multipliers and divisors The distribution function for Cauchy transforms The backward shift on

COMPACT COMPOSITION OPERATORS ON BMOA

H^2

The solution of the diffusion equation in two space variables subject to the specification of mass

Clark measures The normalized Cauchy transform Other operators on the Cauchy transforms List of symbols Bibliography Index.

A numerical procedure for diffusion subject to the specification of mass

We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk U that are Poisson integrals of functions on ∂U , that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator Cφ : BMOA → BMOA relates to the shape of φ(U) near ∂U , introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.

Completely continuous composition operators

The Problem ut = uxx+uyy, 0 < x,y < 1,0 < t ≤ T; u(x,y,0) = ƒ(x,y), 0 ≤ x,y ≤ 1; u (o,y,t) = g0(y,t), u(1,y,t) = g1(y,t), 0<y1,0<t≤T;u(x,1,t)=h1(x,t), u(x,0,t)= μ(t)h0(x), 0<x < 1,0 < t ≤ T; and , where ƒ g0,g1, h0, h1, s, and m are known functions while the function u and μ are unknown, is reduced to an equivalent integral equation for the unknown function μ(t). Existence and unicity are demonstrated. A numerical procedure is discussed along with some results of numerical experiments.

Cauchy transforms of characteristic functions and algebras generated by inner functions

Abstract An application of the maximum principle yields an a priori estimate for the derivative u x of the solution u of u t = u xx +s 0 , subject to u(x, 0) = f(x), 0 , and the specification of mass ∫ 0 b(t) u(x,t) d x = m(t), 0 . From this a priori estimate the continuous dependence of the solution u on the data is established. The maximum principle can also be applied to a numerical scheme for the derivative of u . Thus convergence is shown for an elementary numerical scheme. The article concludes with the results of some numerical experiments.

COMPUTABLE ANALYSIS AND BLASCHKE PRODUCTS

A composition operator Tbf = f o b is completely continuous on H1 if and only if lbl < 1 a.e. If the adjoint operator Tb is completely continuous on VMOA, then Tb is completely continuous on H1 . Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying Banach space.

On Carleson Embeddings Of Star-invariant Supspaces

We prove that Cauchy transforms of characteristic functions of subsets of positive measure of the unit circle are equidistributed in the unit disk in the sense that the L P -closure of the polynomial algebra in these Cauchy transforms coincides with the L p -closure of the polynomial algebra in a canonical inner function. As a corollary to this result we find conditions describing when the polynomial algebra in two singular inner functions determined by point masses is dense in the Hardy spaces H P .

Boundary spectra of uniform Frostman Blaschke products

We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.

An Observation about Frostman Shifts

Embeddings of star-invariant subspaces K p θ of H p determined by an inner function θ are studied. Using a method of Aleksandrov, it is shown that the embedding into L p (μ) is compact whenever μ satisfies a certain vanishing condition with respect to the singular set of θ. Conversely, if the embedding is compact, and θ is a so-called one-component inner function, the measure is shown to satisfy a different vanishing condition.