John B. Garnett

Positive length but zero analytic capacity

and A(aj, 8j) is the disc {z -ajI 0 but y(E) = 0. However Vituskins proof is quite complicated and contains many typographical errors. We give a simpler counterexample, and compare ours to Vituskins.

Pointwise bounded approximation and Dirichlet algebras

Abstract Those open sets U of S2 for which A(U) is pointwise boundedly dense in H∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in CR(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H∞(U).

Constructive techniques in rational approximation

This paper consists of several loosely organized remarks on the constructive methods for rational approximation developed by Vitushkin in [13]. These remarks are grouped under three headings. The first topic, taken up in ?1, illustrates the simplest case of the approximation scheme, and shows how it can be used to give simple proofs of rational approximation theorems on a class of infinitely connected compact sets. Special cases of these theorems have been proved by other methods by Fisher [8] and Zalcman [16]. This section also serves as an introduction to the integral operator TO, which is used in ??2 and 3. The second topic, involving pointwise bounded approximation, occupies ??2 through 5. In ?2, necessary and sufficient conditions are given on an open plane set U in order that every bounded analytic function on U be a pointwise limit on U of a bounded sequence of uniformly continuous analytic functions on U. This result, together with Mergelyans theorem, yields the Farrell-Rubel-Shields theorem (cf. [12]) on pointwise bounded approximation by polynomials, and its extension to finitely connected sets (cf. Corollary 3.3) by Ahern and Sarason [1]. It will be noted, however, that the constructive techniques alone do not give the best possible bounds on the norms of the approximating functions. In ?3 we obtain partial results on pointwise bounded approximation by rational functions. ??4 and 5 are devoted to constructing an example of a set for which pointwise bounded approximation by uniformly continuous analytic functions obtains, whereas approximation by rational functions fails. The final topic, relegated to ?6, involves extending Vitushkins techniques to vector-valued functions, in order to obtain results on uniform approximation by analytic functions of several complex variables. A related result has been given by

Characterization of radially symmetric finite time blowup in multidimensional aggregation equations

This paper studies the transport of a mass

Algebras generated by inner functions

\mu

A doubling measure on ℝ^{} can charge a rectifiable curve

in

The Regularity of the Boundary of a Multidimensional Aggregation Patch

\mathbb{R}^d, d \geq 2,

A convex minimization model in image restoration via one-dimensional Sobolev norm profiles

by a flow field

Large sets of zero analytic capacity

v= -\nabla K*\mu

Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

. We focus on kernels