Abstract
We apply the higher order Cauchy transforms to describe the closures of rational modules with respect to the LP norms, the uniform norm and different Lipschitz norms on a compact set in the plane.
Highlights
Let X be a compact subset of the complex lane
In recent years the problem of approximation by rational functions in the areal mean has received great attention as well as the problem of uniform approximation, but little work has been devoted to the corresponding problem of approximation by functions in a rational module due to the lack of analyticity, perhaps
The concept of rational modules arises in a natural fashion when one attempts to study rational approximation in Lipschitz norms
Summary
Let X be a compact subset of the complex lane. Let the module R(X)P be the m m space. Is a rational function with poles off X. [i], [3], [8]) as well as the problem of uniform approximation (see [5], [12]), but little work has been devoted to the corresponding problem of approximation by functions in a rational module due to the lack of analyticity, perhaps. In [i0], O’Farrell studied the relation of the problems of approximation by rational modules in different Lipschitz norms, and in the uniform norms, etc., to one another.
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