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On Weierstrass’ monsters in the disc algebra. (English) Zbl 1407.30031

Starting with the example given by Weierstrass, along the time a lot of studies were done on functions which are continuous everywhere on the domain of definition, but nowhere differentiable, the so-called Weierstrass monsters. In this paper the study of Weierstrass monsters is extended not only in the topological sense, but also in the algebraic sense. In the beginning of the paper, some notation and necessary preliminaries are presented. Also some auxiliary results are given. The main sections of the paper are the third and the fourth one.
In the third section Weierstrass monsters in the disc algebra \(A(\mathbb D)\) are studied. There the authors prove the existence of a dense vector subspace of functions which are nowhere differentiable on the unit circle.
In the fourth section the authors prove that, for Jordan domains \(\Omega\) with piecewise analytic boundary, there exist dense linear subspaces of infinitely generated subalgebras of \(A(\Omega)\) all of whose nonzero members are not differentiable at almost any point of the boundary.
The paper ends with an appendix where the study is considered under a measure-theoretical point of view.

MSC:

30H50 Algebras of analytic functions of one complex variable
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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Full Text: Euclid