Orientationpreserving mappings, a semigroup of geometric transformations, and a class of integral operators
Author:
Antonio O. Farias
Journal:
Trans. Amer. Math. Soc. 167 (1972), 279289
MSC:
Primary 57D40; Secondary 43A80
MathSciNet review:
0295374
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Abstract: A Titus transformation is a linear operator on the vector space of mappings from the circle into the plane given by , where is a nonnegative, function on the circle . Let denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of of a degenerate curve, , where is a function on and is fixed in the plane . A mapping is called properly extendable if there is a mapping , D the open unit disk and its closure, such that on near the boundary of and . A mapping is called normal if it is an immersion with no triple points and all its double points are transversal. The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping. An application is made to a class of integral operators of the convolution type, . It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert's transform of potential theory, , which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202953746
PII:
S 00029947(1972)02953746
Keywords:
Normal immersions,
extendable maps,
holomorphic maps,
Hilbert transform
Article copyright:
© Copyright 1972
American Mathematical Society
