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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators
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by Antonio O. Farias PDF
Trans. Amer. Math. Soc. 167 (1972), 279-289 Request permission

Abstract:

A Titus transformation $T = \langle \alpha ,v\rangle$ is a linear operator on the vector space of ${C^\infty }$ mappings from the circle into the plane given by $(Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,f’(t)]v$, where $\alpha$ is a nonnegative, ${C^\infty }$ function on the circle ${S^1}$. Let $\tau$ denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of $\tau$ of a degenerate curve, ${\alpha _0}{v_0}$, where ${\alpha _0}$ is a ${C^\infty }$ function on ${S^1}$ and ${v_0}$ is fixed in the plane ${R^2}$. A ${C^\infty }$ mapping $f:{S^1} \to {R^2}$ is called properly extendable if there is a ${C^\infty }$ mapping $F:{D^ - } \to {R^2}$, D the open unit disk and ${D^ - }$ its closure, such that ${J_F} \geqq 0$ on $D,{J_F} > 0$ near the boundary ${S^1}$ of ${D^ - }$ and $F{|_{{s^1}}} = f$. A ${C^\infty }$ mapping $f:{S^1} \to {R^2}$ 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, $y(t) = - \smallint _0^{2\pi } {k(s)x(t - s)ds}$. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert’s transform of potential theory, $y(t) = \smallint _0^{2\pi } {\cot (s/2)x(t - s)ds}$, 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
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 167 (1972), 279-289
  • MSC: Primary 57D40; Secondary 43A80
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0295374-6
  • MathSciNet review: 0295374