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 $(Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,fβ(t)]v$, 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 Hilbertβs 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.References
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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