International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 758694, 11 pages
http://dx.doi.org/10.1155/2012/758694
Research Article

A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets

Felipe A. Apolonio,1 Daniel H. T. Franco,1 and Fábio N. Fagundes2

1Departamento de Física, Universidade Federal de Viçosa, Avenida Peter Henry Rolfs s/n, Campus Universitário Viçosa, 36570-000 Vioçsa, MG, Brazil
2Universidade Federal de Mato Grosso, Campus Sinop, ICNHS, Avenida Alexandre Ferronato 1.200, Distrito Industrial, 78557-267 Sinop, MT, Brazil

Received 9 February 2012; Revised 5 April 2012; Accepted 19 April 2012

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2012 Felipe A. Apolonio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the 𝑘 -space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution 𝑓 ∈ 𝒮 ( ℝ 𝑛 ) , the continuous wavelet transform of 𝑓 with respect to a conical wavelet is defined in such a way that the directional wavelet transform of 𝑓 yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of 𝑓 .