Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces.

Li, Chun, McIntosh, Alan, and Qian, Tao. "Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces.." Revista Matemática Iberoamericana 10.3 (1994): 665-721. <http://eudml.org/doc/39468>.

@article{Li1994,
abstract = {In the Fourier theory of functions of one variable, it is common to extend a function and its Fourier transform holomorphically to domains in the complex plane C, and to use the power of complex function theory. This depends on first extending the exponential function eixξ of the real variables x and ξ to a function eizζ which depends holomorphically on both the complex variables z and ζ .Our thesis is this. The natural analog in higher dimensions is to extend a function of m real variables monogenically to a function of m+1 real variables (with values in a complex Clifford algebra), and to extend its Fourier transform holomorphically to a function of m complex variables. This depends on first extending the exponential function ei(x,ξ) of the real variables x ∈ Rm and ξ ∈ Rm to a function e(x,ζ) which depends monogenically on x = x + xLeL ∈ Rm+1 and holomorphically on ζ = ξ + iη ∈ Cm.},
author = {Li, Chun, McIntosh, Alan, Qian, Tao},
journal = {Revista Matemática Iberoamericana},
keywords = {Algebras de Clifford; Transformada de Fourier; Dominios de Lipschitz; Convolución; Funciones de variable compleja; Lipschitz surfaces; Clifford algebra; Fourier transform; singular convolution operators},
language = {eng},
number = {3},
pages = {665-721},
title = {Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces.},
url = {http://eudml.org/doc/39468},
volume = {10},
year = {1994},
}

TY - JOUR
AU - Li, Chun
AU - McIntosh, Alan
AU - Qian, Tao
TI - Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces.
JO - Revista Matemática Iberoamericana
PY - 1994
VL - 10
IS - 3
SP - 665
EP - 721
AB - In the Fourier theory of functions of one variable, it is common to extend a function and its Fourier transform holomorphically to domains in the complex plane C, and to use the power of complex function theory. This depends on first extending the exponential function eixξ of the real variables x and ξ to a function eizζ which depends holomorphically on both the complex variables z and ζ .Our thesis is this. The natural analog in higher dimensions is to extend a function of m real variables monogenically to a function of m+1 real variables (with values in a complex Clifford algebra), and to extend its Fourier transform holomorphically to a function of m complex variables. This depends on first extending the exponential function ei(x,ξ) of the real variables x ∈ Rm and ξ ∈ Rm to a function e(x,ζ) which depends monogenically on x = x + xLeL ∈ Rm+1 and holomorphically on ζ = ξ + iη ∈ Cm.
LA - eng
KW - Algebras de Clifford; Transformada de Fourier; Dominios de Lipschitz; Convolución; Funciones de variable compleja; Lipschitz surfaces; Clifford algebra; Fourier transform; singular convolution operators
UR - http://eudml.org/doc/39468
ER -