Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces
- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 10, Nº 3, 1994, págs. 665-721
- Idioma: inglés
- DOI: 10.4171/rmi/164
- Títulos paralelos:
- Algebras de Clifford, transformadas de Fourier y operadores de convolución singulares sobre superficies de Lipschitz.
- Enlaces
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Resumen
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 eix? of the real variables x and ? to a function eiz? which depends holomorphically on both the complex variables z and ? .
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,?) of the real variables x Î Rm and ? Î Rm to a function e(x,?) which depends monogenically on x = x + xLeL Î Rm+1 and holomorphically on ? = ? + i? Î Cm.
