Hardy, Paley-Wiener and Bernstein spaces in Clifford analysis

• Published in: Complex variables and elliptic equations Vol. 62; no. 9; pp. 1314 - 1328
• Main Authors: Franklin, D. J., Hogan, J. A., Larkin, K. G.
• Format: Journal Article
• Language: English
• Published: Colchester Taylor & Francis 02-09-2017; Taylor & Francis Ltd
• Subjects: Clifford algebra; Clifford Fourier transform; Fourier transforms; Hardy spaces; Integrals; Mathematical functions; Paley-Wiener theorem
• ISSN: 1747-6933; 1747-6941
• DOI: 10.1080/17476933.2016.1250411

We describe the relationship between the growth conditions of monogenic extensions of square-integrable functions f in terms of pointwise bounds or bounds on integral averages on the one hand, and the support of the Fourier transform
or its annihilation by certain higher-dimensional analogues of the signum function on the other. We review known results involving a function's monogenic extension and their classical Fourier transform. These results are extended to the Clifford-Fourier transform of Brackx, De Schepper and Sommen. The equivalence of the pointwise bounds and the bounds on the integral averages is observed as a consequence.