Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On finite Hilbert transforms

Author: Kevin F. Clancey
Journal: Trans. Amer. Math. Soc. 212 (1975), 347-354
MSC: Primary 47G05; Secondary 44A15
MathSciNet review: 0377598
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator $ {H_E}$ defined on one of the spaces $ {L^p}(E)(1 < p < \infty )$ by

$\displaystyle {H_E}f(x) = {(\pi i)^{ - 1}}\int_E {f(t){{(t - x)}^{ - 1}}\;dt;} $

here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for $ {H_E}$ to be Fredholm on $ {L^p}(E)$, when $ p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $ p < 2$ it is shown that the operator $ {H_E}$ has an infinite dimensional null space. The method of proof is constructive.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47G05, 44A15

Retrieve articles in all journals with MSC: 47G05, 44A15

Additional Information

Article copyright: © Copyright 1975 American Mathematical Society