nth order integral operators associated with Hilbert transforms

G. O. Okikiolu

Vol. 18 (1966), No. 2, 343–360

DOI: 10.2140/pjm.1966.18.343

Abstract

In this paper, we study the n-th order analogues of certain integral operators allied to the Hilbert transform and to Dirichlet’s integrals. Most of the results known to be true for n = 0 are proved for the general case. Some cases in which the analogy fails are also considered. Among the integrals considered are transforms B_α⁽ⁿ⁾(f) and I_α⁽ⁿ⁾(f) defined by

Bα(n)(f)	= (−1)n(P.V.) ∫ −∞∞f(t)
	× dt
Iα(n)(f)	= (−1)n(P.V.) ∫ −∞∞f(t)
	× dt.

Inversion processes by which f may be expressed in terms of the B_α⁽ⁿ⁾ and I_α⁽ⁿ⁾ operators are also obtained. The results proved in the paper are also shown to be true for integrals defined with respect to a continuous parameter ν. These integrals reduce to the original ones when ν is an integer.

Mathematical Subject Classification

Primary: 44.30

Milestones

Received: 15 April 1965

Published: 1 August 1966