Vol. 18, No. 2, 1966

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nth order integral operators associated with Hilbert transforms

G. O. Okikiolu

Vol. 18 (1966), No. 2, 343–360

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 Bα(n)(f) and Iα(n)(f) defined by

Bα(n)(f) = (1)n(2n +1)!
---π----(P.V.) −∞f(t)
×            ∑n −1 (− 1)m(αt− αx)2m+1
(t− x)2n+2 dt
Iα(n)(f) = (1)n(2n +2)!
---π----(P.V.) −∞f(t)
×∑n    (−1)m(αt−αx)2m
(t− x)2n+3 dt.
Inversion processes by which f may be expressed in terms of the Bα(n) and Iα(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
Received: 15 April 1965
Published: 1 August 1966
G. O. Okikiolu