Vol. 25, No. 1, 1968

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ISSN: 0030-8730
On a class of convolution transforms

Zeev Ditzian

Vol. 25 (1968), No. 1, 83–107
Abstract

In this paper the convolutian transform

      ∫
∞                   ∗
f(x) =  −∞G (x− t)φ (t)dt ≡ (G φ )(x)
(1.1)

whose kernel G(t) is the Fourier transform of [E(iy)]1 where E(s) is defined by

       bs ∞∏                 − 1
E(s) = e    (1 − s∕ak)exp(sReak ),
k=1                                 ∑
Reb = b and   |ak|−2 < ∞   (1.2)
will be studied. An inversion theory similar to that achieved when ak of (1.2) are real will be obtained. The results will show that under certain rather weak conditions, an infinite subsequence ak(i) of ak can satisfy
                          π-
min{|arg ak(i)|,|arg− ak(i)|} ≧ 4.

Classes of transforms will be introduced that allow the occurrence of min{|arg ak|,|arg ak|}π∕4 for all k.

Mathematical Subject Classification
Primary: 44.25
Milestones
Received: 20 December 1966
Published: 1 April 1968
Authors
Zeev Ditzian