There are two different ways by
which one obtains representation theorems for the Laplace transform. One way is to
impose integral conditions on the inverse operator; and the other way is to
impose summation conditions without referring to the inverse operator.
Representation theorems for the convolution transform have hitherto been
obtained by imposing integral conditions on the inverse operator, and no
attempt has been made to impose summation conditions. We obtain here some
representation theorems, which involve summation conditions, for convolution
transforffls wilh kernels in Class IH. A representation theorem for convolution
transforms of Class II with determining functions of hounded variation in
(−∞,∞), is given. A# so, represenlation theorems involving determining
functions which are integrals of functions in the Orlicz class LM(−∞,∞) are
obtained.
