A new approach to representation theory for convolution transforms

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 transforms with kernels in Class II. A representation theorem for convolution transforms of Class II with determining functions of bounded variation in (—∞, ∞), is given. Also, representation theorems involving determining functions which are integrals of functions in the Orlicz class L_M(—∞, ∞) are obtained.