Vol. 35, No. 2, 1970

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A new approach to representation theory for convolution transforms

Dany Leviatan

Vol. 35 (1970), No. 2, 441–449

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.

Mathematical Subject Classification
Primary: 44.25
Received: 7 August 1969
Published: 1 November 1970
Dany Leviatan