Convolution Transform for Boehmians

Abstract

The theory of Boehmians was initiated by J. Mikusinski and P. Mikusinski in 1981 and later, several applications of Boehmians were discovered by P. Mikusinski, D. Nemzer and others. The main aim of this paper is to study certain properties of integral transform, which carries f(t) into F(x) as a convolution, through a kernel G(x - y), given by the map f(t) → F(x) = ∫ R f(t)G(x - t) dt. We treat the convolution transform as a continuous linear operator on a suitably defined Boehmian space. In this paper, we construct a suitable Boehmian space on which the convolution transform can be defined and the generalized function space L' c,d can be imbedded. In addition to this, our definition extends the convolution transform to more general spaces and that the definition remains consistent for L' c,d elements under a suitable condition on c and d. We also discuss the operational properties of the convolution transform on Boehmians and finally end with an example of a Boehmian which is not in any L' c,d but is convolution transformable.