Communications of the
Korean Mathematical Society CKMS

In this paper, we investigate a modified $G^{2}$ transform on a class of Boehmians. We prove the axioms which are necessary for establishing the $G^{2}$ class of Boehmians. Addition, scalar multiplication, convolution, differentiation and convergence in the derived spaces have been defined. The extended $G^{2}$ transform of a Boehmian is given as a one-to-one onto mapping that is continuous with respect to certain convergence in the defined spaces. The inverse problem is also discussed.

Keywords: modified $G^{2}$ transform, Mellin-Barnes type integral, $G$ function, summable space, Fox's $H$ function, Boehmians

MSC numbers: Primary 54C40, 14E20; Secondary 46E25, 20C20