Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.
A class of distributions supported by certain noncompact regular sets K are identified with continuous linear functionals on . The proof is based on a parameter version of the Seeley extension theorem.
Solving a problem of L. Schwartz, those constant coefficient partial differential operators are characterized that admit a continuous linear right inverse on or , an open set in . For bounded with -boundary these properties are equivalent to being very hyperbolic. For they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial .
The commutative neutrix convolution product of the functions and is evaluated for and all . Further commutative neutrix convolution products are then deduced.