Unifying discrete and integral transforms through the use of a Banach algebra
Integral Transforms and Special Functions
A weighted (Formula presented.) space is introduced and some of its properties are highlighted. This is done with the aim of defining a space such that the transform of a class of functions exists. Certain properties of a new class of discrete and integral transforms are highlighted, including the codomain of the transform being a set of continuous functions and continuity of the operator associated with the (Formula presented.) space. The shifting and convolution properties are introduced and it is proven that the weighted (Formula presented.) space is closed under the convolution operation. Several examples which fall in this class of transforms are given, including the Dirichlet series, the Z transform and the Laplace transform. Some conditions are stated which imply that any member of a class of discrete operators is injective. Following this, it is established that the weighted (Formula presented.) space is a Banach algebra where the convolution is the underlying product.