Unifying discrete and integral transforms through the use of a Banach algebra. Issue 12 (2nd December 2022) - Futcher, Thomas; Rodrigo, Marianito R.
ABSTRACT: A weighted L 1 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 L 1 space. The shifting and convolution properties are introduced and it is proven that the weighted L 1 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 L 1 space is a Banach algebra where the convolution is the underlying product.