Unifying discrete and integral transforms through the use of a Banach algebra

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.

KEYWORDS:

• Integral transforms
• convolution
• shifting property
• Banach algebra

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 44A35
• 44A05
• 45P05

Disclosure statement

No potential conflict of interest was reported by the author(s).