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"Unifying discrete and integral transforms through the use of a Banach " by Thomas Futcher and Marianito R. Rodrigo

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. ....

Banach Algebra , Ntegral Transforms , Hifting Property ,

"A general class of integral transforms and an expression for their con" by Thomas Futcher and Marianito R. Rodrigo

A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms which are yet to be named. The assumptions allow for certain properties to exist for these integral transforms, such as the shifting and convolution properties. Further conditions are presented which allow for the integral transform operator to be injective on a specific class of functions. Applications of the results are provided and analytical solutions to differential equations are obtained. Numerical solutions are then compared to the analytical solutions to demonstrate the usefulness of this class of transforms for solving a particular family of differential equations. ....

Differential Equations , Ntegral Transforms , Nversion Formula , Hifting Property ,