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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. ...

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. Numerica...