Basic Transforms of Generalized Fourier Analysis

In this chapter, we will develop the generalized analogs of several basic operators from classical Fourier analysis and calculus. Naturally, we will start by generalizing the Fourier transforms, but we will also generalize the operations of variable scaling, translation and differentiation, and we will explore how these different operations interact.

You will probably notice that the opening paragraphs of the first few sections are strikingly (numbingly?) similar to each other. You might even notice that certain technical issues regarding whether the generalized transforms are “well defined” or “unique” are only lightly touched upon in these sections. The reason for all this is that each of ...

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