We are now ready to develop the basic elements of a generalized function theory that will support, in a reasonably natural fashion, a much more general way of defining Fourier transforms. Naturally, we do not want this new theory to be something completely different from what we already know. It should extend the well-known classical theory of functions, and should, in spirit and notation, be similar enough to the classical theory that we can begin using the new theory with a minimum of psychological pain. This gives us the following design parameters for our new theory:
1. The set of generalized functions should contain all classical functions of interest (in our case, all exponentially integrable functions).
2. The ...
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