Principles of Fourier Analysis, 2nd Edition

31

A Starting Point for the Generalized Theory

The theory we have developed thus far for the Fourier transform (the “classical theory”) provides a number of mathematical tools that could be useful in solving many problems of real interest — provided those problems only involve classically transformable functions. Unfortunately, this is often not the case. Even fairly simple real-world problems are likely to involve constant functions, polynomials and exponential functions, none of which can be “Fourier transformed” using the classical theory. Since we want to deal with such functions, let us now turn our attention to finding a more general way of defining the Fourier transform.

31.1  Starting Points

Suppose f is a classically transformable ...