"A journey of a thousand miles begins with a nice cup of coffee."
The Fourier transform, first developed by J. Fourier in the early 1800’s to solve heat flow problems, is one of the most widely used integral transforms and has found numerous applications in mathematics, engineering and science. Not all functions are Fourier transformable, however, and, in some situations, this can restrict the applicability of the Fourier transform as a tool for solving problems.
If f(t) is a suitably integrable function, then its Fourier transform is defined by
A more general, though less obvious, way to define F is to require that it be the function satisfying
where
The second approach only requires that the product
be "suitably integrable."
In the early 1950’s L. Schwartz produced a space of test functions such that the above could define the Fourier transform of any locally integrable, polynomially bounded function on the real line. More recently, Professor Howell has developed a space of analytic test functions that allowed the definition of the Fourier transform for any locally integrable, exponentially bounded function. This is exciting because many problems in which the Fourier transform could be applicable involve functions that are exponentially (but not polynomially) bounded. Moreover, because these test functions are analytic, certain techniques and ideas from complex analysis can be introduced to Fourier analysis in a straightforward manner. For example, in this new theory there is an analog to the classical "translation" identity of Fourier analysis in which the translations may be complex, even though the functions may not be naturally defined off the real axis.