ABSTRACT
We are now ready for the first official set of definitions for the Fourier transforms. These definitions are directly inspired by formulas (17.8) and (17.9) on page 249 and require the computation of integrals over R . Accordingly, we will refer to these transforms as the Fourier integral transforms. Also, to ensure our integrals are well defined, we will only use these definitions for the Fourier transforms of functions in A , the set of piecewise continuous, absolutely integrable functions on the entire real line, R .1 This will not be completely satisfactory. Many functions of interest are not absolutely integrable. Consequently, one of our goals will later be to intelligently extend the basic definitions given in this chapter so that we can deal with interesting functions that are not absolutely integrable.
