Principles of Fourier Analysis, 2nd Edition

19

The Fourier Integral Transforms

We are now ready for the first official set of definitions for the Fourier transforms. These definitions are directly inspired by the integral formulas (17.8) and (17.9) on page 255, and will, accordingly, be called the Fourier integral transforms. 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.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 ...