ABSTRACT
This chapter presents selected results on the Fourier theory. It starts with 1D-Fourier asymptotics, in particular the method of the stationary phase. The chapter focuses on multi-dimensional orthonormal periodic polynomials and their role in Fourier (orthogonal) expansions. For our purposes of lattice point summation, it points out that the convergence criteria to justify the process of periodization in Euclidean space is different from those developed in the literature, e.g., by E. M. Stein, G. Weiss. The integral transform for discontinuous functions possessing a “potato-like” regular region as a local support remains critical for Fourier inversion. In this respect, it can be shown that the Hankel transform provides a way out which allows it to handle alternating, not absolutely convergent series expansions in terms of Bessel functions. The chapter presents some basic results on the multi-dimensional Fourier transform on Euclidean spaces.
