ABSTRACT
This chapter presents basic material on the classical Fourier theory in higher dimensions. It starts with multi-dimensional orthonormal periodical polynomials and their role in Fourier expansions (in Section 7.1). The Fourier transform is discussed in the standard reference space of (Lebesgue) absolutely integrable functions over the Euclidean space Rq (in Section 7.2). Our main interest is the relation between functions being absolutely integrable over Rq
as well as periodical with respect to the unit lattice Zq ⊂ Rq. In consequence, we are immediately led to the process of “periodization” as “bridging tool”, i.e., the Poisson summation formula in Rq. The results are formulated for dimensions q ≥ 2 (in Section 7.3). For our lattice point summation (as intended, e.g., in Chapters 10 and 14), however, it must be emphasized that our technique of realizing the process of periodization in Rq, q ≥ 2, is different from those developed in the classical literature, e.g., by E.M. Stein, G. Weiss [1971]. In fact, these authors verify the Poisson summation formula under the strong assumption of the absolute convergence of all occurring sums. The essential calamity in lattice point theory, however, is the convergence behavior of the Fourier series expansion of the lattice function in dimensions q ≥ 2. This fact is a striking difference to the one-dimensional theory. Moreover, in contrast
formula. As a powerful remedy, the inversion formula of Fourier integrals can be understood in the terminology of certain means (such as Gauß-Weierstraß or Abel-Poisson transforms in Section 7.4). Furthermore it turns out that the integral transform for discontinuous functions possessing a “potato-like” regular region as a local support is critical for Fourier inversion. Nevertheless, at least in the case of spherical geometry, the Hankel transform provides a way out for handling alternating, not absolutely convergent series expansions in terms of Bessel functions.
