ABSTRACT
This chapter presents selected results on the Fourier theory: It starts with 1D-Fourier asymptotics, in particular the method of the stationary phase. Then we go over to multi-dimensional orthonormal periodic polynomials and their role in Fourier (orthogonal) expansions. Our main interest concerning the Fourier transform in the Euclidean space R q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429355103/509b923d-963e-4e6d-a3f5-f127f3172564/content/inline-math8_1.jpg"/> is the relation between functions being not-necessarily absolutely integrable over R q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429355103/509b923d-963e-4e6d-a3f5-f127f3172564/content/inline-math8_2.jpg"/> as well as periodic with respect to a lattice Λ ⊂ R q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429355103/509b923d-963e-4e6d-a3f5-f127f3172564/content/inline-math8_3.jpg"/> . As a consequence, we are immediately led to the process of “periodization” as the bridging tool to the Poisson summation formula in R q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429355103/509b923d-963e-4e6d-a3f5-f127f3172564/content/inline-math8_4.jpg"/> . For our purposes of lattice point summation, however, it must be pointed out that the convergence criteria to justify the process of periodization in R q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429355103/509b923d-963e-4e6d-a3f5-f127f3172564/content/inline-math8_5.jpg"/> is different from those developed in the 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. However, this assumption is not relevant for our lattice point and resulting sampling identities because of the specific alternating character of the constituting sums.
