ABSTRACT
In this chapter we provide well known one-dimensional tools and methods of basic importance for this work. The point of departure is the Gamma function. A central topic is the Stirling formula. Particular attention is paid to generalizations of the Riemann-Lebesgue theorem known from the Fourier theory. We continue with some procedures of the stationary phase that turn out to be extremely helpful to secure the convergence of weighted lattice point sums including Fourier integrals (as discussed, for example, in Subsection 13.2). Finally, our considerations are dedicated to Abel-Poisson and GaußWeierstraß limit relations as canonical preparations for the Abel-Poisson and Gauß-Weierstraß transforms in multi-dimensional Euclidean spaces Rq (as studied in Section 7.4 and applied to the lattice point theory in Section 12.1).
First our purpose is to introduce the classical Gamma function. Its essential properties are explained (for a more detailed discussion the reader is referred, e.g., to N. Nielsen [1906], E.T. Whittaker, G.N. Watson [1948], N.N. Lebedev [1973], C. Mu¨ller [1998], and the references therein). In particular, the Stirling
