On the number of lattice points in a ball (Q6594638)

The aim of this interesting paper is to elucidate of connection between the problems and methods of finding the number of lattice points in a ball in Euclidean space considered by the author and \textit{J. J. Holt} [Duke Math. J. 83, No. 1, 203--248 (1996; Zbl 0859.30029)], by \textit{D. R. Heath-Brown} [in: Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30--July 9, 1997. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter. 883--892 (1999; Zbl 0929.11040)], by \textit{J. E. Mazo} and \textit{A. M. Odlyzko} [Monatsh. Math. 110, No. 1, 47--61 (1990; Zbl 0719.11063)], and the problems and methods of extremal functions associated with the Beurling-Selberg problem and its generalizations considered by the author [Bull. Am. Math. Soc., New Ser. 12, 183--216 (1985; Zbl 0575.42003)], by the author and Holt [loc. cit.], by \textit{E. Carneiro} and \textit{F. Littmann} [Am. J. Math. 139, No. 2, 525--566 (2017; Zbl 1366.41016)]. \NTo do this it is necessary to construct of entire functions of exponential type in one and several variables with prescribed behaviour on the real line and to ``establish a special case of the Poisson summation formula (essentially \textit{E. M. Stein} and \textit{G. Weiss} [Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press (1971; Zbl 0232.42007)] Theorem 2.4) that applies to integrable functions \(F: {\mathbb R}^N \to {\mathbb R}\) which are the restriction to \({\mathbb R}^N\) of a real entire function \({\mathbb C}^N \to {\mathbb C}\) of exponential type''. Let \(V_N\) and \(\omega_{N-1}\) be the volume and surface area, respectively, of the unit ball in \({\mathbb R}^N\), \(|A| = \sup \{ |A{\mathbf x}| : {\mathbf x} \in {\mathbb R}^N, |{\mathbf x}| \le 1\}\), \(\chi_R: {\mathbb R}^N \to \{0, \frac{1}{2}, 1\}\) be the normalized characteristic function of a ball of radius \(R\) centered at \(0\). Below \(J_{\nu}(x)\) is the Bessel function of the index \(\nu\). \N\NThe main result of this paper is the following Theorem 1.1: ``Let \(A\) be an \(M \times N\) matrix with real entries, and \(1 \le N = rank \; A \le M\). Let \(-1 < \nu\) and \(0 < \delta\) be real parameters such that \(2\nu + 2 = N\). Then for \(0 < R\) and \(|A| \le \delta^{-1}\) we have \N\[ \vert (\det A^T A)^{\frac{1}{2}} \sum_{m \in {\mathbb Z}} \chi_R( A({\mathbf m} + {\mathbf x})) - V_NR^N \vert \le \omega_{N - 1} u_{\nu}(R, \delta) \] \Nfor all \({\mathbf x}\) in \({\mathbb R}^N\), where \(u_{\nu}(R, \delta)\) is positive and defined by \N\[ u_{\nu}(R, \delta) = {\delta}^{-1} R^{N-1} \left( 1 - \frac{\pi}{2} (N - 1) \int_{\pi \delta R}^{\infty} x^{-1} J_{\nu}(x)J_{\nu + 1}(x) dx\right)^{-1}.\text{''} \] \N\NThis Theorem 1.1 is a generalization of the author's and Holt [loc. cit.] previous work, which considered the case of a ball centered at the origin in Euclidean space. The paper consists of five sections. After the Introductory section the Section 2 is devoted to entire functions of exponential type. Section 3 deals with the Beurling-Selberg extremal problem for a ball. In Section 4 the author of the paper under review establishes a special case of the Poisson summation formula, and Section 5 contains the proof of Theorem 1.1.