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Location of points on a torus and extremal properties of polynomials. (English. Russian original) Zbl 1053.42500

Proc. Steklov Inst. Math. 219, 447-457 (1997); translation from Tr. Mat. Inst. Steklova 219, 453-463 (1997).
Let \(x=(x_1,\cdots,x_n)\in\mathbb R^n\); the value \(\sqrt{x_1^2+\cdots+x_n^2}\) is denoted by \(|x|\). Let \(\mathbb Z^n\) be the lattice of integer numbers in \(\mathbb R^n\), and let \(B_n=\{x\in\mathbb R^n\colon |x|\leq1\}\) be the unit ball in \(\mathbb R^n\). Denote by \(J_\nu(z)\), \(\nu=n/2-1\), the Bessel function, and let \(q_i\) be positive zeros of this function arranged in ascending order.
For a finite set of points \(W=\{x^{(k)}\}_{k=1}^N\) in the torus \(\mathbb T^n=\{x\colon -1/2<x_k\leq1/2\) \((k=1,\cdots,n)\}\), denote \[ R(W)=\sup_{r>1}\left\{r\colon \sum_{k=1}^Ne^{2\pi i\nu x^{(k)}}=0, \nu\in\mathbb Z^n, 0<|\nu|<R\right\}. \] \(W\) is called a spherical design of order \(R\) if \(R(W)\geq R\). Also, if \(W\) contains at least two points, denote \[ \varepsilon(\mathbb R)=\min_{x,y\in W,x\neq y}|x-y|. \] Clearly, \(0<\varepsilon(W)\leq\sqrt n/2\).
The author proves that for \(R(W)\geq2q_2/\pi\sqrt{n}\), \(\varepsilon(W)R(W)\leq q_2/\pi\). The inequality is obtained as a result of the study of the following extremal problem: it is required to find a trigonometric polynomial \(f(x)\) satisfying the inequality \(f(x)\geq0,\quad|x|\geq\varepsilon, x\in\mathbb T^n\), with minimal \(\varepsilon\) among the set of real trigonometric polynomials \[ f(x)=\sum_{\nu\in\mathbb Z^n,|\nu|<R}\widehat f_\nu e^{2\pi i\nu/x},\quad \widehat f_\nu=\int_{\mathbb T^n}f(x)e^{-2\pi i\nu x}\,dx \] for which \(\widehat f_0=0,\quad \widehat f_\nu\geq0\) for all \(\nu\in\mathbb Z^n, 0<|\nu|<R\).
{Unfortunately, there are some typos in the English translation of the paper (\(\Leftrightarrow\) instead of \(-\)).}
For the entire collection see [Zbl 0907.00017].

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
11C08 Polynomials in number theory