On a Turán problem for \(\ell-1\)-radial, positive-definite functions
The notion of Turán's problem is based on a paper by \textit{S. B. Stechkin} [Acta Math. Acad. Sci. Hung. 23, 289--291 (1972; Zbl 0259.42015)] who studied an extremal problem which was posed to him by P. Turán. Up to now, several variations of the original problem have been studied (cf. [\textit{N. N. Andreev}, Mosc. Univ. Math. Bull. 52, No. 1, 29--32 (1997); translation from Vestn. Mosk. Univ., Ser. I 1997, No. 1, 29--32 (1997; Zbl 0913.42009)], [\textit{D. V. Gorbachev} and \textit{A. S. Manoshina}, Math. Notes 76, No. 5, 640--652 (2004); translation from Mat. Zametki 76, No. 5, 688--700 (2004; Zbl 1074.42002)], [\textit{M. N. Kolountzakis} and \textit{S. G. Révész}, Proc. Am. Math. Soc. 131, No. 11, 3423--3430 (2003; Zbl 1042.42004)], [\textit{V. V. Arestov} and \textit{E. E. Berdysheva}, East J. Approx. 8, No. 3, 381--388 (2002)]). The present paper deals with a two-dimensional version of Turán's problem for \(\ell\)-1 radial positive definite functions. Let \({\mathcal P}_\pi^{(2)}\) denote the set of positive definite functions in \({\mathbb R}^2\) with support in the set \(\{x\in{\mathbb R}^2:| x| _1\leq \pi\}\) and \(f(0)=1\). Let further \({\mathcal K}_2^{(2)}\) denote the subset of \(\ell-1\) radial functions in \(C(\overline{\mathbb R}_+)\) with support in \(\{x\in{\mathbb R}^2:| x| _1\leq \pi\}\), i.e., of functions \(\varphi\) with \(\varphi(| \,\cdot\,| _1)\in{\mathcal P}_\pi^{(2)}\). The problem then is to determine the constant \[ \alpha_\pi^{(2)} \, = \, \sup_{\varphi\in{\mathcal K}_2^{(2)}} \, 2\int_0^\pi u\varphi(u)\, du= \sup_{\varphi\in{\mathcal K}_2^{(2)}} \, \frac{\psi(0)}{h_2(0)}, \] where \[ \psi(\rho)= 2\int_0^\infty \varphi(u)h_2(u)u\, du. \] Hereby, \(\psi\) can be interpreted as the analog of the Fourier-Bessel transform of \(\ell-1\) radial functions (cf. [\textit{H. Berens} and \textit{Y. Xu}, Math. Proc. Camb. Philos. Soc. 122, No. 1, 149--172 (1997; Zbl 0881.42007)]), where the kernel is given by \[ h_2(t)=\sin t/(1+\cos t), \quad t\in{\mathbb R}_+. \] A first upper bound \(\alpha_\pi^{(2)}\leq \pi^2/2\) follows from work by \textit{N. N. Andreev} [loc. cit.]. Using the explicit representation of the kernel \(h_2\), the authors prove a closer upper bound as well as a lower bound for \(\alpha_\pi^{(2)}\). A nice geometric argument leads to the lower bound, which is given for arbitrary dimension, based on \textit{R. E. Williamson}'s integral representation for \(n\)-times monotone functions on \({\mathbb R}_+\) [Duke Math. J. 23, 189--207 (1956; Zbl 0070.28501)]. Solving a corresponding discrete version of the extremal problem, leads to the upper bound in the two-dimensional case.
scientific article; zbMATH DE number 2174558