On the doubling condition for non-negative positive definite functions on on the half-line with power weight

D. V. Gorbachev ; V. I. Ivanov

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 90-100

Résumé

Continuous non-negative positive definite functions satisfy the following property: \begin{equation} \int_{-R}^{R}f(x)\,dx\le C(R)\int_{-1}^{1}f(x)\,dx,\quad R\ge 1, \tag{*} \end{equation} where the smallest positive constant $C(R)$ does not depend on $f$. For $R=2$, this property is well known as the doubling condition at zero. These inequalities have applications in number theory. In the one-dimensional case, the inequality ($*$) was studied by B.F. Logan (1988), as well as recently by A. Efimov, M. Gaál, and Sz. Révész (2017). It has been proven that $2R-1\le C(R)\le 2R+1$ for $R=2,3,\ldots$, whence it follows that $C(R)\sim 2R$. The question of exact constants is still open. A multidimensional version of the inequality ($*$) for the Euclidean space $\mathbb{R}^{n}$ was investigated by D.V. Gorbachev and S.Yu. Tikhonov (2018). In particular, it was proved that for continuous positive definite functions $f\colon \mathbb{R}^{n}\to \mathbb{R}_{+}$ $$ \int_{|x|\le R}f(x)\,dx\le c_{n}R^{n}\int_{|x|\le 1}f(x)\,dx, $$ where $c_{n}\le 2^{n}n\ln n\,(1+o(1))(1+R^{-1})^{n}$ при $n\to \infty$. For radial functions, we obtain the one-dimensional weight inequality $$ \int_{0}^{R}f(x)x^{n-1}\,dx\le c_{n}R^{n}\int_{0}^{1}f(x)x^{n-1}\,dx,\quad n\in \mathbb{N}. $$ We study the following natural weight generalization of such inequalities: $$ \int_{0}^{R}f(x)x^{2\alpha+1}\,dx\le C_{\alpha}(R)\int_{0}^{1}f(x)x^{2\alpha+1}\,dx,\quad \alpha\ge -1/2, $$ where $f\colon \mathbb{R}_{+}\to \mathbb{R}_{+}$ is an even positive definite function with respect to the weight $x^{2\alpha+1}$. This concept has been introduced by B.M. Levitan (1951) and means that for arbitrary $x_{1},\ldots,x_{N}\in \mathbb{R}_{+}$ matrix $(T_{\alpha}^{x_i}f(x_j))_{i,j=1}^{N}$ is semidefinite. Here $T_{\alpha}^{t}$ is the Bessel–Gegenbauer generalized translation. Levitan proved an analogue of the classical Bochner theorem for such functions according to which $f$ has the nonnegative Hankel transform (in the measure sense). We prove that for every $\alpha\ge -1/2$ $$ c_{1}(\alpha)R^{2\alpha+2}\le C_{\alpha}(R)\le c_{2}(\alpha)R^{2\alpha+2},\quad R\ge 1. $$ The lower bound is trivially achieved on the function $f(x)=1$. To prove the upper bound we apply lower estimates of the sums $\sum_{k=1}^{m}a_{k}T^{x_{k}}\chi(x)$, where $\chi$ is the characteristic function of the segment $[0,1]$, and also we use properties of the Bessel convolution.

Détail

Keywords: positive definite function, doubling condition, Hankel transform, Bessel generalized translation.