
On the doubling condition for nonnegative positive definite functions on on the halfline with power weight
D. V. Gorbachev^{}, V. I. Ivanov^{} ^{} Tula State University
Abstract:
Continuous nonnegative 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 onedimensional 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 $2R1\le C(R)\le 2R+1$ 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 onedimensional weight inequality
$$
\int_{0}^{R}f(x)x^{n1} dx\le c_{n}R^{n}\int_{0}^{1}f(x)x^{n1} dx,\quad n\in
\mathbb{N}.
$$
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.
Keywords:
positive definite function, doubling condition, Hankel transform, Bessel generalized translation.
DOI:
https://doi.org/10.22405/22268383201819290100
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UDC:
517.5 Received: 29.05.2018 Accepted:17.08.2018
Citation:
D. V. Gorbachev, V. I. Ivanov, “On the doubling condition for nonnegative positive definite functions on on the halfline with power weight”, Chebyshevskii Sb., 19:2 (2018), 90–100
Citation in format AMSBIB
\Bibitem{GorIva18}
\by D.~V.~Gorbachev, V.~I.~Ivanov
\paper On the doubling condition for nonnegative positive definite functions on on the halfline with power weight
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 2
\pages 90100
\mathnet{http://mi.mathnet.ru/cheb641}
\crossref{https://doi.org/10.22405/22268383201819290100}
\elib{https://elibrary.ru/item.asp?id=37112141}
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