Abstract
In the paper, the following problem is considered for positive definite functions on \(\mathbb{R}^n\) (the class \(\Phi(\mathbb{R}^n)\) ). Let a function \(h\) be continuous on \([0,+\infty)\) , differentiable on the interval \((0,+\infty)\) , \(th'(t)\to 0\) as \(t\to+0\) , \(h(t)\not\equiv h(0)\) , and \(h(\rho(x))\in\Phi(\mathbb{R}^n)\) . Here the function \(\rho\) is continuous on \(\mathbb{R}^n\) , \(\rho(x)>0\) for \(x\ne0\) , and \(\rho(tx)=|t|\rho(x)\) , for \(x\in \mathbb{R}^n\) , \(t\in\mathbb{R}\) . For \(\beta\in\mathbb{R}\) , we define the function \(H_\beta(t):=h(t)+\beta th'(t)\) for \(t>0\) and set \(H_\beta(0):=h(0)\) . It is required to find the set of \(\beta\in\mathbb{R}\) for which \(H_\beta(\rho(x))\in\Phi(\mathbb{R}^n)\) . Under the above assumptions, this set is a closed interval \([-\beta(h,\mathbb{R}^n,\rho), \widetilde{\beta}(h,\mathbb{R}^n,\rho)]\) which contains the point \(0\) . In Theorem 1, formulas for the ends of this closed interval are found. In the case of the Euclidean norm, when \((\mathbb{R}^n,\rho)=\ell_{2}^{n}\) , in Theorem 2, for a wide class of functions \(h\) , the exact value for the right end is found: \(\widetilde{\beta}(h,\ell_{2}^{n})=1/n\) . In Theorem 3, for the function \(h_p(t)=\exp(-t^p)\) in the case of \((\mathbb{R}^n,\rho)=\ell_{q}^{n}\) , exact values for the right end and, in several cases, for the left one are found: if \(0<p\le q\le 2\) , then \(\widetilde{\beta}(h_p,\ell_{q}^{n})=1/n\) , \({\beta}(h_q,\ell_{q}^{n})={\beta}(h_q,\ell_{q}^{1})/n\) , \({\beta}(h_1,\ell_{1}^{n})=1/n\) , \({\beta}(h_1,\ell_{2}^{n})=1\) , and \({\beta}(h_2,\ell_{2}^{n})=0\) .
