On a Turán problem for \(\ell-1\)-radial, positive-definite functions. II
In an earlier paper [Result. Math. 47, No. 1--2, 17--32 (2005; Zbl 1096.42002)], the authors treated Turán's problem (the treatest possible value for \(\int_{\mathbb{R}^d} f(x)\,dx/f(0)\) for positive definite \(f\) on \(\mathbb{R}^d\)) for the casd \(d= 2\), \(f(x)= \varphi(\| x\|_1)\). Here they extend their results to general \(d\). They introduce \[ K^{(d)}:= \{\varphi\in C_0(\mathbb{R}_+): \varphi(0)= 1,\,\text{supp\,}\varphi\subset [0,\pi],\;\psi\geq 0\text{ on }\mathbb{R}_+\} \] with \(\psi(v):= 2\int^\infty_0 u^{d-1} h_d(vu)\,du\), \(v\in\mathbb{R}_+\), ``\(h_d\)-transform of \(\varphi\)'', \(h_d\) recursively defined (or explicit with hypergeometric function \({_1F_2}\); \(h_1=\cos\), \(h+2= h_1+ (\sin u)/u) C_0K^{(d)}:= \{\varphi\in C_0(\mathbb{R}_+): \varphi(0)= 1\), \(\text{supp\,}\varphi\subset [0,\pi]\), \(\varphi\) ``\(2d\)-monotone'' on \(\mathbb{R}_+\}\) is contained in \(K^{(d)}\), with maximal element \((1- u/\pi)^{2d-1}\); \(K^{(d)}\subset JC^{d-1}(\mathbb{R}_+)\). Here 2-monotone = convex, \(\varphi\) \(n\)-monotone: \((-1)^k\varphi^{(k)}\geq 0\), nonincreasing and convex on \(\mathbb{R}_+\) for \(0\leq k\leq n-1\). For the \(h_d\)-transform also a formula with spherical Bessel functions as kernels is given.
scientific article; zbMATH DE number 5641724
scientific article