Best constants for tensor products of Bernstein type operators
Let \(I=[0,1]\,\text{or}\,[0,\infty)\), and let \(L\) be a Bernstein-type operator on \(I\), defined by \(Lf(x):=Ef(\xi(x))\), \(x\in I\), \(f\in\mathcal L\), where \(\{\xi(x):x\in I\}\) is an integrable stochastic process with values in \(I\), and the set \(\mathcal L\) is a suitable collection of real-valued measurable functions on \(I\), so that the operator is well defined. Also, let \(\xi^{(k)}(\mathbf x):=(\xi_1(x_1),\dots,\xi_k(x_k))\), \(\mathbf x:=(x_1,\dots,x_k)\in I^k:=I\otimes\cdots\otimes I\), the tensor product of \(k\) copies of \(I\), where \(\{\xi_i(x):x\in I\}\), \(i=1,\dots,k\), are \(k\) stochastically independent copies of \(\{\xi(x):x\in I\}\). Given a function \(f\in\mathcal L^k\), i.e., \(f(\mathbf x):=\prod_{i=1}^kf_i(x_i)\), \(f_i\in\mathcal L\), \(1\leq i\leq k\), and \(\mathbf x:=(x_1,\dots,x_k)\in I^k\), the authors associate with the operator \(L\) the operator \(L^{(k)}f(\mathbf x):=\prod_{i=1}^kLf_i(x_i)\), and they consider the problem of obtaining estimates of the form \(\omega_p(L^{(k)}f,\delta)\leq C(\delta)\omega_p(f,\delta)\) where \(C(\delta)\) is independent of \(f\in\mathcal L^k\), and \(\omega_p(f,\delta):=\sup\{|f(\mathbf x)-f(\mathbf y)|:\mathbf x,\mathbf y\in I^k, \|\mathbf x-\mathbf y\|_p<\delta\}\), \(1\leq p\leq\infty\). Further, the authors are interested in finding the best constant \[ C^{(k)}_p(\delta):=\sup\frac{\omega_p(L^{(k)}f,\delta)}{\omega_p(f,\delta)}, \] where the supremum is taken over all \(f\in\mathcal L^k\) such that \(0<\omega_p(f,\delta)<\infty\). For centered operators \(L\), that is, when \(E\xi(x)=x\), \(x\in I\), and such that the process \(\xi(x)\) has stationary increments, one of the authors (with another coauthor), has obtained the estimate \(C^{(k)}_p(\delta)\leq1+k^{1-1/p}\), for \(\delta\in(0,k^{1/p}]\) if \(I=[0,1]\), and for \(\delta>0\) if \(I=[0,\infty)\). In this paper the authors give necessary and sufficient conditions for equality to hold in the above estimate. It is proved that necessary and sufficient conditions are that \(\xi(x)/x\to0\) in probability, as \(x\downarrow0\), and that \(P(\xi(x)=0)=0\), for all \(0<x\in I\). Further they provide sufficient conditions that guarantee that the above best constant is achieved for a family of operators instead of one. The paper ends with applying the two results to various examples like the beta and gamma operators, and the Baskakov operators.
scientific article; zbMATH DE number 2131165