Treffer: A best constant for bivariate Bernstein and Szász-Mirakyan operators

For \(n=1,2,\dots\) we consider the tensor product \(B_n^{\langle 2 \rangle}: =B_n\otimes B_n\), where \(B_n\) denotes the classical Bernstein operator over the interval \([0,1]\) given by \(B_nf(x): =\sum^n_{k=0} f({k\over n})\cdot ({n \over k}) \times^k(1-x)^{n-k}\). Let \(c_n^{\langle 2\rangle} (\delta)\) be the best constant in preservation of the usual modulus of continuity for the \(\ell_\infty\)-norm in \(\mathbb{R}^2\), that is \(c_n^{\langle 2\rangle} (\delta): = \sup_{f \in{\mathcal F}_2} {\omega(B_n^{\langle 2\rangle}f; \delta)\over \omega(f; \delta)}\), \(00\) we consider the tensor product \(S_t^{\langle 2 \rangle}: =S_t\otimes S_t\), where \(S_t\) denotes the Szász-Mirakyan operator over the interval \([0,\infty)\) given by \(S_tg(x):=\sum^\infty_{k=0} g({k\over n})e^{-t_x}{(tx)^k \over k!}\). Let us denote by \(D_t^{\langle 2\rangle} (\delta)\) the corresponding best constant, i.e. \(D_t^{\langle 2\rangle} (\delta) :=\sup_{g\in {\mathcal G}_2}{\omega(S_t^{\langle 2\rangle} g;\delta) \over\omega (g; \delta)}\), \(\delta>0\), where \({\mathcal G}_2\) is the set of all real non-constant functions \(g\) on \([0,\infty)^2\) such that \(\omega(g; \delta) 0\). The main result is the following theorem: We have \[ \sup_{_\delta> 0}D_1^{\langle 2\rangle} (\delta)= \sup_{n\geq 1}\sup_{0< \delta\leq 1}c_n^{ \langle 2\rangle} (\delta)= 1-e^{-2}+\sum^\infty_{j=0} \left[1-e^{-2}\left( \sum^j_{i=0} {1\over i!} \right)^2\right]= 2.3884423\dots \] The proof involves both probabilistic and analytic arguments, as well as numerical computation of some specific values.