Bivariate Right Fractional Pseudo-Polynomial Monotone Approximation

Abstract

In this article we deal with the following general two-dimensional problem: Let f be a two variable continuously differentiable real valued function of a given order, let $\overline{L}$ be a linear right fractional mixed partial differential operator and suppose that $\overline{L}\left( f\right) \ge 0$ on a critical region. Then for sufficiently large $n,m\in \mathbb {N}$ , we can find a sequence of pseudo-polynomials $Q_{n,m}^{*}$ in two variables with the property $\overline{L}\left( Q_{n,m}^{*}\right) \ge 0$ on this critical region such that f is approximated with rates right fractionally and simultaneously by $Q_{n,m}^{*}$ in the uniform norm on the whole domain of f. This restricted approximation is given via inequalities involving the mixed modulus of smoothness $\omega _{s,q}$, $s,q\in \mathbb {N}$, of highest order integer partial derivative of f.