Bivariate Left 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 L ^∗ be a linear left fractional mixed partial differential operator and suppose that $L^{{\ast}}\left (f\right ) \geq 0$ 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 $L^{{\ast}}\left (Q_{n,m}^{{\ast}}\right ) \geq 0$ on this critical region such that f is approximated with rates 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 ω _s, q , $s,q \in \mathbb{N}$, of highest order integer partial derivative of f.