Bivariate Right Fractional Polynomial Monotone Approximation

Abstract

Let $f \in C^{r,p}\left (\left [0,1\right ]^{2}\right )$, $r,p \in \mathbb{N}$, and let $\overline{L}$ be a linear right fractional mixed partial differential operator such that $\overline{L}\left (f\right ) \geq 0$, for all $\left (x,y\right )$ in a critical region of $\left [0,1\right ]^{2}$ that depends on $\overline{L}$. Then there exists a sequence of two-dimensional polynomials $Q_{\overline{m_{1}},\overline{m_{2}}}\left (x,y\right )$ with $\overline{L}\left (Q_{\overline{m_{1}},\overline{m_{2}}}\left (x,y\right )\right ) \geq 0$ there, where $\overline{m_{1}},\overline{m_{2}} \in \mathbb{N}$ such that $\overline{m_{1}}> r$, $\overline{m_{2}}> p$, so that f is approximated right fractionally simultaneously and uniformly by $Q_{\overline{m_{1}},\overline{m_{2}}}$ on $\left [0,1\right ]^{2}$. This restricted right fractional approximation is accomplished quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity.