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