Bivariate abstract fractional monotone constrained approximation by polynomials

Abstract

Let $f\in C^{r,p}([0,1]^{2})$, $r,p\in \mathbb{N}$, and let $L^{\ast}$ be an abstract linear left or right fractional mixed partial differential operator such that $L^{\ast}(f) \geq 0$ for all $(x,y)$ in a critical region of $[0,1]^{2}$that depends on $L^{\ast}$. Then there exists a sequentce of two-dimensional polynomials $Q_{\overline{m_{1}},\overline{m_{2}}}(x,y)$ with $L^{\ast}(Q_{\overline{m_{1}},\overline{m_{2}}}(x,y)) \geq 0$ 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 left or right abstract fractionally simultaneously and uniformly by $Q_{\overline{m_{1}},\overline{m_{2}}}$ on $[0,1]^{2}$. This restricted left or right abstract fractional approximation is achieved quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity. This monotone constrained fractional approximation applies to a wide range of Caputo type fractional calculi of singular or non-singular kernels.