Bivariate left fractional polynomial monotone approximation

Let f ∈ Cr,p ([0, 1]2), r, p ∈ ℕ, and let L∗ be a linear left fractional mixed partial differential operator such that L∗ (f) ≥ 0, for all (x, y) in a critical region of [0, 1]2 that depends on L∗. Then there exists a sequence of two-dimensional polynomials (Formula presented.) with (Formula presented.) there, where (Formula presented.) such that (Formula presented.), so that f is approximated left fractionally simultaneously and uniformly by (Formula presented.) on [0, 1]2. This restricted left fractional approximation is accomplished quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity