Bivariate right fractional polynomial monotone approximation


Let f ∈ Cr,p ([O, 1]2), r, p ∈ ℕ, and let L be a linear right 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 Qm1,m2 (x, y) with L (Qm1,m2 (x, y)) ≥ 0 there, where m1, m2 ∈ ℕ such that m1 > r, m2 > p, so that f is approximated right fractionally simultaneously and uniformly by Qm1,m2 on [0, 1]2. This restricted right fractional approximation is accomplished quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity.

Publication Title

Springer Proceedings in Mathematics and Statistics