Bivariate Left Fractional Polynomial Monotone Approximation

摘要

Let f ∈ C~(r,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 (χ, y) in a critical region of [0, 1]~2 that depends on L*. Then there exists a sequence of two-dimensional polynomials Q{formula} (χ, y) with L* (Q{formula}(χ,y)) - 0 there, where {formula}, {formula} ∈ N such that {formula} > r, {formula} > p, so that f is approximated left fractionally simultaneously and uniformly by Q{formula} 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.