Bivariate monotone approximation

Anastassiou, George A.

doi:10.1090/S0002-9939-1991-1069682-2

Bivariate monotone approximation
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by George A. Anastassiou PDF

Proc. Amer. Math. Soc. 112 (1991), 959-964 Request permission

Abstract:

Let $f$ be a two variable continuously differentiable real-valued function of certain order on ${[0,1]^2}$ and let $L$ be a linear differential operator involving mixed partial derivatives and suppose that $L(f) \geq 0$. Then there exists a sequence of two-dimensional polynomials ${Q_{m,n}}(x,y)$ with $L({Q_{m,n}}) \geq 0$, so that $f$ is approximated simultaneously and uniformly by ${Q_{m,n}}$. This approximation is accomplished quantitatively by the use of a suitable two-dimensional first modulus of continuity.

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