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Bivariate monotone approximation


Author: George A. Anastassiou
Journal: Proc. Amer. Math. Soc. 112 (1991), 959-964
MSC: Primary 41A29; Secondary 41A25
DOI: https://doi.org/10.1090/S0002-9939-1991-1069682-2
MathSciNet review: 1069682
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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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DOI: https://doi.org/10.1090/S0002-9939-1991-1069682-2
Article copyright: © Copyright 1991 American Mathematical Society

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