Bivariate Monotone Approximation

Abstract

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

Negative theorem for monotone bivariate approximation

“…డ௫ డ௬ ೞ .I would like to say that we prove our result in [2] with the help of the theorem introduced by George A. Anastassiou in [1]. Also in [1], George A. Anastassiou asked: Could we involve in our inequalities a higher order modulus of smoothness? In this paper we answer this question.…”

“…The first result of multi variable polynomial monotone approximation is introduced by George A. Anastassiou in [1]. He used the ‫ܮ‬ ஶ -norm and raised a remark what about ‫ܮ‬ -case.…”