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Global smoothness preservation by multivariate approximation operators. (English) Zbl 0795.41011

Baron, S. (ed.) et al., Approximation, interpolation, and summability. In honor of Amnon Jakimovski on his sixty-fifth birthday. Proceedings of an international conference, held at Tel-Aviv University, Tel Aviv and Bar- Ilan University, Ramat-Gan, Israel, June 4-8, 1990. Providence, RI: American Mathematical Society. Isr. Math. Conf. Proc. 4, 31-44 (1991).
In a previous paper [Analysis 11, No. 1, 43-57 (1991; Zbl 0722.41021)] the authors investigated global smoothness presentation by certain univariate linear operators; in the present paper the multivariate case is discussed. Extending a result of M. K. Khan and M. A. Peters [J. Approximation Theory 59, 307-315 (1989; Zbl 0693.41019)], the authors prove a general result for operators having the splitting property; more complete inequalities for Bernstein operators over a \(k\)- dimensional simplex and cube are given. It is shown how tensor product operators inherit global smoothness properties from their univariate building blocks. Finally some applications are given in the context of stochastic approximation.
For the entire collection see [Zbl 0771.00020].

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A36 Approximation by positive operators
41A63 Multidimensional problems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A16 Lipschitz (Hölder) classes
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