Inequalities for a class of multivariate operators
© Zhao; licensee Springer 2012
Received: 22 January 2012
Accepted: 1 August 2012
Published: 7 August 2012
This paper introduces and studies a class of generalized multivariate Bernstein operators defined on the simplex. By means of the modulus of continuity and so-called Ditzian-Totik’s modulus of function, the direct and inverse inequalities for the operators approximating multivariate continuous functions are simultaneously established. From these inequalities, the characterization of approximation of the operators follows. The obtained results include the corresponding ones of the classical Bernstein operators.
MSC:41A25, 41A36, 41A60, 41A63.
Here, is the modulus of continuity of first order of the function f. In , some approximation properties for the operators were further investigated.
In this paper, we will introduce and study the multivariate version defined on the simplex of the generalized Bernstein operators given by (1). The main aim is to establish the direct and inverse inequalities of approximation, which will imply the characterization of approximation of the operators.
here and in the following C denotes a positive constant independent of f and n, but its value may be different at a different occurrence.
Now we state the main results of this paper as follows.
From Theorem 1.2 and Theorem 1.3, we easily obtain the following corollaries, which characterize the approximation feature of the multivariate operators given by (8).
implies that and ().
From Corollary 1.2 and Corollary 1.3, we have the following.
is and ().
2 Some lemmas
In this section, we prove some lemmas.
where , .
The proof of Lemma 2.1 is completed. □
Secondly, we need prove two Bernstein type inequalities.
So, the proof of Lemma 2.3 is complete. □
Hence, the proof of Lemma 2.4 is complete. □
We also need the following two interesting results related to nonnegative numerical sequence. The proof of the first result can be found in , and the proof of the other is similar to Lemma 2.1 of  where the proof of case and was given.
3 The proof of main results
This completes the proof of Theorem 1.2.
The proof of Theorem 1.3 is complete.
This research was supported by the National Nature Science Foundation of China (No. 61101240, 61272023), the Zhejiang Provincial Natural Science Foundation of China (No. Y6110117), and the Science Foundation of the Zhejiang Education Office (No. Y201122002).
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