Relations between anisotropic Besov spaces and multivariate Bernstein-Durrmeyer operators
© Feng and Feng; licensee Springer 2013
Received: 15 July 2011
Accepted: 10 April 2013
Published: 23 April 2013
In this paper, we use the multivariate Bernstein-Durrmeyer operators defined on the simplex to characterize anisotropic Besov spaces.
1 Introduction and some notations
Let , , , , be the space consisting of all Lebesgue measurable functions f on T for which the norm is finite. Let , be the space consisting of all continuous functions f on T for which the norm is finite.
, , , , , , , .
where the derivatives are in the sense of distributions, and is the interior of T.
where , .
where , , , .
By  and the definition of anisotropic Besov spaces, it is not difficult to get the following.
For convenience, throughout this paper, M denotes a positive constant independent of x, n and f which may be different in different places.
2 Auxiliary lemmas
To prove the theorems, we need the following lemmas. The following two lemmas were proved in .
According to the definition of K-functional , Lemma 2.3 has been proved. □
According to the definition of K-functional , Lemma 2.4 has been proved. □
3 Main results
In this section we will prove our main results.
The necessity has been proved.
By Theorem 1.2, the sufficiency has also been proved. The proof is completed. □
Remark 1 For other integral-type operators, the method and the results are similar.
The authors would like to thank the anonymous referees for their valuable comments, remarks and suggestions which greatly helped us to improve the presentation of this paper and make it more readable. Project supported by the Natural Science Foundation of China (Grant No. 10671019), the Zhejiang Provincial Natural Science Foundation (Grant No. LY12A01008), and the Cultivation fund of Taizhou University.
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