The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap
© Y. Wang and F. Cao. 2009
Received: 10 July 2009
Accepted: 23 October 2009
Published: 25 October 2009
Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, the so-called Jackson-type operator is constructed to approximate the function defined on the spherical cap . We thus establish the direct and inverse inequalities and obtain saturation theorems for on the cap . Using methods of -functional and multiplier, we obtain the inequality and that the saturation order of these operators is , where is the modulus of smoothness of degree 2, the constants and are independent of and .
In the past decades, many mathematicians dedicated to establish the Jackson and Bernstein-type theorems on the sphere (see [1–9]). Early works, such as Butzer and Johnen , Nikol'skii and Lizorkin [8, 9], and Lizorkin and Nikol'skii  had successfully established the direct and inverse theorems on the sphere. In 1991, Li and Yang  constructed Jackson operators on the sphere and obtained the Jackson and Bernstein-type theorems for the Jackson operators.
Jackson operator on the sphere is defined by (see )
is the classical Jackson kernel, is measurable function of degree on the sphere in , is the elementary surface piece, is the measurement of . For , ( is the collection of continuous functions on ), Li and Yang  proved that
Naturally, we desire to obtain the similar results on the spherical caps. To achieve the goal, a key issue is to establish the inverse inequality on the cap.
Recently, Belinsky et al.  constructed th translation operator when discussing the averages of functions on the sphere. This inspires us to construct the th Jackson-type operator on the spherical cap. We then prove a strong-type converse inequality for , which helps us get the direct and inverse theorems of approximation on the spherical cap. Also, we obtain that the saturation order for the constructed Jackson-type operator is , the same to that of the Jackson operator on the sphere.
2. Definitions and Auxiliary Notations
Throughout this paper, we denote by the letters and ( is either positive integers or variables on which depends only) positive constants depending only on the dimension . Their value may be different at different occurrences, even within the same formula. We will denote the points in by and , and the elementary surface piece on by . If it is necessary, we will write referring to the variable of the integration. The notation means that there exists a positive constant such that where is independent of and .
Ultraspherical polynomials are defined in terms of the generating function (see ):
For any , we have (see )
When (see ),
Besides, for any and , (see ).
The projection operators is defined by
It follows from (2.10) and (2.13) that
Using the method of , we have
It has been proved that (see )
Finally, we introduce the definition of saturation for operators (see ).
3. Some Lemmas
We need the following lemma.
For Jackson-type operator, we have the following lemma.
The following lemma is useful in the proof of the converse inequality for Jackson-type operator.
Lemma 3.4 (see ).
4. Main Results and Their Proofs
where the Minkowski inequality, (4.3), and Lemma 3.1 are used in the first inequality, and the second and third one are deduced from (3.3) and Lemma 3.1. From (2.22) and (i) of Lemma 3.3, it is easy to deduce (4.2).
In order to prove (4.6), we have to show that
Now we can complete the proof of (4.6). Let
This completes the proof.
We thus obtain the corollary of Theorems 4.1 and 4.2.
then the proof is completed.
The research was supported by the National Natural Science Foundation of China (no. 60873206),the Natural Science Foundation of Zhejiang Province of China (no. Y7080235), the Key Foundation of Department of Education of Zhejiang Province of China (no. 20060543), and the Innovation Foundation of Post-Graduates of Zhejiang Province of China (no. YK2008066).
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