- Research Article
- Open Access
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 .
- Measurable Function
- Simple Calculation
- Projection Operator
- Elementary Surface
- Constant Independent
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.
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 ).
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 ).
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).
- Antonela T: Generating functions for the mean value of a function on a sphere and its associated ball in . Journal of Inequalities and Applications 2008, 2008:-14.Google Scholar
- Belinsky E, Dai F, Ditzian Z: Multivariate approximating averages. Journal of Approximation Theory 2003,125(1):85–105. 10.1016/j.jat.2003.09.005MathSciNetView ArticleMATHGoogle Scholar
- Butzer PL, Johnen H: Lipschitz spaces on compact manifolds. Journal of Functional Analysis 1971, 7: 242–266. 10.1016/0022-1236(71)90034-6MathSciNetView ArticleMATHGoogle Scholar
- Li LQ, Yang RY: Approximation by spherical Jackson polynomials. Journal of Beijing Normal University (Natural Science) 1991, 27: 1–12.MATHGoogle Scholar
- Lizorkin IP, Nikol'skii SM: A theorem concerning approximation on the sphere. Analysis Mathematica 1983,9(3):207–221. 10.1007/BF01989806MathSciNetView ArticleMATHGoogle Scholar
- Lizorkin PL, Nikol'skii SM: Function spaces on a sphere that are connected with approximation theory. Matematicheskie Zametki 1987, 41: 509–515. English translation in Mathematical Notes, vol. 41, pp. 286–291, 1987 English translation in Mathematical Notes, vol. 41, pp. 286–291, 1987MathSciNetGoogle Scholar
- Wang KY, Li LQ: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing, China; 2000.Google Scholar
- Nikol'skii SM, Lizorkin PI: Approximation by spherical polynomials. Trudy Matematicheskogo Instituta imeni V.A. Steklova 1984, 166: 186–200.MathSciNetMATHGoogle Scholar
- Nikol'skii SM, Lizorkin PI: Approximation of functions on the sphere. Izvestiya Akademii Nauk SSSR. Seriya. Matematiíeskaya 1987,51(3):635–651.MathSciNetMATHGoogle Scholar
- Müller C: Spherical Harmonics, Lecture Notes in Mathematics. Volume 17. Springer, Berlin, Germany; 1966.Google Scholar
- Stein EM, Weiss G: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ, USA; 1971.MATHGoogle Scholar
- Rustamov KV: On equivalence of different moduli of smoothness on the sphere. Proceedings of the Steklov Institute of Mathematics 1994,204(3):235–260.MathSciNetMATHGoogle Scholar
- Berens H, Butzer PL, Pawelke S: Limitierungsverfahren von reihen mehrdimensionaler kugelfunktionen und deren saturationsverhalten. Publications of the Research Institute for Mathematical Sciences A 1968,4(2):201–268. 10.2977/prims/1195194875MathSciNetView ArticleMATHGoogle Scholar
- van Wickeren E: Steckin-marchaud-type inequalities in connection with bernstein polynomials. Constructive Approximation 1986,2(4):331–337.MathSciNetView ArticleMATHGoogle Scholar
- Butzer PL, Nessel RJ, Trebels W: On summation processes of Fourier expansions in Banach spaces. II. Saturation theorems. The Tohoku Mathematical Journal 1972,24(4):551–569. 10.2748/tmj/1178241446MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.