Open Access

The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap

Journal of Inequalities and Applications20092009:205298

DOI: 10.1155/2009/205298

Received: 10 July 2009

Accepted: 23 October 2009

Published: 25 October 2009

Abstract

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 .

1. Introduction

In the past decades, many mathematicians dedicated to establish the Jackson and Bernstein-type theorems on the sphere (see [19]). Early works, such as Butzer and Johnen [3], Nikol'skii and Lizorkin [8, 9], and Lizorkin and Nikol'skii [5] had successfully established the direct and inverse theorems on the sphere. In 1991, Li and Yang [4] 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 [4])

(1.1)

where and are positive integers,

(1.2)

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 [4] proved that

(1.3)

and the saturation order for is , where and are independent of positive integer and , and is the modulus of smoothness of degree 2 on the unit sphere .

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. [2] 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 .

Next, we introduce some concepts and properties of sphere as well as caps (see [7, 10]). The volume of is

(2.1)

Corresponding to , the inner product on is defined by

(2.2)

Denote by the space of -integrable functions on endowed with the norms

(2.3)

We denote by the spherical cap with center and angle , that is,

(2.4)

and by the volume of , that is,

(2.5)

Then for fixed and , is a Banach space endowed with the norm defined by

(2.6)

For any , we note

(2.7)

and clearly, and . This allows us to introduce some operators on spherical cap using existing operators on the sphere.

Definition 2.1.

Suppose that is an operator on , then
(2.8)

is called the operator on introduced by . We may use the notation instead of for convenience without mixing up.

We now make a brief introduction of projection operators by ultraspherical (Gegenbauer) polynomials for discussion of saturation property of Jackson operators.

Ultraspherical polynomials are defined in terms of the generating function (see [11]):

(2.9)

where , .

For any , we have (see [11])

(2.10)

When (see [7]),

(2.11)

where is the Legendre polynomial of degree . Particularly,

(2.12)

Therefore,

(2.13)

Besides, for any and , (see [10]).

The projection operators is defined by

(2.14)

It follows from (2.10) and (2.13) that

(2.15)

In the same way, we define the inner product on as follows:

(2.16)

We denote by the Laplace-Beltrami operator

(2.17)

by which we define a -function on as

(2.18)

For , the translation operator is defined by

(2.19)

where denotes the the elementary surface piece on the sphere . Then we have

(2.20)

The modulus of smoothness of is defined by

(2.21)

Using the method of [3], we have

(2.22)

We introduce th translation operator in terms of multipliers (see [6, 7, 12])

(2.23)

It has been proved that (see [7])

(2.24)

With the help of , we can construct Jackson-type operator on .

Definition 2.2.

For , the th Jackson-type operator of degree on is defined by
(2.25)

where , and satisfying

Remark 2.3.

We may notice that is a bit different from classical Jackson kernel
(2.26)
This difference will help us to prove the converse inequality for . For sake of ensuring that the converse inequality for holds, has to be no more than . Particularly, for , we have
(2.27)

Finally, we introduce the definition of saturation for operators (see [13]).

Definition 2.4.

Let be a positive function with respect to , , tending monotonely to zero as . For , is a sequence of operators. If there exists such that:

(i)If , then ;

(ii) if and only if ;

then is said to be saturated on with order and is called its saturation class.

3. Some Lemmas

In this section, we show some lemmas on both and as the preparation for the main results. For , we have the following.

Lemma 3.1.

For , , ,

(i)for , ;

(ii)for , ;

(iii)for , ;

(iv)for , , , where , as ;

(v)for , and which satisfies , .

Proof.

(i), (ii) and (iii) are clear. Using [2, Remark ], we can obtain (iv). For (v), we have
(3.1)
which implies
(3.2)

We need the following lemma.

Lemma 3.2.

For , , , , and , one has
(3.3)

where .

Proof.

A simple calculation gives, for ,
(3.4)
Therefore,
(3.5)

For Jackson-type operator, we have the following lemma.

Lemma 3.3.

For , , there hold

(i) ;

(ii)for which satisfies , ;

(iii)for , and , .

Proof.

From the definition and (ii) and (v) of Lemma 3.1, (i) and (ii) are clear. We just have to add the proof of (iii). In fact, using Minkowski inequality, (iv) of Lemmas 3.1 and 3.2, we have
(3.6)

where the constant in the approximation is independent of and .

The following lemma is useful in the proof of the converse inequality for Jackson-type operator.

Lemma 3.4 (see [14]).

Suppose that for nonnegative sequences , with the inequality
(3.7)
is satisfied for any positive integer . Then one has
(3.8)

The following lemma gives the multiplier representation of , which follows from Definition 2.2 and (2.23).

Lemma 3.5.

For , has the representation
(3.9)
where
(3.10)

The following lemma is useful for determining the saturation order. It can be deduced by the methods of [13, 15].

Lemma 3.6.

Suppose that is a sequence of operators on , and there exists function series with respect to , such that
(3.11)
for every . If for any there such that
(3.12)

then is saturated on with the order and the collection of all constants is the invariant class for on .

4. Main Results and Their Proofs

In this section, we will discuss the main results, that is, the lower and upper bounds as well as the saturation order for Jackson-type operator on .

The following theorem gives the Jackson-type inequality for .

Theorem 4.1.

For any integer and , is the series of Jackson-type operators on defined previously, and .

Then

(4.1)
Therefore, for ,
(4.2)

where is independent of and .

Proof.

Since we have (see [13])
(4.3)
and it is true that
(4.4)
Therefore (explained below),
(4.5)

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).

Next, we prove the Bernstein-type inequality for for .

Theorem 4.2.

Assume that , are th Jackson-type operators on . For , , then there exits a constant independent of and such that
(4.6)

holds for every and every integer k.

Proof.

Li and Yang [4] have proved the Marchaud-Ste kin inequality for Jackson operator on the sphere. Following the method in [4], we first prove the Marchaud-Ste kin inequality for :
(4.7)
Let
(4.8)
then for , by Lemma 3.3,
(4.9)
so we can deduce from Lemma 3.4 (where is set to be ) that
(4.10)
that is,
(4.11)
Since there exists such that
(4.12)
then,
(4.13)
By (2.22), we obtain that
(4.14)
So (4.7) holds, and it implies that
(4.15)

In order to prove (4.6), we have to show that

(4.16)
We first prove
(4.17)
It follows from (4.2) and (4.15) that
(4.18)
Then we prove
(4.19)
In fact, the proof is similar to that of (4.17)
(4.20)
Hence,
(4.21)

Now we can complete the proof of (4.6). Let

(4.22)
It follows from (4.16) that
(4.23)

Thus, . Since , then , we obtain from (4.16) that

(4.24)
Noticing that , we may rewrite the previous inequality as
(4.25)

This completes the proof.

We thus obtain the corollary of Theorems 4.1 and 4.2.

Corollary 4.3.

Suppose that , , are Jackson-type operators on the spherical cap , , then the following are equivalent for any , ,

(i) ;

(ii) .

Theorem 4.4.

Suppose that , are Jackson-type operators on the spherical cap , . Then are saturated on with order and the collection of constants is their invariant class.

Proof.

We obtain from Lemma 3.2 that, for
(4.26)
By Lemma 3.6, if it is true that for
(4.27)

then the proof is completed.

In fact, for any , it follows from (3.3) that

(4.28)

We deduce from (2.15) that, for any , there exists , for , it holds that

(4.29)
Then it follows that
(4.30)
So,
(4.31)

Therefore, we obtain by Lemma 3.6 that the saturation order for is .

Declarations

Acknowledgments

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).

Authors’ Affiliations

(1)
Institute of Metrology and Computational Sciences, China Jiliang University

References

  1. 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
  2. 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
  3. 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
  4. Li LQ, Yang RY: Approximation by spherical Jackson polynomials. Journal of Beijing Normal University (Natural Science) 1991, 27: 1–12.MATHGoogle Scholar
  5. Lizorkin IP, Nikol'skii SM: A theorem concerning approximation on the sphere. Analysis Mathematica 1983,9(3):207–221. 10.1007/BF01989806MathSciNetView ArticleMATHGoogle Scholar
  6. 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
  7. Wang KY, Li LQ: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing, China; 2000.Google Scholar
  8. Nikol'skii SM, Lizorkin PI: Approximation by spherical polynomials. Trudy Matematicheskogo Instituta imeni V.A. Steklova 1984, 166: 186–200.MathSciNetMATHGoogle Scholar
  9. Nikol'skii SM, Lizorkin PI: Approximation of functions on the sphere. Izvestiya Akademii Nauk SSSR. Seriya. Matematiíeskaya 1987,51(3):635–651.MathSciNetMATHGoogle Scholar
  10. Müller C: Spherical Harmonics, Lecture Notes in Mathematics. Volume 17. Springer, Berlin, Germany; 1966.Google Scholar
  11. Stein EM, Weiss G: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ, USA; 1971.MATHGoogle Scholar
  12. 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
  13. 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
  14. van Wickeren E: Steckin-marchaud-type inequalities in connection with bernstein polynomials. Constructive Approximation 1986,2(4):331–337.MathSciNetView ArticleMATHGoogle Scholar
  15. 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

Copyright

© Y. Wang and F. Cao. 2009

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.