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


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])


where and are positive integers,


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


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


Corresponding to , the inner product on is defined by


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


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


and by the volume of , that is,


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


For any , we note


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


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]):


where ,.

For any , we have (see [11])


When (see [7]),


where is the Legendre polynomial of degree . Particularly,




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

The projection operators is defined by


It follows from (2.10) and (2.13) that


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


We denote by the Laplace-Beltrami operator


by which we define a -function on as


For , the translation operator is defined by


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


The modulus of smoothness of is defined by


Using the method of [3], we have


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


It has been proved that (see [7])


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


where , and satisfying

Remark 2.3.

We may notice that is a bit different from classical Jackson kernel


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


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 , .


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


which implies


We need the following lemma.

Lemma 3.2.

For , , , , and , one has


where .


A simple calculation gives, for ,




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

Lemma 3.3.

For , , there hold


(ii)for which satisfies , ;

(iii)for , and , .


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


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


is satisfied for any positive integer . Then one has


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

Lemma 3.5.

For , has the representation




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


for every . If for any there such that


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 .



Therefore, for ,


where is independent of and .


Since we have (see [13])


and it is true that


Therefore (explained below),


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


holds for every and every integer k.


Li and Yang [4] have proved the Marchaud-Stekin inequality for Jackson operator on the sphere. Following the method in [4], we first prove the Marchaud-Stekin inequality for :




then for , by Lemma 3.3,


so we can deduce from Lemma 3.4 (where is set to be ) that


that is,


Since there exists such that




By (2.22), we obtain that


So (4.7) holds, and it implies that


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


We first prove


It follows from (4.2) and (4.15) that


Then we prove


In fact, the proof is similar to that of (4.17)




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


It follows from (4.16) that


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


Noticing that , we may rewrite the previous inequality as


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 , ,



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.


We obtain from Lemma 3.2 that, for


By Lemma 3.6, if it is true that for


then the proof is completed.

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


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


Then it follows that




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


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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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Correspondence to Feilong Cao.

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Wang, Y., Cao, F. The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap. J Inequal Appl 2009, 205298 (2009).

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  • Measurable Function
  • Simple Calculation
  • Projection Operator
  • Elementary Surface
  • Constant Independent