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Direct and converse results in the Ba space for Jackson-Matsuoka polynomials on the unit sphere
Journal of Inequalities and Applications volume 2014, Article number: 419 (2014)
In this paper, we introduce K-functional and modulus of smoothness of the unit sphere in the Ba space, establish their relations and obtain the direct and converse theorem of approximation in the Ba space for Jackson-Matsuoka polynomials on the unit sphere of
MSC:41A25, 41A35, 41A63.
Let denote the unit sphere in (), , where denotes the usual Euclidean norm, the set of nonnegative integers, and ℕ the set of positive integers. We denote by , , the space of functions defined on with the finite norm
where , and dϖ is the measure element on , and is the surface area of .
The conception of Ba space was first put forward by Ding and Luo (see ) in their discussion of the prior estimate of Laplace operator in some classical domains and in their study of the embedding theorem of Orlicz-Sobolev spaces, higher dimensional singular integrals, and harmonic function etc.
Definition 1.1 (see )
Let be a sequence of linear normed function spaces, be a sequence of nonnegative numbers. For , we form the power series of
If has a non-zero radius of convergence, we say .
The norm in Ba is defined by
As proved in , Ba is a Banach space if is a Banach space. Evidently, if , then Ba space is an Orlicz space. If , , then a Ba space is a classical Lebesgue space.
Hereafter the space of spherical harmonics of degree k is denoted by . The Laplace-Beltrami operator on the unit sphere is denoted by
which has eigenvalue corresponding to the eigenspace with , namely, . For the properties of the space of spherical harmonics and the Laplace-Beltrami operators, see [2–4]. The standard Hilbert space theory shows that . The orthogonal projection takes the form
where , denotes hyperspherical polynomials of degree k which satisfies , .
The spherical means are denoted by
Based on the classical Jackson-Matsuoka kernel (see ) we define a new kernel
where , is chosen such that . It is well known that is an even nonnegative operator. In particular, it is an even and nonnegative trigonometric polynomial of degree at most for and the Jackson polynomial for . Using we consider spherical convolution:
It is called the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka kernel. In particular, for . The classical Jackson-Matsuoka polynomial in classical space has been studied by many authors (see [7, 8]).
In this paper, we consider the approximation of the Jackson-Matsuoka polynomial on the unit sphere in the Ba space. Firstly, we introduce K-functionals, modulus of smoothness on the unit sphere in the Ba space, establish their relations. Then with the help of the relation between K-functionals and modulus of smoothness on the sphere in the Ba space and the properties of the spherical means, we obtain the direct and converse best approximation in the Ba space by Jackson-Matsuoka polynomial on the unit sphere of .
2 K-Functionals and modulus of smoothness
Definition 2.1 For , the modulus of smoothness on the unit sphere is given by
The K-functional of the unit sphere is given by
where , , is a positive constant, denotes the Laplace-Beltrami operator on the unit sphere.
To prove the weak equivalence between the K-functional and the modulus of smoothness on the unit sphere, we need the following lemma.
Lemma 2.2 Let be a sequence of Lebesgue spaces, , , be a sequence of nonnegative numbers, , . If , then
Proof Since , we may let . From , we may let . Then .
In view of the , the exists. Let
By the definition of supremum, for any , there exists , such that . By the definition of , for any , there exists , such that holds. Therefore . Namely
By the arbitrariness of δ,
and also ε is arbitrary, therefore
which implies that for any , we have
The proof is completed. □
We will establish the weak equivalence between the K-functional and the modulus of smoothness on the unit sphere in the Ba space.
Theorem 2.3 Let be a sequence of Lebesgue spaces, , , be a sequence of nonnegative numbers. If , . Then for , , the weak equivalence
holds, where the weakly equivalent relation means that and , and relation means that there is a positive constant C independent on n such that holds.
Throughout this paper, C denotes a positive constant independent on n and f and denotes a positive constant dependent on a, which may be different according to the circumstances.
Proof For , , note that 
By the definition of the Ba-norm and (2.3), we have
Let , then . Consequently . Therefore, we have
The proof is similar to that of (2.6), we get
The triangle inequality gives
which shows that . On the other hand, we define
with . Then , this also gives
Since for , the inequality shows that . Moreover,
Consequently, we get
By (2.8) and (2.9), similar to the proof of (2.6), we obtain
Combining (2.10), (2.11), and the definition of K-functional, we have
Corollary 2.4 For , there is a constant C such that
Proof By the weakly equivalent relation between the modulus of smoothness and K-functional, and the definition of , we have
Corollary 2.4 has been proved. □
3 Some lemmas
Lemma 3.1 Let . Then the weak equivalence
holds for , .
Proof As , and for , we have
since , . Lemma 3.1 has been proved. □
Lemma 3.2 For , , , there is a constant such that
Proof Since , and for , by , we have
Lemma 3.3 (see )
Suppose that . Then, for and , we have
Lemma 3.4 Let , , , , be the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka kernel, . Then there is a constant such that
Proof For , by (3.5), we have
Using Lemma 3.3, and the expression of , we obtain
By Lemma 3.2, and the Hölder-Minkowski inequality we get
Consequently, by (3.7), (3.8), and (3.9), we get
By Lemma 2.2, we have
The proof is completed. □
4 Main results
Theorem 4.1 Suppose that , , , be the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka kernel, , , . Then
Proof Since for , Therefore, we have
Splitting the integral on into two integrals on and , respectively, and using the definition of , we conclude that
From Corollary 2.4 we have, for ,
By (4.3), (4.4), and Lemma 3.1, we get
Therefore, by (4.2), (4.5), we have
Theorem 4.2 Suppose that , , is the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka kernel, , , , . Then the following statements are equivalent:
Proof By Theorem 4.1, we have (2) ⇒ (1). Now, we prove (1) ⇒ (2). Let r be a fixed positive integer, defined by
By orthogonality of the orthogonal projector , we have
Let , by (4.9) we get
On the other hand,
Note that 
For , from (2.4) we have
holds for . For , by Lemma 3.2, we get
Consequently, the inequality
holds uniformly for . Thereby
Without loss of generality, we may assume , . Using Lemma 3.4 and (4.9), we have
Consequently, considering , by the definition of , and Theorem 2.3, we have
In view of (4.7), we get
Let , we have
The proof is completed. □
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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 made it more readable. Project was 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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Feng, G., Feng, Y. Direct and converse results in the Ba space for Jackson-Matsuoka polynomials on the unit sphere. J Inequal Appl 2014, 419 (2014). https://doi.org/10.1186/1029-242X-2014-419
- Jackson-Matsuoka polynomials
- Ba space
- modulus of smoothness
- spherical means