# Strong Converse Inequality for a Spherical Operator

- 894 Downloads

## Abstract

In the paper titled as "Jackson-type inequality on the sphere" (2004), Ditzian introduced a spherical nonconvolution operator Open image in new window , which played an important role in the proof of the well-known Jackson inequality for spherical harmonics. In this paper, we give the lower bound of approximation by this operator. Namely, we prove that there are constants Open image in new window and Open image in new window such that Open image in new window for any Open image in new window th Lebesgue integrable or continuous function Open image in new window defined on the sphere, where Open image in new window is the Open image in new window th modulus of smoothness of Open image in new window .

### Keywords

Equivalence Class Invariant Measure Unit Sphere Spherical Harmonic Haar Measure## 1. Introduction

In the following, Open image in new window will always be one of the spaces Open image in new window for Open image in new window , or Open image in new window for Open image in new window .

where Open image in new window means the Open image in new window -dimensional surface area of sphere embedded into Open image in new window . Here we integrate over the family of points Open image in new window whose spherical distance from the given point Open image in new window (i.e., the length of minor arc between Open image in new window and Open image in new window on the great circle passing through them) is equal to Open image in new window . Thus Open image in new window can be interpreted as the mean value of the function Open image in new window on the surface of a Open image in new window -dimensional sphere with radius Open image in new window .

Clearly, the modulus is meaningful to describe the approximation degree and the smoothness of Open image in new window , which has been widely used in the study of approximation on sphere.

Throughout this paper, we denote by Open image in new window the positive constants independent of Open image in new window and Open image in new window and by Open image in new window the positive constants depending only on Open image in new window . Their value will be different at different occurrences, even within the same formula. By Open image in new window we denote that there are positive constants Open image in new window and Open image in new window such that Open image in new window .

In [3], Ditzian gave a converse inequality for Open image in new window as follows.

Theorem A.

In this paper, we improve this result. Motivated by [8, 9], we obtain the following Theorem 1.1.

Theorem 1.1.

## 2. The Proof of Main Result

Before proceeding the proof, we state some useful lemmas at first. The first one can be find in [3, page 6].

Lemma 2.1.

The following three lemmas reveal some important properties of Open image in new window . Their proofs can be found in [3, Theorem 6.1], [3, Theorem 6.2], and [3, equation (4.8)], respectively.

Lemma 2.2.

where Open image in new window .

Lemma 2.3.

where Open image in new window is a polynomial of degree Open image in new window in Open image in new window . Moreover, Open image in new window only for Open image in new window

Lemma 2.4.

From (1.8) and [10, Theorem 3.2] (see also [3, page 16]) we deduce the following Lemma 2.5 easily.

Lemma 2.5.

Now, we give the last lemma, which can easily be deduced from [10, Theorem 3.1].

Lemma 2.6.

The above inequality together with (2.13) and (2.10) yields (2.12). Then we can deduce (2.9) from (2.12) and (2.10) easily. Therefore (2.8) holds. This completes the proof of Theorem 1.1.

## Notes

### Acknowledgment

The research was supported by the National Natural Science Foundation of China (no. 60873206).

### References

- 1.Rudin W:
**Uniqueness theory for Laplace series.***Transactions of the American Mathematical Society*1950,**68:**287–303. 10.1090/S0002-9947-1950-0033368-1MATHMathSciNetCrossRefGoogle Scholar - 2.Berens H, Butzer PL, Pawelke S:
**Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten.***Publications of the Research Institute for Mathematical Sciences*1968,**4:**201–268. 10.2977/prims/1195194875MATHMathSciNetCrossRefGoogle Scholar - 3.Ditzian Z:
**Jackson-type inequality on the sphere.***Acta Mathematica Hungarica*2004,**102**(1–2):1–35.MATHMathSciNetCrossRefGoogle Scholar - 4.Rustamov KhP:
**On the equivalence of different moduli of smoothness on the sphere.***Proceedings of the Steklov Institute of Mathematics*1993,**204**(3):235–260.MathSciNetGoogle Scholar - 5.Wang K, Li L:
*Harmonic Analysis and Approximation on the Unit Sphere*. Science Press, Beijing, China; 2000.Google Scholar - 6.Müller C:
*Spherical Harmonics, Lecture Notes in Mathematics*.*Volume 17*. Springer, Berlin, Germany; 1966:iv+45.Google Scholar - 7.DeVore RA, Lorentz GG:
**Constructive Approximation.**In*Grundlehren Math. Wiss.*.*Volume 303*. Springer; 1993:x+449.Google Scholar - 8.Dai F, Ditzian Z:
**Strong converse inequality for Poisson sums.***Proceedings of the American Mathematical Society*2005,**133**(9):2609–2611. 10.1090/S0002-9939-05-08089-5MATHMathSciNetCrossRefGoogle Scholar - 9.Yang RY, Cao FL, Xiong JY:
**The strong converse inequalities for de la Vallée Poussin means on the sphere.**Chinese Journal of Contemporary Mathematics, In pressGoogle Scholar - 10.Ditzian Z, Felten M:
**Averages using translation induced by Laguerre and Jacobi expansions.***Constructive Approximation*2000,**16**(1):115–143. 10.1007/s003659910005MATHMathSciNetCrossRefGoogle Scholar

## Copyright information

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.