# Strong Converse Inequality for a Spherical Operator

- Shaobo Lin
^{1}and - Feilong Cao
^{1}Email author

**2011**:434175

https://doi.org/10.1155/2011/434175

© Shaobo Lin and Feilong Cao. 2011

**Received: **2 July 2010

**Accepted: **8 February 2011

**Published: **2 March 2011

## Abstract

In the paper titled as "Jackson-type inequality on the sphere" (2004), Ditzian introduced a spherical nonconvolution operator , 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 and such that for any th Lebesgue integrable or continuous function defined on the sphere, where is the th modulus of smoothness of .

## 1. Introduction

In the following, will always be one of the spaces for , or for .

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

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

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

In [3], Ditzian gave a converse inequality for 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 . Their proofs can be found in [3, Theorem 6.1], [3, Theorem 6.2], and [3, equation (4.8)], respectively.

Lemma 2.2.

Lemma 2.3.

where is a polynomial of degree in . Moreover, only for

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.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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