Strong Converse Inequality for a Spherical Operator

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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

Let Open image in new window be the unit sphere of Open image in new window endowed with the usual rotation invariant measure Open image in new window . We denote by Open image in new window the space of all spherical harmonics of degree Open image in new window on Open image in new window and Open image in new window the space of all spherical harmonics of degree at most Open image in new window . The spaces Open image in new window Open image in new window are mutually orthogonal with respect to the inner product
so there holds
By Open image in new window and Open image in new window , Open image in new window , we denote the space of continuous, real-value functions and the space of (the equivalence classes of) Open image in new window -integrable functions defined on Open image in new window endowed with the respective norms

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 .

For an arbitrary number Open image in new window , Open image in new window , we define the spherical translation operator with step Open image in new window as (see [1, 2])

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 .

By the help of translation operator, we can define the modulus of smoothness of Open image in new window as (see [3, Chapter 10] or [4])

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.

The Laplace-Beltrami operator Open image in new window is defined by (see [5, 6])
where Open image in new window . For the modulus of smoothness and Open image in new window -functional, the following equivalent relationship has been proved (see [3, Section 10.6])

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 introduced a spherical operator Open image in new window and used it to prove the well-known Jackson type inequality for spherical harmonics. Before giving the definition of Open image in new window , we need to introduce some preliminaries. Denote
where Open image in new window denotes the group of orthogonal matrices on Open image in new window with determinants 1. We denote further
For an orthogonal matrix Open image in new window with determinant 1, we define
Now we are in the position to define the operator Open image in new window . At first we define the averaging operator Open image in new window by (see [3])
where Open image in new window represents the Haar measure on Open image in new window normalized so that
where the definition of the Haar measure can be found in [7]. Furthermore, for a measure Open image in new window supported in Open image in new window ( Open image in new window being fixed and Open image in new window is the variable) such that Open image in new window Open image in new window for Open image in new window , the operator Open image in new window is defined by

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.

Let Open image in new window be defined in (2.3) and Open image in new window , then one has

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

Lemma 2.6.

Let Open image in new window be defined in (2.3) and Open image in new window , then one has
We now give the proof of Theorem 1.1. It has been shown in (1.15) and (1.8) that there exists a constant Open image in new window such that
hence we only need to prove that there exists a constant Open image in new window such that
From (2.5) it is sufficient to prove that, for Open image in new window , there holds
In order to prove (2.9), we first prove
Indeed, from (2.1), we have
Now we turn to prove
In fact, from (2.3), we obtain
In order to estimate Open image in new window , we use (2.6) and obtain that
Using (2.2) again and (2.10), we have

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).

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Copyright information

© Shaobo Lin and Feilong Cao. 2011

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.

Authors and Affiliations

  1. 1.Institute of Metrology and Computational ScienceChina Jiliang UniversityHangzhou, Zhejiang ProvinceChina

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