Strong Converse Inequality for a Spherical Operator

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

Let be the unit sphere of endowed with the usual rotation invariant measure . We denote by the space of all spherical harmonics of degree on and the space of all spherical harmonics of degree at most . The spaces are mutually orthogonal with respect to the inner product

(1.1)

so there holds

(1.2)

By and , , we denote the space of continuous, real-value functions and the space of (the equivalence classes of) -integrable functions defined on endowed with the respective norms

(1.3)

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

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

(1.4)

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 .

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

(1.5)

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.

The Laplace-Beltrami operator is defined by (see [5, 6])

(1.6)

where , . We also need a -functional on sphere defined by (see [3])

(1.7)

where . For the modulus of smoothness and -functional, the following equivalent relationship has been proved (see [3, Section 10.6])

(1.8)

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 introduced a spherical operator and used it to prove the well-known Jackson type inequality for spherical harmonics. Before giving the definition of , we need to introduce some preliminaries. Denote

(1.9)

where denotes the group of orthogonal matrices on with determinants 1. We denote further

(1.10)

For an orthogonal matrix with determinant 1, we define

(1.11)

Now we are in the position to define the operator . At first we define the averaging operator by (see [3])

(1.12)

where represents the Haar measure on normalized so that

(1.13)

where the definition of the Haar measure can be found in [7]. Furthermore, for a measure supported in ( being fixed and is the variable) such that for , the operator is defined by

(1.14)

In [3], Ditzian gave a converse inequality for as follows.

Theorem A.

For any , , and some fixed , there holds

(1.15)

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

Theorem 1.1.

For any , , there holds

(1.16)

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.

For any , , there exists a constant depending only on such that

(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.

For , , and , one has

(2.2)

where .

Lemma 2.3.

For , and , there holds

(2.3)

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

Lemma 2.4.

For any , any , and , there holds

(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 be defined in (2.3) and , then one has

(2.5)

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

Lemma 2.6.

Let be defined in (2.3) and , then one has

(2.6)

We now give the proof of Theorem 1.1. It has been shown in (1.15) and (1.8) that there exists a constant such that

(2.7)

hence we only need to prove that there exists a constant such that

(2.8)

From (2.5) it is sufficient to prove that, for , there holds

(2.9)

In order to prove (2.9), we first prove

(2.10)

Indeed, from (2.1), we have

(2.11)

Now we turn to prove

(2.12)

In fact, from (2.3), we obtain

(2.13)

In order to estimate , we use (2.6) and obtain that

(2.14)

Using (2.2) again and (2.10), we have

(2.15)

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.