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
so there holds
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
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])
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 )
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])
where , . We also need a -functional on sphere defined by (see )
where . For the modulus of smoothness and -functional, the following equivalent relationship has been proved (see [3, Section 10.6])
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 , 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
where denotes the group of orthogonal matrices on with determinants 1. We denote further
For an orthogonal matrix with determinant 1, we define
Now we are in the position to define the operator . At first we define the averaging operator by (see )
where represents the Haar measure on normalized so that
where the definition of the Haar measure can be found in . Furthermore, for a measure supported in ( being fixed and is the variable) such that for , the operator is defined by
In , Ditzian gave a converse inequality for as follows.
For any , , and some fixed , there holds
In this paper, we improve this result. Motivated by [8, 9], we obtain the following Theorem 1.1.
For any , , there holds