Equivalence between Modulus of Smoothness and K-Functional on Rotation Group SO(3)*


In this paper we obtain the equivalence between modulus of smoothness and K-functional on rotation group SO(3).

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Yang, Z. and Yang, Z. (2017) Equivalence between Modulus of Smoothness and K-Functional on Rotation Group SO(3)*. Journal of Applied Mathematics and Physics, 5, 341-345. doi: 10.4236/jamp.2017.52031.

1. Introduction

Many results of approximation are based on Euclid spaces or their compact subsets. Periodic approximation is based on compact group {exp(ix)}, whereas matrix group U(n) is the generalization of {exp(ix)}. We know homomorphism between SU(2) and rotation group SO(3), which has many applications in Physics and Chemistry. Some approximation problems on compact groups have been studied since in 1920s F. Peter and H. Weyl proved the approximation theorem on compact group, that is, the irreducible character generate a dense subspace of the space of continuous classes function. For instance, Gongsheng (see [1]) studied the basic problems of Fourier analysis on unitary and rotation groups, including the degree of convergence of Abel sum based on Poisson kernel. Xue-an Zheng (see [2] [3]) studied the polynomial approximation on compact Lie groups. D. I. Cartwright et al. studied Jackson’s theorem for compact connected Lie groups (see [4]), and so on. In this paper, we study the modulus of smoothness and K-functional on rotation group SO(3) and as classical casein Euclid space we will obtain the equivalence between them.

Let be the rotation group, where is the group of invertible real (n × n) matrices. For 1 ≤ p < +∞, , where μ is the normalized Harr measure on G. For, the Lie algebra of G = SO(3), i = 1, 2, 3, Let , i = 1, 2, 3, where denote the r-order derivative of g in direction.

We also write the difference of function f and modulus of smoothness in the direction Di as follows



where is the norm induced by Killing inner product on g.

We denote


Accordingly, we denote K-functional as follows

Further, for the isotropic case.

Let multi-indice, and, , , , here is the unit vector in the i-th direction. Define






The corresponding K-functional is defined by


where, ,.

In the next paragraph we denote by C or Ci the positive constants but are not the same in the different formula. And means there exist two positive constants C1, C2 satisfying C1A ≤ B ≤ C2A.

2. Theorems and Their Proofs

We will use the next lemma 1.

Lemma 1 [5] [6]. If, then

, where Nr denotes the normalized B-spline of order r (degree r-1).

Theorem 1. If, , , , then


Proof. For i = 1, 2, 3, we first construct the approximation operators as follows

By Lemma 1,



Obviously, is a bounded operator from Lp to Lp().

If we differentiate r times, then



. (1)



We get




Conversely, for, using (see [7])


we have



Theorem 2. For, , then


Proof. Noting that for,


we get


Writing and using the last inequality will give


Moreover, we construct the approximation operator as follows



It easy to see that by using the boundedness of, i = 1, 2, 3.

It is similarly to (1), we have



Remark: Theorem 1 and theorem 2 can be easily generalized to SO(n) (n > 3).


*This research was supported by the national natural science foundation of China (11361076).

Conflicts of Interest

The authors declare no conflicts of interest.


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