Strong Converse Inequality for a Spherical Operator
© Shaobo Lin and Feilong Cao. 2011
Received: 2 July 2010
Accepted: 8 February 2011
Published: 2 March 2011
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 .
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 .
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 gave a converse inequality for as follows.
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].
Now, we give the last lemma, which can easily be deduced from [10, Theorem 3.1].
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.
The research was supported by the National Natural Science Foundation of China (no. 60873206).
- Rudin W: Uniqueness theory for Laplace series. Transactions of the American Mathematical Society 1950, 68: 287–303. 10.1090/S0002-9947-1950-0033368-1MATHMathSciNetView ArticleGoogle Scholar
- Berens H, Butzer PL, Pawelke S: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publications of the Research Institute for Mathematical Sciences 1968, 4: 201–268. 10.2977/prims/1195194875MATHMathSciNetView ArticleGoogle Scholar
- Ditzian Z: Jackson-type inequality on the sphere. Acta Mathematica Hungarica 2004,102(1–2):1–35.MATHMathSciNetView ArticleGoogle Scholar
- Rustamov KhP: On the equivalence of different moduli of smoothness on the sphere. Proceedings of the Steklov Institute of Mathematics 1993,204(3):235–260.MathSciNetGoogle Scholar
- Wang K, Li L: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing, China; 2000.Google Scholar
- Müller C: Spherical Harmonics, Lecture Notes in Mathematics. Volume 17. Springer, Berlin, Germany; 1966:iv+45.Google Scholar
- DeVore RA, Lorentz GG: Constructive Approximation. In Grundlehren Math. Wiss.. Volume 303. Springer; 1993:x+449.Google Scholar
- Dai F, Ditzian Z: Strong converse inequality for Poisson sums. Proceedings of the American Mathematical Society 2005,133(9):2609–2611. 10.1090/S0002-9939-05-08089-5MATHMathSciNetView ArticleGoogle Scholar
- Yang RY, Cao FL, Xiong JY: The strong converse inequalities for de la Vallée Poussin means on the sphere. Chinese Journal of Contemporary Mathematics, In pressGoogle Scholar
- Ditzian Z, Felten M: Averages using translation induced by Laguerre and Jacobi expansions. Constructive Approximation 2000,16(1):115–143. 10.1007/s003659910005MATHMathSciNetView ArticleGoogle Scholar
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