Approximation by semigroup of spherical operators

Abstract

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere {\mathbb{S}}^{\mathrm{n}} of the (n + 1)-dimensional Euclidean space for n≥2.We prove that such operators form a strongly continuous contraction semigroup of class (C_0) and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator {\oplus }^r{V}_t^{\gamma } and the rth Boolean of the generalized spherical Weierstrass operator {\oplus }^r{W}_t^k for integer r≥1 and reals γ, \kappa \in (0, 1] have errors \mathit{\parallel }{\oplus }^r{V}_t^{\gamma }f-f{\mathit{\parallel }}_X\approx {\omega }^{r\gamma }(f,t^{\mathrm{1}/\gamma }{)}_X and \mathit{\parallel }{\oplus }^r{W}_t^{k}f-f{\mathit{\parallel }}_X\approx {\omega }^{\mathrm{2}rk}{(f,t^{\mathrm{1}/(\mathrm{2}k)})}_X for all f\in X and 0≤t≤2π, where Xis the Banach space of all continuous functions or all L^pintegrable functions, 1≤p<+∞, on {\mathbb{S}}^{\mathrm{n}} with norm \mathit{\parallel }\cdot {\mathit{\parallel }}_X, and {\omega }^s{(f,t)}_Xis the modulus of smoothness of degree s>0 for f\in X. Moreover, {\oplus }^r{V}_t^{\gamma } and {\oplus }^r{W}_t^k have the same saturation class if \gamma =\mathrm{2}\kappa.

Key words： Sphere; semigroup; approximation; modulus of smoothness; multiplier

References

1 AskeyR, WaingerS. On the behavior of special classes of ultraspherical expansions, I. J d’Analyse Math, 1965, 15: 193-220

2 AskeyR, WaingerS. On the behavior of special classes of ultraspherical expansions, II. J d’Analyse Math, 1965, 15: 221-244

3 BerensH, ButzerP L, PawelkeS. Limitierungsverfahren von reihen mehrdimensionaler kugelfunktionen und deren saturationsverhalten. Publ Res Inst Math Sci Ser A, 1968, 4(2): 201-268

4 BochnerS. Quasi analytic functions, Laplace operator, positive kernels. Ann Math, 1950, 51(1): 68-91

5 BochnerS. Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions. In: Proceedings of the Conference on Differential Equations. University of Maryland, 1955, 23-48

6 ButzerP L, BerensH. Semi-groups of Operators and Approximation. Berlin: Springer, 1967

7 DaiF. Some equivalence theorems with K-functionals. J Approx Theory, 2003, 121: 143-157

8 DaiF, DitzianZ. Strong converse inequality for Poisson sums. Proc Amer Math Soc, 2005, 133(9): 2609-2611

9 DitzianZ, IvanovK. Strong converse inequalities. J d’Analyse Math, 1993, 61: 61-111

10 DunklC F. Operators and harmonic analysis on the sphere. Trans Amer Math Soc, 1966, 125(2): 250-263

11 FavardJ. Sur l’approximation des fonctions d’une variable reelle. Colloque d’Anal Harmon Publ CNRS, Paris, 1949, 15: 97-110

12 KaczmarzS, SteinhausH. Theorie der Orthogonalreihen. Warsaw: Instytut Matematyczny Polskiej Akademi Nauk, 1935

13 KuttnerB. On positive Riesz and Abel typical means. Proc Lond Math Soc Ser 2, 1947, 49(1): 328-352

14 RiemenschneiderS, WangK Y. Approximation theorems of Jackson type on the sphere. Adv Math (China), 1995, 24(2): 184-186

15 SzegöG. Orthogonal Polynomials. Providence: Amer Math Soc, 2003

16 WangK Y, LiL Q. Harmonic Analysis and Approximation on the Unit Sphere. Beijing: Science Press, 2006