Frontiers of Mathematics in China

, Volume 9, Issue 2, pp 387–416 | Cite as

Approximation by semigroup of spherical operators

Research Article


This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere \(\mathbb{S}^n \) 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 ⊕ r V t γ and the rth Boolean of the generalized spherical Weierstrass operator ⊕ r W t κ for integer r ⩾ 1 and reals γ, κ ∈ (0, 1] have errors \(\left\| { \oplus ^r V_t^\gamma f - f} \right\|_X \asymp \omega ^{r\gamma } (f,t^{1/\gamma } )_X \) and \(\left\| { \oplus ^r W_t^\kappa f - f} \right\|_X \asymp \omega ^{r\kappa } (f,t^{1/(2\kappa )} )_X \) for all f\(X\) and 0 ⩽ t ⩽ 2π, where \(X\) is the Banach space of all continuous functions or all p integrable functions, 1 ⩽ p < +∞, on \(\mathbb{S}^n \) with norm \(\left\| \cdot \right\|_X \), and \(\omega ^s (f,t)_X \) is the modulus of smoothness of degree s > 0 for f\(X\). Moreover, ⊕ r V t γ and ⊕ r W t κ have the same saturation class if γ = 2κ.


Sphere semigroup approximation modulus of smoothness multiplier 


42C10 41A25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Askey R, Wainger S. On the behavior of special classes of ultraspherical expansions, I. J d’Analyse Math, 1965, 15: 193–220CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Askey R, Wainger S. On the behavior of special classes of ultraspherical expansions, II. J d’Analyse Math, 1965, 15: 221–244CrossRefMathSciNetGoogle Scholar
  3. 3.
    Berens H, Butzer P L, Pawelke S. Limitierungsverfahren von reihen mehrdimensionaler kugelfunktionen und deren saturationsverhalten. Publ Res Inst Math Sci Ser A, 1968, 4(2): 201–268CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bochner S. Quasi analytic functions, Laplace operator, positive kernels. Ann Math, 1950, 51(1): 68–91CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bochner S. 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, 23–48Google Scholar
  6. 6.
    Butzer P L, Berens H. Semi-groups of Operators and Approximation. Berlin: Springer, 1967CrossRefMATHGoogle Scholar
  7. 7.
    Dai F. Some equivalence theorems with K-functionals. J Approx Theory, 2003, 121: 143–157CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dai F, Ditzian Z. Strong converse inequality for Poisson sums. Proc Amer Math Soc, 2005, 133(9): 2609–2611CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ditzian Z, Ivanov K. Strong converse inequalities. J d’Analyse Math, 1993, 61: 61–111CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dunkl C F. Operators and harmonic analysis on the sphere. Trans Amer Math Soc, 1966, 125(2): 250–263CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Favard J. Sur l’approximation des fonctions d’une variable reelle. Colloque d’Anal Harmon Publ CNRS, Paris, 1949, 15: 97–110MathSciNetGoogle Scholar
  12. 12.
    Kaczmarz S, Steinhaus H. Theorie der Orthogonalreihen. Warsaw: Instytut Matematyczny Polskiej Akademi Nauk, 1935Google Scholar
  13. 13.
    Kuttner B. On positive Riesz and Abel typical means. Proc Lond Math Soc Ser 2, 1947, 49(1): 328–352CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Riemenschneider S, Wang K Y. Approximation theorems of Jackson type on the sphere. Adv Math (China), 1995, 24(2): 184–186MATHGoogle Scholar
  15. 15.
    Szegö G. Orthogonal Polynomials. Providence: Amer Math Soc, 2003Google Scholar
  16. 16.
    Wang K Y, Li L Q. Harmonic Analysis and Approximation on the Unit Sphere. Beijing: Science Press, 2006Google Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsChina Jiliang UniversityHangzhouChina

Personalised recommendations