Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-08T15:22:04.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Singular Integrals on Ultraspherical Series

Published online by Cambridge University Press:  20 November 2018

Charles F. Dunkl
Charles F. Dunkl*
Affiliation:
University of Virginia, Charlottesville, Virginia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.

We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 60 - 71
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Abramowitz, M. and Stegun, I., Handbook of mathematical functions, National Bureau of Standards (Dover, New York, 1964).Google Scholar
2. Askey, R. and Wainger, S., A transplantation theorem between ultraspherical series, Illinois J. Math. 10 (1966), 322–344.Google Scholar
3. Askey, R. and Wainger, S., On the behaviour of special classes of ultraspherical expansions. I, J. d'Anal. Math. 15 (1965), 193–220.Google Scholar
4. Boerner, H., Representations of groups (North-Holland, Amsterdam, 1963).Google Scholar
5. Calderón, A. P. and Zygmund, A., On singular integrals, Amer. J. Math. 78 (1956), 289–309.Google Scholar
6. Dunkl, C., Linear functionals on homogeneous polynomials, Can. Math. Bull. 11 (1968), 465–468.Google Scholar
7. Dunkl, C. and Ramirez, D., Topics in harmonic analysis (Appleton-Century-Crofts, New-York, 1971).Google Scholar
8. Hua, L. K., Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc. Translation (Amer. Math. Soc, Providence, R.I., 1963).Google Scholar
9. Levine, D., Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493–522.Google Scholar
10. Muckenhoupt, B. and Stein, E. M., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.Google Scholar
11. Seeley, R. T., Singular integrals on compact manifolds, Amer. J. Math. 81 (1959), 658–690.Google Scholar
12. Seeley, R. T., Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204.Google Scholar