Abstract
In this paper the Lebesque constants (L KR (G))R>0 of Fourier series on compact Lie groups G corresponding to general one-dimensional groupings on the dual object G^ are estimated from below by the associated (abelian) Lebesgue constants (L KR (T))R>0 on a maximal torus T in G. For spherical groupings this leads to the estimate L ⊙R (G)≧const.R(l-1)/2, l=dimT≧2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
ADAMS, J.F.: Lectures on Lie groups. New York: Benjamin 1969
CAHN, R.S.: Lattice points and Lie groups I. Trans. Amer. Math. Soc.183, 119–129 (1973)
DOOLEY, A.H.: Norms of characters and lacunarity for compact Lie groups. J. Funct. Anal. (to appear)
DRESELER, B.: Lebesque constants for certain partial sums of Fourier series on compact Lie groups. In: Linear spaces and approximation. P. L. Butzer and B. Sz.-Nagy (eds.). Stuttgart: Birkhäuser 1978, 203–211
DRESELER, B.: Symmetrization formulas and norm estimates of projections in multivariate polynomial approximation. In: Proc. of the Conf. “Mehrdimensionale konstruktive Funktionentheorie” at the Res. Inst. Oberwolfach, Black Forest 1979. Stuttgart: Birkhäuser (to appear)
GIULINI, S., SOARDI, P. M., TRAVAGLINI, G.: A Cohen type inequality for compact Lie groups. Preprint
IL'IN, V. A.: Problems of localization and convergence for Fourier series with respect to fundamental systems of functions of the Laplace operator Usp. Math. Nauk23, 61–120 (1968)
MEANEY, C.: Divergence of Fourier series on compact Lie groups. Preprint
RAU, H.: Über die Lebesgue Konstanten der Reihenentwicklungen nach Jacobischen Polynomen. J. Reine Angew. Math.161, 237–254 (1929)
RAGOZIN, D.: Uniform convergence of spherical harmonic expansions. Math. Ann.195, 87–94 (1972)
TRAVAGLINI, G.: Dirichlet kernels and failure of localization principle for compact Lie groups. Preprint
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dreseler, B. Estimates from below for Lebesgue constants of Fourier series on compact Lie groups. Manuscripta Math 31, 17–23 (1980). https://doi.org/10.1007/BF01303267
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01303267