Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on n and on compact Lie groups)

Giancarlo Travaglini

Colloquium Mathematicae (1993)

  • Volume: 65, Issue: 1, page 103-116
  • ISSN: 0010-1354

Abstract

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We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity ∑ j = - N N e i j t = s i n ( ( N + ( 1 / 2 ) ) t ) / s i n ( ( 1 / 2 ) t ) . We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.

How to cite

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Travaglini, Giancarlo. "Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $^n$ and on compact Lie groups)." Colloquium Mathematicae 65.1 (1993): 103-116. <http://eudml.org/doc/210195>.

@article{Travaglini1993,
abstract = {We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity $∑_\{j=-N\}^N e^\{ijt\} = sin((N+(1/2))t) / sin((1/2)t)$. We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.},
author = {Travaglini, Giancarlo},
journal = {Colloquium Mathematicae},
keywords = {Fourier series on compact Lie groups; Lebesgue constants; polyhedral Dirichlet kernels; multiple Fourier series; approximation; summability; asymptotic formula},
language = {eng},
number = {1},
pages = {103-116},
title = {Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $^n$ and on compact Lie groups)},
url = {http://eudml.org/doc/210195},
volume = {65},
year = {1993},
}

TY - JOUR
AU - Travaglini, Giancarlo
TI - Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $^n$ and on compact Lie groups)
JO - Colloquium Mathematicae
PY - 1993
VL - 65
IS - 1
SP - 103
EP - 116
AB - We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity $∑_{j=-N}^N e^{ijt} = sin((N+(1/2))t) / sin((1/2)t)$. We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.
LA - eng
KW - Fourier series on compact Lie groups; Lebesgue constants; polyhedral Dirichlet kernels; multiple Fourier series; approximation; summability; asymptotic formula
UR - http://eudml.org/doc/210195
ER -

References

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  15. [15] P. M. Soardi, Serie di Fourier in più variabili, U.M.I., Bologna 1984. 
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  17. [17] V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Prentice-Hall, Englewood Cliffs 1974. Zbl0371.22001
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