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Explicit formulas for the Dunkl dihedral kernel and the ( κ , a )-generalized Fourier kernel

Denis Constales (UGent) , Hendrik De Bie (UGent) and Pan Lian (UGent)
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Abstract
In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (k,a)-generalized Fourier transform for k=0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag–Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the (k,a)-generalized Fourier transform are obtained as well.
Keywords
Dunkl kernel, Generalized Fourier transform, Dihedral group, Bessel function, Poisson kernel

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Chicago
Constales, Denis, Hendrik De Bie, and Pan Lian. 2018. “Explicit Formulas for the Dunkl Dihedral Kernel and the ( κ , a )-generalized Fourier Kernel.” Journal of Mathematical Analysis and Applications 460 (2): 900–926.
APA
Constales, D., De Bie, H., & Lian, P. (2018). Explicit formulas for the Dunkl dihedral kernel and the ( κ , a )-generalized Fourier kernel. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 460(2), 900–926.
Vancouver
1.
Constales D, De Bie H, Lian P. Explicit formulas for the Dunkl dihedral kernel and the ( κ , a )-generalized Fourier kernel. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495  USA: Elsevier BV; 2018;460(2):900–26.
MLA
Constales, Denis, Hendrik De Bie, and Pan Lian. “Explicit Formulas for the Dunkl Dihedral Kernel and the ( κ , a )-generalized Fourier Kernel.” JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 460.2 (2018): 900–926. Print.
@article{8551385,
  abstract     = {In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (k,a)-generalized Fourier transform for k=0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag--Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the (k,a)-generalized Fourier transform are obtained as well.},
  author       = {Constales, Denis and De Bie, Hendrik and Lian, Pan},
  issn         = {0022-247X},
  journal      = {JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS},
  keyword      = {Dunkl kernel,Generalized Fourier transform,Dihedral group,Bessel function,Poisson kernel},
  language     = {eng},
  number       = {2},
  pages        = {900--926},
  publisher    = {Elsevier BV},
  title        = {Explicit formulas for the Dunkl dihedral kernel and the ( \ensuremath{\kappa} , a )-generalized Fourier kernel},
  url          = {http://dx.doi.org/10.1016/j.jmaa.2017.12.018},
  volume       = {460},
  year         = {2018},
}

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