Opuscula Math. 35, no. 5 (2015), 655-664

Opuscula Mathematica

On mean-value properties for the Dunkl polyharmonic functions

Grzegorz Łysik

Abstract. We derive differential relations between the Dunkl spherical and solid means of continuous functions. Next we use the relations to give inductive proofs of mean-value properties for the Dunkl polyharmonic functions and their converses.

Keywords: Dunkl Laplacian, Dunkl polyharmonic functions, mean-values, Pizzetti formula.

Mathematics Subject Classification: 31A30, 31B30, 33C52.

Full text (pdf)

  1. E.F. Beckenbach, M. Reade, Mean values and harmonic polynomials, Trans. Amer. Math. Soc. 51 (1945), 240-245.
  2. C.F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  3. C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  4. C.F. Dunkl, Operators commuting with Coxeter group action on polynomials, [in:] Invariant Theory and Tableaux, IMA Vol. Math. Appl. 19, Springer-Verlag, 1990, 107-117.
  5. C.F. Dunkl, Reflection groups in analysis and applications, Japan J. Math. 3 (2008), 215-246.
  6. C.F. Dunkl, Y. Xu, Ortogonal Polynomials of Several Variables, Cambridge Univ. Press, 2001.
  7. K. Hassine, Mean value property associated with the Dunkl Laplacian, http://arxiv.org/pdf/1401.1949.pdf.
  8. G. Łysik, On the mean-value property for polyharmonic functions, Acta Math. Hungar. 133 (2011), 133-139.
  9. M. Maslouhi, On the generalized Poisson transform, Integral Transforms Spec. Func. 20 (2009), 775-784.
  10. M. Maslouhi, E.H. Youssfi, Harmonic functions associated to Dunkl operators, Monatsh. Math. 152 (2007), 337-345.
  11. M. Maslouhi, R. Daher, Weil's lemma and converse mean value for Dunkl operators, [in:] Operator Theory Adv. Math. 205, Birkhäuser, 2009, 91-100.
  12. H. Mejjaoli, K. Triméche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transforms Spec. Func. 12 (2001), 279-302.
  13. M. Rösler, Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463.
  14. M. Rösler, A positivity radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), 2413-2438.
  15. N.B. Salem, K. Touahri, Pizzetti series and polyharmonicity associated with the Dunkl Laplacian, Mediterr. J. Math. 7 (2010), 455-470.
  16. N.B. Salem, K. Touahri, Cubature formulae associated with the Dunkl Laplacian, Results Math. 58 (2010), 119-144.
  17. K. Triméche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform Spec. Func. 12 (2001), 349-374.
  • Grzegorz Łysik
  • Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, 00-656 Warsaw, Poland
  • Jan Kochanowski University, Faculty of Mathematics and Natural Sciences, ul. Świętokrzyska 15, 25-406 Kielce, Poland
  • Communicated by Semyon B. Yakubovich.
  • Received: 2013-12-13.
  • Revised: 2014-07-04.
  • Accepted: 2014-07-08.
  • Published online: 2015-04-27.
Opuscula Mathematica - cover

Cite this article as:
Grzegorz Łysik, On mean-value properties for the Dunkl polyharmonic functions, Opuscula Math. 35, no. 5 (2015), 655-664, http://dx.doi.org/10.7494/OpMath.2015.35.5.655

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.