On mean-value properties for the Dunkl polyharmonic functions

• Autorzy: Łysik G.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2015.35.5.655

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Dunkl Laplacian; Dunkl polyharmonic functions; mean-values; Pizzetti formula

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 5
• Strony: 655--664
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Łysik G.

• 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

Bibliografia

• [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 Goxeter group action on polynomials, [in:] Invari­ant Theory and Tableaux, IMA Vol. Math. Appl. 19, Springer-Verlag, 1990, 107-117.
• [5] C.F. Dunki, 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, Birkhauser, 2009, 91-100.
• [12] H. Mejjaoli, K. Trimeche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transforms Spec. Func. 12 (2001), 279-302.
• [13] M. Rosier, Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463.
• [14] M. Rosier, 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. Trimeche, The Dunkl intertwining operator on spaces of functions and distribu­tions and integral representation of its dual, Integral Transform Spec. Func. 12 (2001), 349-374.