Integro-differential-difference equations associated with the Dunkl operator and entire functions

Néjib Ben Salem; Samir Kallel

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 4, page 699-725
  • ISSN: 0010-2628

Abstract

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In this work we consider the Dunkl operator on the complex plane, defined by 𝒟 k f ( z ) = d d z f ( z ) + k f ( z ) - f ( - z ) z , k ≥ 0 . We define a convolution product associated with 𝒟 k denoted * k and we study the integro-differential-difference equations of the type μ * k f = ∑ n = 0 ∞ a n , k 𝒟 k n f , where ( a n , k ) is a sequence of complex numbers and μ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

How to cite

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Salem, Néjib Ben, and Kallel, Samir. "Integro-differential-difference equations associated with the Dunkl operator and entire functions." Commentationes Mathematicae Universitatis Carolinae 45.4 (2004): 699-725. <http://eudml.org/doc/249332>.

@article{Salem2004,
abstract = {In this work we consider the Dunkl operator on the complex plane, defined by \[ \mathcal \{D\}\_k f(z)=\frac\{d\}\{dz\}f(z)+k\frac\{f(z)-f(-z)\}\{z\}, k\ge 0. \] We define a convolution product associated with $\mathcal \{D\}_k$ denoted $\ast _k$ and we study the integro-differential-difference equations of the type $\mu \ast _k f=\sum _\{n=0\}^\{\infty \}a_\{n,k\}\mathcal \{D\}^n_k f$, where $(a_\{n,k\})$ is a sequence of complex numbers and $\mu $ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.},
author = {Salem, Néjib Ben, Kallel, Samir},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Dunkl operator; Fourier-Dunkl transform; entire function of exponential type; integro-differential-difference equation; Dunkl operator; Fourier-Dunkl transform; entire function of exponential type; integro-differential-difference equation},
language = {eng},
number = {4},
pages = {699-725},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Integro-differential-difference equations associated with the Dunkl operator and entire functions},
url = {http://eudml.org/doc/249332},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Salem, Néjib Ben
AU - Kallel, Samir
TI - Integro-differential-difference equations associated with the Dunkl operator and entire functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 4
SP - 699
EP - 725
AB - In this work we consider the Dunkl operator on the complex plane, defined by \[ \mathcal {D}_k f(z)=\frac{d}{dz}f(z)+k\frac{f(z)-f(-z)}{z}, k\ge 0. \] We define a convolution product associated with $\mathcal {D}_k$ denoted $\ast _k$ and we study the integro-differential-difference equations of the type $\mu \ast _k f=\sum _{n=0}^{\infty }a_{n,k}\mathcal {D}^n_k f$, where $(a_{n,k})$ is a sequence of complex numbers and $\mu $ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.
LA - eng
KW - Dunkl operator; Fourier-Dunkl transform; entire function of exponential type; integro-differential-difference equation; Dunkl operator; Fourier-Dunkl transform; entire function of exponential type; integro-differential-difference equation
UR - http://eudml.org/doc/249332
ER -

References

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  2. Ben Salem N., Masmoudi W., Integro-differential equations associated with the Bessel operator on the complex domain, C.R. Math. Rep. Acad. Sci. Canada 18 6 257-262 (1996). (1996) Zbl0880.45008MR1441647
  3. Boas R.P., Jr., Entire Functions, Academic Press, New York, 1954. Zbl0293.30026MR0068627
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  8. Mourou M.A., Taylor series associated with a differential-difference operator on the real line, J. Comput. Appl. Math. 153 343-354 (2003). (2003) Zbl1028.34058MR1985705
  9. Mugler D.H., Convolution, differential equations, and entire function of exponential type, Trans. Amer. Math. Soc. 216 145-187 (1976). (1976) MR0387587
  10. Rosenblum M., Generalized Hermite Polynomials and the Bose-like Oscillator Calculus, in: Operator Theory: Advances and Applications, vol. 73, Birkhäuser Verlag, Basel, 1994, pp.369-396. MR1320555
  11. Rösler M., Bessel-type signed hypergroups on ℝ , Probability Measures on Groups and Related Structures XI, Proceedings, Oberwollach, 1994 (H. Heyer and A. Mukherjea, Eds.), World Sci. Publishing, Singapore, 1995. MR1414942

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