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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
Summary:
In this work we consider the Dunkl operator on the complex plane, defined by $$ \Cal D_k f(z)=\frac{d}{dz}f(z)+k\frac{f(z)-f(-z)}{z}, k\geq 0. $$ We define a convolution product associated with $\Cal D_k$ denoted $\ast_k$ and we study the integro-differential-difference equations of the type $\mu \ast_k f=\sum_{n=0}^{\infty}a_{n,k}\Cal D^n_k f$, 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.
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