Hermite interpolation and an inequality for entire functions of exponential type.

Grozev, Georgi R.; Rahman, Qazi I.

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