Infinite order differential operators in spaces of entire functions. (English) Zbl 1039.35146
The authors deal with the differentiation with respect to a single complex variable \(z\in\mathbb{C}\) and the operators constructed by means of the following differential expression
\[
\Delta_{\theta,\omega} = \Delta_\theta +\omega zD \overset {\text{def}} = (\theta+zD)D + \omega zD, \tag{1}
\]
where \(D=\frac {\partial} {\partial z}\) and \(\theta \geq 0\), \(\omega\in \mathbb{R}\) are parameters. Given entire functions \(\varphi, f:\mathbb{C}\to\mathbb{C}\) we set
\[
\bigl(\varphi (\Delta_{\theta, \omega}) f \bigr) (z)= \sum^\infty_{k=1} \frac{1}{k!} \varphi^{(k)} (0)( \Delta^k_{\theta,\omega} f)(z). \tag{2}
\]
It is shown that, for \(\omega \geq 0\), such operators preserve the set of Laguerre entire functions provided the function \(\varphi\) also belongs to this set. Moreover an integral representation of \(\exp(a\Delta_{\theta,\omega})\), \(a>0\) is obtained.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
35R50 | PDEs of infinite order |
30D10 | Representations of entire functions of one complex variable by series and integrals |
30E25 | Boundary value problems in the complex plane |
Keywords:
Fréchet spaces; Exponential type entire functions; Laguerre entire functions; Cauchy problemReferences:
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