Abstract
The PDE-preserving operators O on the space of nuclearly entire functions of bounded type HNb(E) on a Banach space E are characterized. An operator is PDE-preserving when it preserves homogenous solutions to homogeneous convolution equations. We establish a one to one correspondence between O and a set Σ of sequences of entire functionals, i.e. exponential type functions. In this way, algebraic structures on Σ, such as ring structures, can be carried over to O and vice versa. In particular, it follows that O is a non-commutative ring (algebra) with unity with respect to composition and the convolution operators form a commutative subring (subalgebra). We discuss range and kernel properties, for the operators in O, and characterize the projectors (onto polynomial spaces) in O by determining the corresponding elements in Σ.
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Petersson, H. The PDE-preserving operators on nuclearly entire functions of bounded type. Acta Mathematica Hungarica 100, 69–81 (2003). https://doi.org/10.1023/A:1024604100956
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DOI: https://doi.org/10.1023/A:1024604100956