PDE-PRESERVING PROPERTIES

Title & Authors
PDE-PRESERVING PROPERTIES

Abstract
A continuous linear operator T, on the space of entire functions in d variables, is PDE-preserving for a given set $\small{\mathbb{P}\;\subseteq\;\mathbb{C}|\xi_{1},\ldots,\xi_{d}|}$ of polynomials if it maps every kernel-set ker P(D), $\small{P\;{\in}\;\mathbb{P}}$, invariantly. It is clear that the set $\small{\mathbb{O}({\mathbb{P}})}$ of PDE-preserving operators for $\small{\mathbb{P}}$ forms an algebra under composition. We study and link properties and structures on the operator side $\small{\mathbb{O}({\mathbb{P}})}$ versus the corresponding family $\small{\mathbb{P}}$ of polynomials. For our purposes, we introduce notions such as the PDE-preserving hull and basic sets for a given set $\small{\mathbb{P}}$ which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for $\small{\mathbb{P}}$. We also describe PDE-preserving operators via a kernel theorem. We apply Hilberts Nullstellensatz.
Keywords
PDE-preserving;PDE-preserving hull;basic;convolution operator;exponential type;Fourier-Borel transform;algebra;invariant;Hilberts Nullstellensatz;
Language
English
Cited by
1.
HYPERCYCLICITY ON INVARIANT SUBSPACES,;

대한수학회지, 2008. vol.45. 4, pp.903-921
1.
A hypercyclicity criterion with applications, Journal of Mathematical Analysis and Applications, 2007, 327, 2, 1431
2.
Hypercyclic sequences of PDE-preserving operators, Journal of Approximation Theory, 2006, 138, 2, 168
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