Decomposing kernels of iterated operators - a unified approach

For any operator D acting in an Abelian group, we study the kernel of its iterates Dk and describe a general approach for decomposing it through the kernel of the operator D itself and some other given operators T1,...,Tk-1. Due to Almansiʹs famous theorem for polyharmonic functions the different types of decomposition are characterized in terms of strong, weak and restricted Almansi decomposition properties. Sufficient conditions are given for the existence of such decompositions. The case of the iterated Dirac operator (cf. Math. Meth. Appl. Sci. 2002; 25:1541-1552) follows as a special case. Several other special cases are discussed. Finally we prove corresponding decomposition theorems for the iterated weighted Laplacian (|x|a(delta))k, a(element of)(-(infinity), 2), and the iterated Helmholtz type operator ((delta)-(lambda))k, (lambda)(element of)C.