405 research outputs found
The periodic decomposition problem
If a function can be represented as the sum of
periodic functions as with
(), then it also satisfies a corresponding -order difference
equation . The periodic
decomposition problem asks for the converse implication, which may hold or fail
depending on the context (on the system of periods, on the function class in
which the problem is considered, etc.). The problem has natural extensions and
ramifications in various directions, and is related to several other problems
in real analysis, Fourier and functional analysis. We give a survey about the
available methods and results, and present a number of intriguing open
problems
COMMUTATION PROPERTIES OF THE FORM SUM OF POSITIVE, SYMMETRIC OPERATORS
A new construction for the form sum of positive, selfadjoint operators is given in this
paper. The situation is a bit more general, because our aim is to add positive, symmetric
operators. With the help of the used method, some commutation properties of the form sum
extension are observed
Positive forms on Banach spaces
The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case
Operator splitting for nonautonomous delay equations
We provide a general product formula for the solution of nonautonomous
abstract delay equations. After having shown the convergence we obtain
estimates on the order of convergence for differentiable history functions.
Finally, the theoretical results are demonstrated on some typical numerical
examples.Comment: to appear in "Computers & Mathematics with Applications (CAMWA)
Operator splitting for dissipative delay equations
We investigate Lie-Trotter product formulae for abstract nonlinear evolution
equations with delay. Using results from the theory of nonlinear contraction
semigroups in Hilbert spaces, we explain the convergence of the splitting
procedure. The order of convergence is also investigated in detail, and some
numerical illustrations are presented.Comment: to appear in Semigroup Foru
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