2,843 research outputs found
Global functional calculus for operators on compact Lie groups
In this paper we develop the functional calculus for elliptic operators on
compact Lie groups without the assumption that the operator is a classical
pseudo-differential operator. Consequently, we provide a symbolic descriptions
of complex powers of such operators. As an application, we give a constructive
symbolic proof of the G\r{a}rding inequality for operators in
-classes in the setting of compact Lie groups.Comment: 23 pages; minor correction
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
We consider the frequency domain form of proper orthogonal decomposition
(POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is
derived from a space-time POD problem for statistically stationary flows and
leads to modes that each oscillate at a single frequency. This form of POD goes
back to the original work of Lumley (Stochastic tools in turbulence, Academic
Press, 1970), but has been overshadowed by a space-only form of POD since the
1990s. We clarify the relationship between these two forms of POD and show that
SPOD modes represent structures that evolve coherently in space and time while
space-only POD modes in general do not. We also establish a relationship
between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are
in fact optimally averaged DMD modes obtained from an ensemble DMD problem for
stationary flows. Accordingly, SPOD modes represent structures that are dynamic
in the same sense as DMD modes but also optimally account for the statistical
variability of turbulent flows. Finally, we establish a connection between SPOD
and resolvent analysis. The key observation is that the resolvent-mode
expansion coefficients must be regarded as statistical quantities to ensure
convergent approximations of the flow statistics. When the expansion
coefficients are uncorrelated, we show that SPOD and resolvent modes are
identical. Our theoretical results and the overall utility of SPOD are
demonstrated using two example problems: the complex Ginzburg-Landau equation
and a turbulent jet
Extension theory for elliptic partial differential operators with pseudodifferential methods
This is a short survey on the connection between general extension theories
and the study of realizations of elliptic operators A on smooth domains in R^n,
n > 1. The theory of pseudodifferential boundary problems has turned out to be
very useful here, not only as a formulational framework, but also for the
solution of specific questions. We recall some elements of that theory, and
show its application in several cases (including recent results), namely to the
lower boundedness question, and the question of spectral asymptotics for
differences between resolvents.Comment: 26 pages, style changed to LaTeX, new material added at the end, to
appear in the Lecture Notes Series of the London Math. Soc. published by
Cambridge Univ. Pres
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