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Adaptive Harmonic Analysis
In this paper we describe a new approach to the harmonic analysis of the tide. For a number of reasons the harmonic constants are not really constant but vary slowly in time. Therefore, we introduce a narrow-band noise process to model the time-varying behaviour of these harmonic parameters. Furthermore, since the measurements available are not perfect, we also introduce a, possibly time-varying, measurement noise process to model the errors associated with the measurement process. By employing a Kalman filter to estimate the harmonic parameters recursively, the estimates can be adapted contineously to chaning conditions. The adaptive harmonic analysis can be used for the on-line prediction of the astronomical tide or, since the Kalman filter also produces the covariance of the estimation error, to gain quantitative insight into the resolution of tidal constituents
Projective Pseudodifferential Analysis and Harmonic Analysis
We consider pseudodifferential operators on functions on which
commute with the Euler operator, and can thus be restricted to spaces of
functions homogeneous of some given degree. Their symbols can be regarded as
functions on a reduced phase space, isomorphic to the homogeneous space
, and the resulting calculus is a
pseudodifferential analysis of operators acting on spaces of appropriate
sections of line bundles over the projective space : these spaces are
the representation spaces of the maximal degenerate series
of . This new approach to the quantization of
, already considered by other authors, has several advantages: as an
example, it makes it possible to give a very explicit version of the continuous
part from the decomposition of under the quasiregular action of
. We also consider interesting special symbols, which arise from the
consideration of the resolvents of certain infinitesimal operators of the
representation
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