179 research outputs found
Banach algebras of pseudodifferential operators and their almost diagonalization
We define new symbol classes for pseudodifferntial operators and investigate
their pseudodifferential calculus. The symbol classes are parametrized by
commutative convolution algebras. To every solid convolution algebra over a
lattice we associate a symbol class. Then every operator with such a symbol is
almost diagonal with respect to special wave packets (coherent states or Gabor
frames), and the rate of almost diagonalization is described precisely by the
underlying convolution algebra. Furthermore, the corresponding class of
pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener's lemma holds for the underlying convolution algebra,
then the algebra of pseudodifferential operators is closed under inversion. The
theory contains as a special case the fundamental results about Sj\"ostrand's
class and yields a new proof of a theorem of Beals about the H\"ormander class
of order 0.Comment: 28 page
Orthonormal Bases in the Orbit of Square-Integrable Representations of Nilpotent Lie Groups
Let be a connected, simply connected nilpotent group and be a
square-integrable irreducible unitary representation modulo its center
on . We prove that under reasonably weak conditions on
and there exist a discrete subset of and some
(relatively) compact set such that
forms an orthonormal basis of . This construction
generalizes the well-known example of Gabor orthonormal bases in time-frequency
analysis.
The main theorem covers graded Lie groups with one-dimensional center. In the
presence of a rational structure, the set can be chosen to be a
uniform subgroup of
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