29,501 research outputs found
Operational calculus and integral transforms for groups with finite propagation speed
Let be the generator of a strongly continuous cosine family on a complex Banach space . The paper develops an
operational calculus for integral transforms and functions of using the
generalized harmonic analysis associated to certain hypergroups. It is shown
that characters of hypergroups which have Laplace representations give rise to
bounded operators on . Examples include the Mellin transform and the
Mehler--Fock transform. The paper uses functional calculus for the cosine
family which is associated with waves that travel at
unit speed. The main results include an operational calculus theorem for
Sturm--Liouville hypergroups with Laplace representation as well as analogues
to the Kunze--Stein phenomenon in the hypergroup convolution setting.Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868.
Substantial revision to version
Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
A quantum principal bundle is constructed for every Coxeter group acting on a
finite-dimensional Euclidean space , and then a connection is also defined
on this bundle. The covariant derivatives associated to this connection are the
Dunkl operators, originally introduced as part of a program to generalize
harmonic analysis in Euclidean spaces. This gives us a new, geometric way of
viewing the Dunkl operators. In particular, we present a new proof of the
commutativity of these operators among themselves as a consequence of a
geometric property, namely, that the connection has curvature zero
Dynamics in a noncommutative phase space
Dynamics has been generalized to a noncommutative phase space. The
noncommuting phase space is taken to be invariant under the quantum group
. The -deformed differential calculus on the phase space is
formulated and using this, both the Hamiltonian and Lagrangian forms of
dynamics have been constructed. In contrast to earlier forms of -dynamics,
our formalism has the advantage of preserving the conventional symmetries such
as rotational or Lorentz invariance.Comment: LaTeX-twice, 16 page
Functional calculus for generators of symmetric contraction semigroups
We prove that every generator of a symmetric contraction semigroup on a
-finite measure space admits, for , a H\"ormander-type
holomorphic functional calculus on in the sector of angle
. The obtained angle is optimal.Comment: 26 pages, minor corrections and slight changes of notation. Some
changes in Sections 4, 6 and
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