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A Paley-Wiener theorem for step two nilpotent Lie groups.

Sundaram Thangavelu (1994)

Revista Matemática Iberoamericana

It is an interesting open problem to establish Paley-Wiener theorems for general nilpotent Lie groups. The aim of this paper is to prove one such theorem for step two nilpotent Lie groups which is analogous to the Paley-Wiener theorem for the Heisenberg group proved in [4].

A weighted Plancherel formula II. The case of the ball

Genkai Zhang (1992)

Studia Mathematica

The group SU(1,d) acts naturally on the Hilbert space L ² ( B d μ α ) ( α > - 1 ) , where B is the unit ball of d and d μ α the weighted measure ( 1 - | z | ² ) α d m ( z ) . It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

Almost periodic and strongly stable semigroups of operators

Vũ Phóng (1997)

Banach Center Publications

This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators T n : n = 0 , 1 , . . . on a Banach space X (discrete one-parameter semigroups), one-parameter C 0 -semigroups T ( t ) : t 0 on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded...

An extension of deLeeuw’s theorem to the n -dimensional rotation group

Anthony H. Dooley, Garth I. Gaudry (1984)

Annales de l'institut Fourier

We study a method of approximating representations of the group M ( n ) by those of the group S O ( n + 1 ) . As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of L p that applies to the “restrictions” of a function on the dual of M ( n ) to the dual of S O ( n + 1 ) .

Averages of unitary representations and weak mixing of random walks

Michael Lin, Rainer Wittmann (1995)

Studia Mathematica

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; U n converges weakly for every continuous unitary representation of G; U is weakly mixing for any...

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