2,182 research outputs found
Locating Ax, where A is a subspace of B(H)
Let A be a linear space of operators on a Hilbert space H, x a vector in H,
and Ax the subspace of H comprising all vectors of the form Tx with T in A. We
discuss, within a Bishop-style constructive framework, conditions under which
the projection [Ax] of H on the closure of Ax exists. We derive a general
result that leads directly to both the open mapping theorem and our main
theorem on the existence of [Ax]
Grid-free compressive beamforming
The direction-of-arrival (DOA) estimation problem involves the localization
of a few sources from a limited number of observations on an array of sensors,
thus it can be formulated as a sparse signal reconstruction problem and solved
efficiently with compressive sensing (CS) to achieve high-resolution imaging.
On a discrete angular grid, the CS reconstruction degrades due to basis
mismatch when the DOAs do not coincide with the angular directions on the grid.
To overcome this limitation, a continuous formulation of the DOA problem is
employed and an optimization procedure is introduced, which promotes sparsity
on a continuous optimization variable. The DOA estimation problem with
infinitely many unknowns, i.e., source locations and amplitudes, is solved over
a few optimization variables with semidefinite programming. The grid-free CS
reconstruction provides high-resolution imaging even with non-uniform arrays,
single-snapshot data and under noisy conditions as demonstrated on experimental
towed array data.Comment: 14 pages, 8 figures, journal pape
Weak and cyclic amenability for Fourier algebras of connected Lie groups
Using techniques of non-abelian harmonic analysis, we construct an explicit,
non-zero cyclic derivation on the Fourier algebra of the real group. In
particular this provides the first proof that this algebra is not weakly
amenable. Using the structure theory of Lie groups, we deduce that the Fourier
algebras of connected, semisimple Lie groups also support non-zero, cyclic
derivations and are likewise not weakly amenable. Our results complement
earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and
Forrest--Samei--Spronk (IUMJ 2009). As an additional illustration of our
techniques, we construct an explicit, non-zero cyclic derivation on the Fourier
algebra of the reduced Heisenberg group, providing the first example of a
connected nilpotent group whose Fourier algebra is not weakly amenable.Comment: v4: AMS-LaTeX, 26 pages. Final version, to appear in JFA. Includes an
authors' correction added at proof stag
- …