846 research outputs found
Fast and accurate con-eigenvalue algorithm for optimal rational approximations
The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small error.
Specifically, given a rational function with poles in the unit disk, a
rational approximation with poles in the unit disk may be obtained
from the th con-eigenvector of an Cauchy matrix, where the
associated con-eigenvalue gives the approximation error in the
norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the th con-eigenvalue in operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
operations, where is the
approximation error in the norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision
On truncations of Hankel and Toeplitz operators
We study the boundedness properties of truncation operators acting on bounded Hankel (or Toeplitz) infinite matrices. A relation with the Lacey-Thiele theorem on the bilinear Hilbert transform is established. We also study the behaviour of the truncation operators when restricted to Hankel matrices in the Schatten classes
Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups
We prove that defines a
completely bounded semigroup of multipliers on the von Neuman algebra of
hyperbolic groups for all real number . One ingredient in the proof is the
observation that a construction of Ozawa allows to characterize the radial
multipliers that are bounded on every hyperbolic graph, partially generalizing
results of Haagerup--Steenstrup--Szwarc and Wysocza\'nski. Another ingredient
is an upper estimate of trace class norms for Hankel matrices, which is based
on Peller's characterization of such norms.Comment: v2: 28 pages, with new examples, new results, motivations and
hopefully a better presentatio
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