97 research outputs found
Constrained -approximation by polynomials on subsets of the circle
We study best approximation to a given function, in the least square sense on
a subset of the unit circle, by polynomials of given degree which are pointwise
bounded on the complementary subset. We show that the solution to this problem,
as the degree goes large, converges to the solution of a bounded extremal
problem for analytic functions which is instrumental in system identification.
We provide a numerical example on real data from a hyperfrequency filter
Robust identification from band-limited data
Consider the problem of identifying a scalar bounded-input/bounded-output stable transfer function from pointwise measurements at frequencies within a bandwidth. We propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity, the noise level to zero, and asymptotically meeting some gauge constraint outside. Error bounds are derived, and the procedure is illustrated by numerical experiment
Asymptotic estimates for interpolation and constrained approximation in H2 by diagonalization of Toeplitz operators
Sharp convergence rates are provided for interpolation and approximation schemes in the Hardy space H-2 that use band-limited data. By means of new explicit formulae for the spectral decomposition of certain Toeplitz operators, sharp estimates for Carleman and Krein-Nudel'man approximation schemes are derived. In addition, pointwise convergence results are obtained. An illustrative example based on experimental data from a hyperfrequency filter is provided
An Lp Analog to AAK Theory for p⩾2
AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication
Weighted Approximation of Transfer Functions
The aim of this work is to generalize to the case of weighted spaces some results about approximation by analytic and rational functions which are useful to perform the identification of unknown transfer functions of stable (linear causal time--invariant) systems from incomplete frequency data
Decomposition of -vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential
For the boundary of a bounded and connected strongly
Lipschitz domain in with , we prove that any field
decomposes, in an unique way,
as the sum of three silent vector fields---fields whose magnetic potential
vanishes in one or both components of .
Moreover, this decomposition is orthogonal if and only if is
a sphere. We also show that any in is uniquely the sum of two silent fields and a Hardy function,
in which case the sum is orthogonal regardless of ; we express
the corresponding orthogonal projections in terms of layer potentials. When
is a sphere, both decompositions coincide and match what has
been called the Hardy-Hodge decomposition in the literature
Analytic approximation of matrix functions in
We consider the problem of approximation of matrix functions of class
on the unit circle by matrix functions analytic in the unit disk in the norm of
, 2\le p<\be. For an matrix function in , we
consider the Hankel operator , . It
turns out that the space of matrix functions in splits into
two subclasses: the set of respectable matrix functions and the set of weird
matrix functions. If is respectable, then its distance to the set of
analytic matrix functions is equal to the norm of . For weird matrix
functions, to obtain the distance formula, we consider Hankel operators defined
on spaces of matrix functions. We also describe the set of -badly
approximable matrix functions in terms of special factorizations and give a
parametrization formula for all best analytic approximants in the norm of
. Finally, we introduce the notion of -superoptimal approximation and
prove the uniqueness of a -superoptimal approximant for rational matrix
functions.Comment: 43 page
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