JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS VOLUME 8 2010

686 Pages·12.748 MB·English

VOLUME 8, NUMBER 1 JANUARY 2010

ISSN:1548-5390 PRINT,1559-176X ONLINE

JOURNAL

OF CONCRETE

AND APPLICABLE

MATHEMATICS

SPECIAL ISSUE I :APPLIED MATHEMATICS
AND APPROXIMATION THEORY

EUDOXUS PRESS,LLC
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Associate Editors

Editor in -Chief: 21) Gustavo Alberto Perla Menzala
George Anastassiou National Laboratory of Scientific Computation
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EDITOR’S NOTE 
This special issue on “Applied Mathematics and Approximation Theory” contains 
expanded versions of articles that were presented in the international conference 
“Applied Mathematics and Approximation Theory 2008” ( AMAT 08), during  
October 11‐13, 2008 at the University of Memphis, Memphis, Tennessee, USA. 
All articles were refereed.
       The organizer and Editor 

           George Anastassiou
JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS , VOL.8, NO.1, 9-23,2010, COPYRIGHT 2010 EUDOXUS PRESS9, LLC
ITERATIVE RECONSTRUCTION AND STABILITY
BOUNDS FOR SAMPLING MODELS
ERNESTO ACOSTA-REYES
Abstract. This paper studies the reconstruction of a function
f belonging to a shift-invariant space from the set of its non-
uniformly distributed local sampled values. Here it is shown that
if the set of sampling X = {x } satisfies a necessary density
j j∈J
conditions, then we can recover the function f from the set of its
samples geometrically fast using an iterative algorithm. In addi-
tion,thealgorithmisanalyzedwhenthedataisperturbedbynoise,
and it is proved that a small perturbation on the set of samples
causes only a small change of the original function. Moreover, it
is given an upper estimate of the rate of convergence of the algo-
rithm. On the other hand, if we assume that X is a separated set,
then it is shown that X is a set of sampling and explicit stability
bounds are given.
1. Introduction
It is well known that in Sampling Theory there are two main goals
(for an overview see [2], [4], [7]-[8], and [11]): First, given a class of
functions on Rd, to find conditions on the sampling set X = {x } ,
j j∈J
where J is a countable index set, under which a function belonging to
that class can be reconstructed uniquely and stably from its samples
{f(x )} . Second, to find efficient and fast numerical algorithms for
j j∈J
recovering the function from its samples on X.
It is unrealistic to assume that the samples {f(x )} can be mea-
j j∈J
sured exactly. For working with a more realistic model, we consider
that our function(signal) belongs to a shift-invariant space Vp(Φ), for
some 1 ≤ p ≤ ∞, of the form
(cid:110) (cid:88) (cid:111)
(1.1) Vp(Φ) = CTΦ : C ∈ ((cid:96)p(Zd))(r) ,
k k
k∈Zd
and that the samples of the signal have the form
(cid:90)
g (f) = f(x)dµ (x),
xj xj
Rd
Key words and phrases. Irregularsampling,Non-uniformsampling,Reconstruc-
tion, Fast algorithm, Shift-invariant spaces, Stability bounds.
1
10
2 ERNESTO ACOSTA-REYES
µ
where = {µ } is a collection of finite complex Borel measures
xj j∈J
on Rd that acts on the signal f in a neighborhood of x to produce
j
the data {g (f)} . The form of our sampled data generalizes the
xj j∈J
model presented by A. Aldroubi in [2] when for each j ∈ J the Radon-
Nikodym derivative of µ with respect to the Lebesgue measure on Rd
xj
µ
belongs to L2(Rd). On the other hand, if the collection consists of
Dirac measures on Rd concentrated at each point of X, then we obtain
the model presented by A. Aldroubi and K. Gr¨ochenig in [4].
Inthispaperweapplyaniterativealgorithmforrecoveringthesignal
f from its samples values {g (f)} which uses the density proper-
xj j∈J µ
ties of the set X, the support size conditions of the collection , and
the properties of the generator Φ for Vp(Φ). Here we show that the
sequence of functions generated using the algorithm converges to f ge-
ometrically fast. In [12], [14]-[18], this method was used for iterative
reconstruction of band-limited signals, in [2] and [4], it was used for re-
constructing functions belonging to shift-invariant spaces, and in [17]
it was used for reconstructing signals belonging to a weighted multiply
generated shift-invariant spaces. On the other hand, if X is assumed
to be a separated set, then we show that X is also a set of sampling for
µ
Vp(Φ) and , and we give explicit stability bounds in terms of the rate
of convergence of the algorithm, the generator for Vp(Φ), the bounded
projection from Lp(Rd) onto Vp(Φ), and the uniform upper bound for
µ
the total variations of the collection . Moreover, it is given an upper
estimate for the rate of convergence of the iterative algorithm.
The stability of the samplingreconstruction is analyzed when our
local sampled data is perturbed by noise, and we show that a small
perturbation of the sampled data {g (f)} in the (cid:96)p(J) norm pro-
xj j∈J
duces a small perturbation of our original function.
The remainder of this paper has been organized as follows. Section 2
introduces our sampling model, the definitions and notations that we
shall work in this paper. The main results are presented in section 3,
and we provide the proof of some of the results in section 4.
2. Notation and preliminaries
In this section is introduced the sampling model weuse in this paper,
and the notations that will be used later.
The functions we are dealing with in this paper are functions f ∈
Lp(Rd), forsomep ∈ [1,∞]andd ∈ N, whichbelongtoashiftinvariant
space defined in (1.1), where Φ = (φ1,...,φr)T is a vector of functions,
Φ = Φ(·−k), and C = (c1,...,cr)T is a vector of sequences belonging
k