Volume 2009, Article ID 154632, 8 pagesdoi:10.1155/2009/154632 Research Article Approximation of Second-Order Moment Processes from Local Averages 1 School of Science, Tianjin University
Trang 1Volume 2009, Article ID 154632, 8 pages
doi:10.1155/2009/154632
Research Article
Approximation of Second-Order
Moment Processes from Local Averages
1 School of Science, Tianjin University, Tianjin 300072, China
2 Institute of TV and Image Information, Tianjin University, Tianjin 300072, China
Correspondence should be addressed to Weisong Xie,weis xie@tju.edu.cn
Received 6 March 2009; Accepted 8 July 2009
Recommended by Jozef Banas
We use local averages to approximate processes that have finite second-order moments and are continuous in quadratic mean We also provide some insight and generalization of the connection between Bernstein polynomials and Brownian motion, which was investigated by Kowalski in 2006
Copyrightq 2009 Zhanjie Song et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
In the literature, very few researchers considered approximating Brownian motion using Bernstein polynomials Kowalski 1 is the first one who uses this method In fact, if we restrict Brownian motion on0, 1, it is a real process with finite second order moment In this
paper, we will approximate all of the complex second order moment processes ona, b by
Bernstein polynomials and other classical operators by2 Therefore the research obtained generalize that of1
On the other hand, it is well known that the sampling theorem is one of the most powerful tools in signal analysis It says that to recover a function in certain function spaces,
it suffices to know the values of the function on a sequence of points
Due to physical reasons, for example, the inertia of the measurement apparatus, the
measured sampled values obtained in practice may not be values of ft precisely at times
t k k ∈ Z, but only local average of ft near tk Specifically, the measured sampled values
are
f, u k
f tuktdt 1.1
Trang 2for some collection of averaging functions ukt, k ∈ Z, which satisfy the following properties:
supp uk⊂
t k−δ
2, t kδ 2
, u kt ≥ 0,
u ktdt 1. 1.2
Gr ¨ochenig 3 proved that every band-limited signal can be reconstructed exactly by local
averages providing tk1 − tk ≤ δ < 1/√2Ω, where Ω is the maximal frequency of the signal
ft Recently, several average sampling theorems have been established, for example, see
4 7
Since signals are often of random characters, random signals play an important role in signal processing, especially in the study of sampling theorems For this purpose, one usually uses stochastic processes which are stationary in the wide sense as a model8,9 A wide sense stationary process is only a kind of second order moment processes In this paper, we study complex second order moment processes ona, b by some classical operators.
Given a probability spaceA, F, P, a stochastic process {Xt, ω : t ∈ T, T ⊂ R} is said to be a second order moment process on T if E|Xt, ·|2 EXt, ·Xt, · RXt, t < ∞,
∀t ∈ T Now for each n ∈ Z, let tk,n k/n and 0 ≤ δ1n, δ2n ≤ C1/n, where k ∈ Z and
C1is a constant Then for each n ∈ Z, let the averaging functions uk,nt, k ∈ Z, satisfy the
following properties:
supp uk,n ⊂ tk,n − δ1n, tk,n δ2n, uk,nt ≥ 0,
u k,ntdt 1, 1.3
t k,n δ2n
t k,n −δ1n t i u k,ntdt
k n
i
o1
n
, for i 0, 1, 2. 1.4
The local averages of Xt, ω near tk,n k/n are
X·, ω, uk,n
X t, ωuk,ntdt. 1.5
The operator Mnis defined as
Mn X t, ω ∞
k0 X·, ω, uk,n k,nt, 1.6
where Kk,nt ≥ 0 are kernel functions and satisfy the following equations for all constant C
∞
k0
CK k,nt C O
1
n
2 Main Results
In this paper, let T a, b and let Ca, b denote the space of all continuous real functions on
a, b Ma, b denotes the space of all bounded real functions on a, b HA, a, b denotes
Trang 3the space of all second order moment processes ona, b H C A, a, b denotes the space of
all second order moment processes in quadratic mean continuous ona, b Let us begin with
the following proposition
Proposition 2.1 Korovkin 10 Assume that Ln : Ca, b → Ma, b are a sequence of linear
positive operators If for ft 1, t, and t2, one has
lim
n → ∞ L n f
t − ft M 0, 2.1
where
f t M sup
t∈ a,b f t: ft ∈ M
then for any f ∈ Ca, b, one has
lim
n → ∞ L n f
t − ft M 0. 2.3 Notice that for Xt, ω ft ∈ Ca, b, 1.6 can be changed as
M n f
t ∞
k0
f ·, uk,n·K k,nt. 2.4
Then our main result is the following
Theorem 2.2 Let {M n ft, n ≥ 0} be a sequence of operators defined as 2.4 such that for ft
1, t, and t2, one has
lim
n → ∞ M n f
t − ft M 0. 2.5
Then for any second order moment processes in quadratic mean continuous Xt, ω on any finite closed interval a, b , one has
lim
n → ∞ E Mn X t, ω − Xt, ω2 0, 2.6
where {Mn Xt, ω, n ≥ 0} is a sequence of operators defined as 1.6.
Proof Let Xt, ω ∈ H C A, a, b, and let RXt, s be the correlation functions of Xt, ω Then
we have RX t, s ∈ Ca, b × a, b For any fixed ε > 0, there exists δ > 0, such that
|Rt, t − Rt, t∗| < ε
Trang 4whenever|t − t∗| < δ Then there is N > 0 such that 0 ≤ δ1n, δ2n ≤ δ/2 for all n ≥ N Thus when n ≥ N and |tk,n − t| ≤ δ/2, we have
E | X·, ω, uk,n 2
E| X·, ω − Xt, ω, uk,n 2
t k,n δ2n
t k,n −δ1n
R X
x, y
− RXx, t − RXt, y
Rt, t u k,nxuk,ny
dxdy
t k,n δ2n
t k,n −δ1n
R X
x, y
− RXx, t RXx, t − RXx, t
R t, t − RXt, y
u k,nxuk,ny
dx dy
≤ ε.
2.8
At the same time, since Xt, ω ∈ H C A, a, b, E|Xt, ω|2 RXt, t ≤ M < ∞ Then
using2.8, that for any given ε > 0 and any tk,n , t ∈ a, b, we have
E | X·, ω, uk,n 2≤ ε 16M
δ2 t − tk,n2
. 2.9
From1.7, 2.5, and 2.9, we have
E Mn X t, ω − Xt, ω2
E
Mn X t, ω − Xt, ω ∞
k0
K k,nt Xt, ω ∞
k0
K k,nt − Xt, ω
2
≤ 2E
Mn X t, ω − Xt, ω ∞
k0
K k,nt
2
2E
X t, ω ∞
k0
K k,nt − Xt, ω
2
2E
Mn X t, ω − Xt, ω ∞
k0
K k,nt
2
2
∞
k0
1· Kk,nt − 1
2
R Xt, t
2E
∞
k0
X·, ω, uk,n k,nt
2
O
1
n
≤ 2
∞
k0
E | X·, ω, uk,n 2K k,nt
∞
k0
K k,nt
O
1
n
2
∞
k0
E | X·, ω, uk,n 2K k,nt
O
1
n
Trang 5
2
∞
k0
ε 16M
δ2 t − tk,n2
K k,nt
O
1
n
32M
δ2
∞
k0
t2− 2tk,n t t2
k,n
K k,nt
2ε O
1
n
≤ 32M
δ2
t2Mn1t − 2tMn x t M n x2
t2
2ε O1
n
−→ 0 when n −→ ∞. 2.10 This completes the proof
3 Applications
As the application ofTheorem 2.2, we give a new kind of operators
For a signal function defined as
sgnt
⎧
⎨
⎩
1, t ≥ 0,
0, t < 0, 3.1 let{αn , n ∈ R} be a monotonic sequence that satisfies
lim
and let
h nc, t
⎧
⎨
⎩
1 ct −sgnc·α n
Obviously, function hc, t is continuous in R, now we let
b k,nc, t −1 k t k
k! h
k
n c, t. 3.4 Using Gamma-function, bn,kc, t can be noted by
b k,nc, t
⎧
⎪
⎪
⎪
⎪
e −nt nt k
Γsgnc · αn k
Γsgnc · αnk! ct k 1 ct −sgnc·α n −k
, c / 0,
3.5
Trang 6Γsgnc · αn k
Γsgnc · αn
sgnc · αn k − 1sgnc · αn k − 2· · ·sgnc · αn. 3.6
If c < 0, t ∈ 0, −1/c we need {αn , n ∈ Z}; if c ≥ 0, t ∈ 0, ∞ , then {αn , n ∈ R} is
enough Let c −1, 0, 1 and αn n then we have the kernel function of Bernstein polynomials,
Sz´asz-Mirakian operators, and Baskakov operators11
Now we define Gamma-Radom operators by local averages
M∗
n X t, ω, c ∞
k0 X·, ω, uk,n k,nc, t, 3.7
where uk,nt, k ∈ Z satisfy 1.3
Similarly, let
t k,n
⎧
⎪
⎪
⎨
⎪
⎪
⎩
k
−cαn , c < 0, k 0, 1, 2, − cα n , k
n , c 0, k 0, 1, 2, , k
cαn , c > 0, k 0, 1, 2,
3.8
The Nyquist rate is 1/|c|αn or 1/n.
For c −1, 1, let uk,n δ· − k/|c|αn, for c 0, let uk,n δ· − k/n, for example, using
Dirac-function, then for deterministic signals we have the Bernstein polynomials, Sz´asz-Mirakian operators and Baskakov operators11 Let uk,n be a uniform ditributed function
on k/n 1, k 1/n 1 or k/n, k 1/n We can get the BernsteinKantorovich
operators, Sz´asz- Kantorovich operators, and Baskakov-Kantorovich operators 11 For random signals, the following results can be setup
Corollary 3.1 For a second order moment processes Xt, t ∈ 0, D in quadratic mean continuous
on 0, D, one has
lim
n → ∞ E M∗
n X t, ω, c − Xt2 0, 3.9
where D −1/c for c < 0, D > 0 for c ≥ 0, and M n∗Xt, ω, c is defined by 3.7.
Proof A simple computation shows that for c 0, t ∈ 0, ∞, we have
∞
k0
1· bk,n0, t 1,
∞
k0
k
n · bk,n0, t t,
∞
k0
k n
2
· bk,n0, t t2 t
n ,
3.10
Trang 7and for c / 0, t ∈ 0, ∞, we have
∞
k0
1· bk,nc, t 1,
∞
k0
k
|c| · αn · bk,nc, t t,
∞
k0
k
|c| · αn
2
· bk,nc, t t2t 1 ct
|c|n .
3.11
For 0≤ δ1n, δ2n ≤ C1/n → 0 when n → ∞, t ∈ 0, D, we have
M∗
n1t, c 1 O
1
n
,
M∗
n x t, c t O
1
n
,
M∗n x2
t, c t2 O
1
n
.
3.12
UsingTheorem 2.2, we have3.9
Obviously, let c −1, αn n, uk,n δ· − k/n inCorollary 3.1, we get the first result
of Kowalski1
Acknowledgments
The authors would like to express their sincere gratitude to Professors Liqun Wang, Lixing Han, Wenchang Sun, and Xingwei Zhou for useful suggestions which helped them to improve the paper This work was partially supported by the National Natural Science Foundation of China Grant no 60872161 and the Natural Science Foundation of Tianjin
Grant no 08JCYBJC09600
References
1 E Kowalski, “Bernstein polynomials and Brownian motion,” American Mathematical Monthly, vol 113,
no 10, pp 865–886, 2006
2 T K Pog´any, “Some Korovkin-type theorems for stochastic processes,” Theory of Probability and
Mathematical Statistics, no 61, pp 145–151, 1999.
3 K Gr¨ochenig, “Reconstruction algorithms in irregular sampling,” Mathematics of Computation, vol 59,
no 199, pp 181–194, 1992
4 A Aldroubi, “Non-uniform weighted average sampling and reconstruction in shift-invariant and
wavelet spaces,” Applied and Computational Harmonic Analysis, vol 13, no 2, pp 151–161, 2002.
5 F Marvasti, Nonuniform Sampling: Theory and Practice, Information Technology: Transmission,
Processing and Storage, Kluwer Academic/Plenum Publishers, New York, NY, USA, 2001
6 Z Song, S Yang, and X Zhou, “Approximation of signals from local averages,” Applied Mathematics
Letters, vol 19, no 12, pp 1414–1420, 2006.
7 W Sun and X Zhou, “Reconstruction of band-limited signals from local averages,” IEEE Transactions
on Information Theory, vol 48, no 11, pp 2955–2963, 2002.
Trang 88 K Seip, “A note on sampling of bandlimited stochastic processes,” IEEE Transactions on Information
Theory, vol 36, no 5, p 1186, 1990.
9 Z Song, W Sun, X Zhou, and Z Hou, “An average sampling theorem for bandlimited stochastic
processes,” IEEE Transactions on Information Theory, vol 53, no 12, pp 4798–4800, 2007.
10 P P Korovkin, Linear Operators and Approximation Theory, Gosizdat Fizmatlit, Moscow, Russia, 1959.
11 Z Ditzian and V Totik, Moduli of Smoothness, vol 9 of Springer Series in Computational Mathematics,
Springer, New York, NY, USA, 1987
... Zhou, ? ?Approximation of signals from local averages,” Applied MathematicsLetters, vol 19, no 12, pp 1414–1420, 2006.
7 W Sun and X Zhou, “Reconstruction of band-limited... when n −→ ∞. 2.10 This completes the proof
3 Applications
As the application ofTheorem 2.2, we give a new kind of operators
For a signal function defined... n ≥ 0} is a sequence of operators defined as 1.6.
Proof Let Xt, ω ∈ H C A, a, b, and let RXt, s be the correlation functions of Xt, ω Then
we