Journal of Inequalities and Applications Volume 2009, Article ID 154632,8pages doi:10.1155/2009/154632
Research Article
Approximation of Second-Order
Moment Processes from Local Averages
Zhanjie Song,
1, 2Ping Wang,
1and Weisong Xie
11School of Science, Tianjin University, Tianjin 300072, China
2Institute of TV and Image Information, Tianjin University, Tianjin 300072, China
Correspondence should be addressed to Weisong Xie,weis [email protected]
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.
Copyrightq2009 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, bby 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 offtprecisely at times
tkk ∈Z, but only local average offtneartk. Specifically, the measured sampled values are
f, uk
for some collection of averaging functionsukt, k∈Z, which satisfy the following properties:
suppuk⊂
tk−δ
2, tk
δ
2
, ukt≥0,
uktdt1. 1.2
Gr ¨ochenig 3proved that every band-limited signal can be reconstructed exactly by local averages providingtk1−tk ≤ δ < 1/
√
2Ω, whereΩis 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, bby 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 onT ifE|Xt,·|2EXt,·Xt,· RXt, t<∞,
∀t ∈T. Now for eachn∈ Z, lettk,n k/nand 0 ≤δ1n, δ2n≤ C1/n, wherek ∈Zand
C1is a constant. Then for eachn∈Z, let the averaging functionsuk,nt, k ∈Z, satisfy the following properties:
suppuk,n⊂tk,n−δ1n, tk,nδ2n, uk,nt≥0,
uk,ntdt1, 1.3
tk,nδ2n
tk,n−δ1n
tiuk,ntdt
k n
i
o
1
n
, fori0,1,2. 1.4
The local averages ofXt, ωneartk,nk/nare
X·, ω, uk,n·
Xt, ωuk,ntdt. 1.5
The operatorMnis defined as
MnXt, ω
∞
k0
X·, ω, uk,n·Kk,nt, 1.6
whereKk,nt≥0 are kernel functions and satisfy the following equations for all constantC
∞
k0
CKk,nt CO
1
n
. 1.7
2. Main Results
In this paper, letT a, band letCa, bdenote the space of all continuous real functions on
the space of all second order moment processes ona, b.HCA,a, bdenotes the space of
all second order moment processes in quadratic mean continuous ona, b. Let us begin with the following proposition.
Proposition 2.1Korovkin10. Assume thatLn : Ca, b → Ma, bare a sequence of linear positive operators. If forft 1, t, andt2, one has
lim
n→ ∞Lnf
t−ftM0, 2.1
where
ft
M sup t∈a,b
ft:ft∈M
, 2.2
then for anyf∈Ca, b, one has
lim
n→ ∞Lnf
t−ftM0. 2.3
Notice that forXt, ω ft∈Ca, b,1.6can be changed as
Mnft
∞
k0
f·, uk,n·Kk,nt. 2.4
Then our main result is the following.
Theorem 2.2. Let{Mnft, n≥0}be a sequence of operators defined as2.4such that forft
1, t, andt2, one has
lim
n→ ∞Mnf
t−ftM0. 2.5
Then for any second order moment processes in quadratic mean continuousXt, ω on any finite closed intervala, b, one has
lim
n→ ∞EMnXt, ω−Xt, ω 20
, 2.6
where{MnXt, ω, n≥0}is a sequence of operators defined as1.6.
Proof. LetXt, ω∈HCA,a, b, and letRXt, sbe the correlation functions ofXt, ω. Then
we haveRXt, s∈Ca, b×a, b. For any fixedε >0, there existsδ >0, such that
|Rt, t−Rt, t∗|< ε
whenever|t−t∗|< δ. Then there isN >0 such that 0≤δ1n, δ2n≤δ/2 for alln≥N. Thus
whenn≥Nand|tk,n−t| ≤δ/2, we have
E| X·, ω, uk,n· −Xt, ω|2
E| X·, ω−Xt, ω, uk,n·|2
tk,nδ2n
tk,n−δ1n
RXx, y−RXx, t−RXt, yRt, tuk,nxuk,nydxdy
tk,nδ2n
tk,n−δ1n
RXx, y−RXx, t RXx, t−RXx, t
Rt, t−RXt, yuk,nxuk,nydx dy ≤ε.
2.8
At the same time, sinceXt, ω∈HCA,a, b,E|Xt, ω|2 RXt, t≤M <∞. Then
using2.8, that for any givenε >0 and anytk,n, t∈a, b, we have
E| X·, ω, uk,n· −Xt, ω|2≤ε 16M
δ2 t−tk,n 2
. 2.9
From1.7,2.5, and2.9, we have
EMnXt, ω−Xt, ω2
E
MnXt, ω−Xt, ω
∞
k0
Kk,nt Xt, ω
∞
k0
Kk,nt−Xt, ω 2
≤2E
MnXt, ω−Xt, ω
∞
k0
Kk,nt 2
2E
Xt, ω
∞
k0
Kk,nt−Xt, ω 2
2E
MnXt, ω−Xt, ω
∞
k0
Kk,nt 2
2 ∞
k0
1·Kk,nt−1
2
RXt, t
2E
∞
k0
X·, ω, uk,n· −Xt, ωKk,nt
2 O 1 n ≤2 ∞
k0
E| X·, ω, uk,n· −Xt, ω|2Kk,nt
∞
k0
Kk,ntO 1 n 2 ∞
k0
E| X·, ω, uk,n· −Xt, ω|2Kk,ntO
1
2 ∞
k0
ε 16M
δ2 t−tk,n 2
Kk,ntO
1
n
32M
δ2
∞
k0
t2−2tk,ntt2
k,n
Kk,nt2εO
1
n
≤ 32M δ2
t2Mn1t−2tMnxt Mnx2t22εO 1
n
−→0 whenn−→ ∞. 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
n→∞αn ∞, 3.2
and let
hnc, t ⎧ ⎨ ⎩
e−nt, c0,
1ct−sgnc·αn
, c /0. 3.3
Obviously, functionhc, tis continuous inR, now we let
bk,nc, t −1kt
k
k!h k
n c, t. 3.4
Using Gamma-function,bn,kc, tcan be noted by
bk,nc, t ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
e−ntntk
k! , c0,
Γsgnc·αnk
Γsgnc·αnk! ct
k1ct−sgnc·αn−k
, c /0,
where
Γsgnc·αnk
Γsgnc·αn
sgnc·αnk−1sgnc·αnk−2· · ·sgnc·αn. 3.6
Ifc < 0,t ∈0,−1/cwe need{αn, n ∈Z}; ifc ≥0,t∈0,∞, then{αn, n ∈R}is enough. Letc−1,0,1 andαnnthen 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∗nXt, ω, c
∞
k0
X·, ω, uk,n·bk,nc, t, 3.7
whereuk,nt, k∈Zsatisfy1.3. Similarly, let
tk,n ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
k
−cαn, c <0, k0,1,2, . . .−cαn,
k
n, c0, k0,1,2, . . . , k
cαn, c >0, k0,1,2, . . . .
3.8
The Nyquist rate is 1/|c|αnor 1/n.
Forc−1,1, letuk,nδ· −k/|c|αn, forc0, letuk,nδ· −k/n, for example, using Dirac-function, then for deterministic signals we have the Bernstein polynomials, Sz´asz-Mirakian operators and Baskakov operators11. Letuk,n be a uniform ditributed function on k/n1,k 1/n1 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 processesXt, t∈0, Din quadratic mean continuous on0, D, one has
lim
n→ ∞EM ∗
nXt, ω, c−Xt20, 3.9
whereD−1/cforc <0,D >0forc≥0, andMn∗Xt, ω, cis defined by3.7.
Proof. A simple computation shows that forc0, 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,
and forc /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 t2t1|c|nct.
3.11
For 0≤δ1n, δ2n≤C1/n → 0whenn → ∞,t∈0, D, we have
Mn∗1t, c 1O
1
n
,
M∗nxt, c tO
1
n
,
M∗nx2
t, c t2O
1
n
.
3.12
UsingTheorem 2.2, we have3.9.
Obviously, letc−1,αn n,uk,n δ· −k/ninCorollary 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. 60872161and the Natural Science Foundation of Tianjin
Grant no. 08JCYBJC09600.
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