# Approximation of Second-Order Moment Processes from Local Averages

- Zhanjie Song
^{1, 2}, - Ping Wang
^{1}and - Weisong Xie
^{1}Email author

**2009**:154632

https://doi.org/10.1155/2009/154632

© Zhanjie Song et al. 2009

**Received: **6 March 2009

**Accepted: **8 July 2009

**Published: **17 August 2009

## Abstract

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.

## 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 on , it is a real process with finite second order moment. In this paper, we will approximate all of the complex second order moment processes on by Bernstein polynomials and other classical operators by [2]. Therefore the research obtained generalize that of [1].

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.

Gröchenig [3] proved that every band-limited signal can be reconstructed exactly by local averages providing , where is the maximal frequency of the signal . 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 model [8, 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 on by some classical operators.

where are kernel functions and satisfy the following equations for all constant

## 2. Main Results

In this paper, let and let denote the space of all continuous real functions on . denotes the space of all bounded real functions on . denotes the space of all second order moment processes on . denotes the space of all second order moment processes in quadratic mean continuous on . Let us begin with the following proposition.

Proposition 2.1 (Korovkin [10]).

Then our main result is the following.

Theorem 2.2.

where is a sequence of operators defined as (1.6).

Proof.

This completes the proof.

## 3. Applications

As the application of Theorem 2.2, we give a new kind of operators.

If , we need ; if , , then is enough. Let and then we have the kernel function of Bernstein polynomials, Szász-Mirakian operators, and Baskakov operators [11].

For , let , for , let , for example, using Dirac-function, then for deterministic signals we have the Bernstein polynomials, Szász-Mirakian operators and Baskakov operators [11]. Let be a uniform ditributed function on or . We can get the BernsteinKantorovich operators, Szász- Kantorovich operators, and Baskakov-Kantorovich operators [11]. For random signals, the following results can be setup.

Corollary 3.1.

where for , for , and is defined by (3.7).

Proof.

Using Theorem 2.2, we have (3.9).

Obviously, let , , in Corollary 3.1, we get the first result of Kowalski [1].

## Declarations

### 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).

## Authors’ Affiliations

## References

- Kowalski E:
**Bernstein polynomials and Brownian motion.***American Mathematical Monthly*2006,**113**(10):865–886. 10.2307/27642086MathSciNetView ArticleMATHGoogle Scholar - Pogány TK:
**Some Korovkin-type theorems for stochastic processes.***Theory of Probability and Mathematical Statistics*1999, (61):145–151.MATHMathSciNetGoogle Scholar - Gröchenig K:
**Reconstruction algorithms in irregular sampling.***Mathematics of Computation*1992,**59**(199):181–194.MathSciNetView ArticleMATHGoogle Scholar - Aldroubi A:
**Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces.***Applied and Computational Harmonic Analysis*2002,**13**(2):151–161. 10.1016/S1063-5203(02)00503-1MathSciNetView ArticleMATHGoogle Scholar - Marvasti F:
*Nonuniform Sampling: Theory and Practice, Information Technology: Transmission, Processing and Storage*. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2001:xxvi+924.MATHGoogle Scholar - Song Z, Yang S, Zhou X:
**Approximation of signals from local averages.***Applied Mathematics Letters*2006,**19**(12):1414–1420. 10.1016/j.aml.2006.01.018MathSciNetView ArticleMATHGoogle Scholar - Sun W, Zhou X:
**Reconstruction of band-limited signals from local averages.***IEEE Transactions on Information Theory*2002,**48**(11):2955–2963. 10.1109/TIT.2002.804047MathSciNetView ArticleMATHGoogle Scholar - Seip K:
**A note on sampling of bandlimited stochastic processes.***IEEE Transactions on Information Theory*1990,**36**(5):1186. 10.1109/18.57226View ArticleGoogle Scholar - Song Z, Sun W, Zhou X, Hou Z:
**An average sampling theorem for bandlimited stochastic processes.***IEEE Transactions on Information Theory*2007,**53**(12):4798–4800.MathSciNetView ArticleMATHGoogle Scholar - Korovkin PP:
*Linear Operators and Approximation Theory*. Gosizdat Fizmatlit, Moscow, Russia; 1959.MATHGoogle Scholar - Ditzian Z, Totik V:
*Moduli of Smoothness, Springer Series in Computational Mathematics*.*Volume 9*. Springer, New York, NY, USA; 1987:x+227.Google Scholar

## Copyright

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