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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Approximation of Second-Order Moment Processes from Local Averages | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 154632 8 pages doi 10.1155 2009 154632 Research Article Approximation of Second-Order Moment Processes from Local Averages Zhanjie Song 1 2 Ping Wang 1 and Weisong Xie1 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@tjii.edii.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. Copyright 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 on 0 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 on a b 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. Due to physical reasons for example the inertia of the measurement apparatus the measured sampled values obtained in practice may not be values of f f precisely at times fk k e Z but only local
