On restoration of noisy signals by a regularization method
Keywords:regularization, Hilbert space, unbounded operator, weak convergence, convergence in norm, Euler inequalities, Euler identities
The problem for restoration of noisy signals is treated as a problem of calculation of certain values for an unbounded operator. In this connection, a Tikhonov regularization method is used. Completely theoretically justified approaches to the choice of the regularization parameter as well as pragmatic approaches based on intuitive reasons are considered. The usage of some a priori information on the structure of the required «useful» signal in the «temporary» area as well as in the «frequent» one is allowed. The discussion is given in terms of functional analysis. This permits a strict substantiation of all considerations and provides a further range of possible applications in various fields of knowledge connected with processing of experimental data. The formulation of the problem corresponds to the case of direct measurements. The effect of «large» noises is also analyzed. This work was supported by the Russian Foundation for Basic Research (project N 10-01-00297a).
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