Regression operator in infinite dimensional vector spaces and its application to some identification problems

  • Andrzej Szymanski
System Modeling And Identification
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3)


The main purpose of this paper was the transfer of the notion of regression into the theory of Banach valued random variables. This approach enables the formulation of identification problem in the domain of the infinite dimensional probability theory.

In the special case, when the random variables are Hilbert valued we state that the regression operator minimizes the square mean error. This makes the proof of the approximation theorem possible in the case of the linear bounded regression operator. Given further the statistical approximation theorem enables practically the identification of linear systems.

The problem unsolved in this paper is as well the approximation of the nonlinear regression operator.

It seems to the author that the general formulation of identification problem presented in his paper should help with solving also these problems.


Conditional Expectation Approximation Theorem Separable Hilbert Space Separable Banach Space Borelian Mapping 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Andrzej Szymanski
    • 1
  1. 1.Institut of MathematicsUniversity of LodzPoland

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