2000, Volume 3, Number 2, pp.135--152

Experimental data are considered in terms of Hilbert space and measure theory. As illustrations, we take the three tasks: least squares approximation of data, prediction of time series, and estimation of probability distributions. To these ends, we use data-tuned bases composed of functions which are orthonormal with respect to a specially defined inner product. This greatly reduces computations to be performed and enables one to calculate corrections to improve an available description (fit) of data. Two methods of constructing above mentioned bases are presented: a generalization of Chebyshev polynomial expansions and a generalization of the Gram--Schmidt orthogonalization procedure. It is demonstrated that the Dirac representation formalism of quantum theory can be useful to process data owing to a convenient technique of transitions between various orthogonal bases in a measurable rigged separable Hilbert space. As a more wide tool, we suggest a generalization of generalized frames involved in wavelets analyses of signals. It is shown that entropy-like functionals can be introduced as quantities associated with an arbitrary Hermitian operator and any partition of its spectrum. We define also entropy-like functionals connected with expansions of a signal vector over orthonormal basis functions and over generalized frames.
Key words: Hilbert space, prediction of time series, Gram-Schmidt orthogonalization, entropy-like functional.

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