Effective rank of a problem of function estimation based on measurement with an error of finite number of its linear functionals

Abstract

The problem of restoration of an element f of Euclidean functional space L2(X) based on the results of measurements of a finite set of its linear functionals, distorted by (random) error is solved. A priori data aren't assumed. Family of linear subspaces of the maximum (effective) dimension for which the projections of element f to them allow estimates with a given accuracy, is received. The effective rank () of the estimation problem is defined as the function equal to the maximum dimension of an orthogonal component Pf of the element f which can be estimated with a error, which is not surpassed the value . The example of restoration of a spectrum of radiation based on a finite set of experimental data is given.

Keywords

• mathematical model of measurement
• measurement reduction
• spectrometry
• optimum decisions
• singular decomposition
• effective rank