The possibility of estimating the values of a function at given points of the measurement results of a finite number of its linear functionals

The possibility of estimating the values of a function at given points of its domain is investigated. The estimation uses results of the measurements of a finite number of linear functionals; the results of the measurements are distorted by an error. It is shown that only the component of the functions from the linear finite-dimensional subspace can be estimated with a finite error. A method for estimating this component with accuracy control is proposed. The mathematical methods of measurement reduction proposed by Yu.P. Pyt’ev are used. An example of the estimation of an emission spectrum that is measured by a double-slit spectrometer is described.