ON SMOOTHING PROBLEMS WITH ONE ADDITIONAL EQUALITY CONDITION

Svetlana Asmuss,Natalia Budkina

doi:10.3846/1392-6292.2009.14.159-168

Svetlana Asmuss, Natalia Budkina

Open Access

https://doi.org/10.3846/1392-6292.2009.14.159-168

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Abstract

Two problems of approximation in Hilbert spaces are considered with one additional equality condition: the smoothing problem with a weight and the smoothing problem with an obstacle. This condition is a generalization of the equality, which appears in the problem of approximation of a histogram in a natural way. We characterize the solutions of these smoothing problems and investigate the connection between them.

Highlights

The idea to use splines for solving the problem of approximation of a histogram firstly appeared in the paper of Boneva, Kendall and Stefanov ([2])
During the last three decades the problem of approximation of the density function of a random value by splines has been investigated in various statements and under different restrictions on its solutions
In the present paper smoothing splines are investigated under one additional condition, which we consider to be very important, as it corresponds to the property of density histogram

Summary

Formulation of Problem of Smoothing Histopolation

The idea to use splines for solving the problem of approximation of a histogram firstly appeared in the paper of Boneva, Kendall and Stefanov ([2]). During the last three decades the problem of approximation of the density function of a random value by splines has been investigated in various statements and under different restrictions on its solutions. In the present paper smoothing splines are investigated under one additional condition, which we consider to be very important, as it corresponds to the property of density histogram (the area under it is equal to 1). Depending on the known information about the frequencies, the problem of smoothing histopolation can be considered with different types of restriction on the solution. Let us consider two problems of smoothing histopolation for given parameters ω > 0, delta > 0. In the case of inexact information this requirement is essential We investigate these two problems, prove the existence of their solutions and show the connection between them in a more general case

A Generalization of the Problem of Smoothing Histopolation

Analysis of the Smoothing Problem with a Weight

Equivalence of Two Smoothing Problems