Optimal recovery of operator sequences

TL;DR

This paper develops optimal methods for recovering operator sequences and scalar products from inexact information, providing exact solutions and applications to Fourier coefficient recovery in Hilbert spaces.

Contribution

It introduces new optimal recovery techniques for operator sequences and scalar products with inexact data, including explicit solutions and methods.

Findings

Exact solutions for recovery problems in sequence spaces

Optimal methods constructed for recovering scalar products

Application to Fourier coefficient recovery in Hilbert spaces

Abstract

In this paper we consider two recovery problems based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\in \ell_q\,:\,\|h\|_q\leqslant 1\}$, where $1\le q < \infty$ and $t_1\geqslant t_2\geqslant \ldots \geqslant 0$, in the space $\ell_q$ when in the capacity of inexact information we know either the first $n\in\mathbb{N}$ elements of a sequence with an error measured in the space of finite sequences $\ell_r^n$, $0 < r \le \infty$, or a sequence itself is known with an error measured in the space $\ell_r$. The second is the problem of optimal recovery of scalar products acting on Cartesian product $W^{T,S}_{p,q}$ of classes $W^T_p$ and $W^S_q$, where $1 < p,q < \infty$, $\frac{1}{p} + \frac{1}{q} = 1$ and $s_1\ge s_2\ge \ldots \ge 0$, when in the capacity of inexact information we know the first $n$ coordinate-wise products $x_1y_1, x_2y_2,\ldots,x_ny_m$ of the element $x\times y \in W^{T,S}_{p,q}$ with an error measured in the space $\ell_r^n$. We find exact solutions to above problems and construct optimal methods of recovery. As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by Fourier coefficients known with an error measured in the space $\ell_p$ with $p > 2$.