Optimal Recovery of Elements of a Hilbert Space and their Scalar Products According to the Fourier Coefficients Known with Errors

Бабенко, В. Ф.; Gun'ko, M.S.; Parfinovych, N. V.

doi:10.1007/s11253-020-01828-4

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…Remark that results of the present work supplement and generalize results of paper [10] on optimal recovery of functions and its derivatives and paper [7].…”

Section: Optimal Recovery Of Operator Sequencessupporting

confidence: 83%

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Optimal recovery of operator sequences

Бабенко¹,

Parfinovych²,

Skorokhodov³

2021

Mat. Stud.

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In this paper we solve two problems of optimal recovery based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,\colon \,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$ are given, in the space $\ell_q$. Information available about a sequence $x\in W^T_q$ is provided either (i) by an element $y\in\mathbb{R}^n$, $n\in\mathbb{N}$, whose distance to the first $n$ coordinates $\left(x_1,\ldots,x_n\right)$ of $x$ in the space $\ell_r^n$, $0 < r \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_\infty$ whose distance to $x$ in the space $\ell_r$ does not exceed $\varepsilon$. We show that the optimal method of recovery in this problem is either operator $\Phi^*_m$ with some $m\in\mathbb{Z}_+$ ($m\le n$ in case $y\in\ell^n_r$), where \smallskip\centerline{$\displaystyle \Phi^*_m(y) = \Big\{y_1\left(1 - \frac{t_{m+1}^q}{t_{1}^q}\Big),\ldots,y_m\Big(1 - \frac{t_{m+1}^q}{t_{m}^q}\Big),0,\ldots\right\},\quad y\in\mathbb{R}^n\text{ or } y\in\ell_\infty,$} \smallskip\noior convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$. The second one is the problem of optimal recovery of the scalar product operator acting on the 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 $s_1\ge s_2\ge \ldots \ge 0$ are given. Information available about elements $x\in W^T_p$ and $y\in W^S_q$ is provided by elements $z,w\in \mathbb{R}^n$ such that the distance between vectors $\left(x_1y_1, x_2y_2,\ldots,x_ny_n\right)$ and $\left(z_1w_1,\ldots,z_nw_n\right)$ in the space $\ell_r^n$ does not exceed $\varepsilon$. We show that the optimal method of recovery is delivered either by operator $\Psi^*_m$ with some $m\in\{0,1,\ldots,n\}$, where \smallskip\centerline{$\displaystyle \Psi^*_m = \sum_{k=1}^m z_kw_k\Big(1 - \frac{t_{m+1}s_{m+1}}{t_ks_k}\Big),\quad z,w\in\mathbb{R}^n,$} \smallskip\noior by convex combination $(1-\lambda)\Psi^*_{m+1} + \lambda\Psi^*_{m}$. As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by the Fourier coefficients of its elements known with an error measured in the space $\ell_p$ with $p > 2$.

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“…Remark that results of the present work supplement and generalize results of paper [10] on optimal recovery of functions and its derivatives and paper [7].…”

Section: Optimal Recovery Of Operator Sequencessupporting

confidence: 83%

“…Recovery of scalar products. Following [3] (see also [4,6,7]), let us consider the problem of optimal recovery of scalar product. Let 1 ≤ p, q ≤ ∞ and given operators T : ℓ p → ℓ p and S : ℓ q → ℓ q be defined as follows: for fixed non-increasing sequences t = {t k } ∞ k=1 and s = {s k } ∞ k=1 ,…”

Section: Case P = ∞mentioning

confidence: 99%

Optimal recovery of operator sequences

Бабенко¹,

Parfinovych²,

Skorokhodov³

2021

Mat. Stud.

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“…Following [3] (see also [4,6,7]), let us consider the problem of optimal recovery of scalar product. Let 1 ≤ p, q ≤ ∞ and given operators T : ℓ p → ℓ p and S : ℓ q → ℓ q be defined as follows: for fixed non-increasing sequences…”

Section: Recovery Of Scalar Productsmentioning

confidence: 99%

Optimal recovery of operator sequences

Babenko,

Parfinovych,

Skorokhodov

2021

Preprint

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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 1 h 1 , t 2 h 2 , . . .) ∈ ℓ q : h q 1}, where 1 ≤ q < ∞ and t 1 t 2 . . . 0, in the space ℓ q when in the capacity of inexact information we know either the first n ∈ N elements of a sequence with an error measured in the space of finite sequences ℓ n r , 0 < r ≤ ∞, or a sequence itself is known with an error measured in the space ℓ 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 < ∞, 1 p + 1 q = 1 and s 1 ≥ s 2 ≥ . . . ≥ 0, when in the capacity of inexact information we know the first n coordinate-wise products x 1 y 1 , x 2 y 2 , . . . , x n y m of the element x × y ∈ W T,S p,q with an error measured in the space ℓ n r . 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 ℓ p with p > 2.

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Optimal Recovery of Elements of a Hilbert Space and their Scalar Products According to the Fourier Coefficients Known with Errors

Cited by 2 publications

References 5 publications

Optimal recovery of operator sequences

Optimal recovery of operator sequences

Optimal recovery of operator sequences

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