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Optimal recovery of fractional powers of the Laplace difference operator E. O. Sivkovaab a Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia b National Research University "Moscow Power Engineering Institute", Moscow, Russia
Bibliographic databases:
    § 1. Problem statementLet $d$ be a natural number and $\mathbb Z$ be the set of integers, and let $h>0$. We let $l_{2}(\mathbb Z_h^d)$ denote the space of functions $f$ on the lattice
$$
\begin{equation*}
\mathbb Z_h^d=\bigl\{h(k_1,\dots,k_d) \colon k=(k_1,\dots,k_d)\in\mathbb Z^d\bigr\}
\end{equation*}
\notag
$$
with the norm
$$
\begin{equation*}
\|f\|_{l_{2}(\mathbb Z^d_h)}=\sqrt{h^d\sum_{k\in\mathbb{Z}^d}f^2(hk)}.
\end{equation*}
\notag
$$
We let $\mathbb T^d_h$ denote the $d$-dimensional torus, which we identify with the cube
$$
\begin{equation*}
\underbrace{\biggl[-\frac{\pi}{h},\frac{\pi}{h}\biggr]\times\dots\times\biggl[-\frac{\pi}{h},\frac{\pi}{h}\biggr]}_{d \text{ times}}.
\end{equation*}
\notag
$$
Since $\mathbb Z_h^d$ is a locally compact Abelian group, the Fourier transform $F$: ${l_{2}(\mathbb Z^d_h)\to L_2(\mathbb{T}^d_h)}$ is defined, which acts in this case as
$$
\begin{equation*}
F[f](\xi)=\sum_{k\in\mathbb{Z}^d}f(hk)e^{-i\langle\xi,hk\rangle},
\end{equation*}
\notag
$$
where $\xi=(\xi_1,\xi_2,\dots,\xi_d)$, $\xi_j\in[-\pi/h,\pi/h]$, $j=1,2,\dots,d$, $\langle\xi,hk\rangle=\sum_{j=1}^d \xi_jhk_j,$ and Plancherel’s theorem is valid:
$$
\begin{equation*}
\frac{1}{(2\pi)^d}\int_{\mathbb{T}^d_h}|F[f](\xi)|^2\, d\xi=h^d\sum_{k\in\mathbb{Z}^d}f^2(hk).
\end{equation*}
\notag
$$
Now we define a difference analogue of the Laplace operator on $\mathbb R^d$ and its fractional powers. It is straightforward to see that the second divided difference of $f\in l_{2}(\mathbb Z^d_h)$, for example, with respect to the first variable has the form
$$
\begin{equation*}
\begin{aligned} \, \Delta^2_{k_1,h}f &=\biggl\{\frac{1}{h^2} \bigl( f((k_1+2)h,k_2h,\dots,k_dh) \\ &\qquad -2f((k_1+1)h,k_2h,\dots,k_dh)+f(k_1h,k_2h,\dots,k_dh)\bigr)\biggr\}_{k\in \mathbb{Z_d}}. \end{aligned}
\end{equation*}
\notag
$$
Similar formulae are true for other variables. We can easily verify that $\Delta^2_{k_j,h}f\in l_{2}(\mathbb Z^d_h)$, $j=1,\dots,d$. The operator $\Delta_h\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)$ that acts according to the rule
$$
\begin{equation*}
\Delta_h f=\sum_{j=1}^d\Delta^2_{k_j,h}f, \qquad f\in l_{2}(\mathbb Z^d_h),
\end{equation*}
\notag
$$
is called the Laplace difference operator with step $h$. We will find the Fourier transform of $\Delta_h f$. First we find, for example, the Fourier transform of the function $\Delta^2_{k_1,h}f$. Simple calculations show that
$$
\begin{equation*}
\begin{aligned} \, F[\Delta^2_{k_1,h}f] &=\sum_{k\in\mathbb{Z}^d}\frac1{h^2}\bigl(f((k_1+2)h,k_2h,\dots,k_dh)-2f((k_1+1)h,k_2h,\dots,k_dh) \\ &\qquad+f(k_1h,k_2h,\dots,k_dh)\bigr)e^{-i\langle\xi,hk\rangle} \\ &=\frac{e^{2i\xi_1h}}{h^2} \sum_{k\in\mathbb{Z}^d}f(k_1h,k_2h,\dots,k_dh)e^{-i\langle\xi,kh\rangle} \\ &\qquad-\frac{2e^{i\xi_1h}}{h^2}\sum_{k\in\mathbb{Z}^d}f(k_1h,k_2h,\dots,k_dh) e^{-i\langle\xi,hk\rangle} \\ &\qquad +\frac{1}{h^2}\sum_{k\in\mathbb{Z}^d}f(k_1h,k_2h,\dots,k_dh)e^{-i\langle\xi,hk\rangle} \\ &=\frac{e^{2i\xi_1h}-2e^{i\xi_1h}+1}{h^2}F[f](\xi)=\frac{(1-e^{i\xi_1h})^2}{h^2}F[f](\xi). \end{aligned}
\end{equation*}
\notag
$$
The Fourier transforms $F[\Delta^2_{k_2,h}f], \dots, F[\Delta^2_{k_d,h}f]$ can be obtained in a similar way, and we arrive at the relation
$$
\begin{equation*}
F[\Delta_h f](\xi)=\biggl(\frac{1}{h^2}\sum_{j=1}^d(1-e^{i\xi_jh})^2\biggr)F[f](\xi).
\end{equation*}
\notag
$$
For short, we introduce the notation
$$
\begin{equation*}
\psi(\xi)=\frac{1}{h^2}\sum_{j=1}^d(1-e^{i\xi_jh})^2\!, \qquad \xi=(\xi_1,\dots,\xi_d)\in\mathbb{T}_h^d.
\end{equation*}
\notag
$$
Let $\alpha\geqslant 0$ and set
$$
\begin{equation*}
(\psi(\xi))^{\alpha/2}=|\psi(\xi)|^{\alpha/2}e^{i(\alpha/2)\arg\psi(\xi)}, \qquad -\pi<\arg\psi(\xi)\leqslant\pi.
\end{equation*}
\notag
$$
We let $\Delta_h^{\alpha/2}$ denote the operator associating with $f\in l_{2}(\mathbb Z_h^d)$ the function ${\Delta_h^{\alpha/2}f\in l_{2}(\mathbb Z_h^d)}$ whose Fourier transform has the form
$$
\begin{equation}
F[\Delta_h^{\alpha/2} f](\xi)=(\psi(\xi))^{\alpha/ 2}F[f](\xi).
\end{equation}
\tag{1.1}
$$
The operator is well defined, since the factor in front of $F[f](\xi)$ is a bounded function and thus the function on the right-hand side of (1.1) is in $L_2(\mathbb T^d_h)$. Since $F$ is an isomorphism, the function $\Delta_h^{\alpha/2}f$ is well defined. The operator $\Delta_h^{\alpha/2}$ is said to be the $\alpha/2$th power of the Laplace difference operator. Clearly, we obtain the Laplace difference operator for $\alpha=2$. We now switch to stating the optimal recovery problem. We introduce the class of functions
$$
\begin{equation*}
W_{2}^\alpha(\mathbb Z_h^d)=\bigl\{f\in l_2(\mathbb Z_h^d) \colon \|\Delta_h^{\alpha/2}f\|_{l_{2}(\mathbb Z_h^d)}\leqslant1\bigr\}.
\end{equation*}
\notag
$$
Assume that functions in this class are known inaccurately. More precisely, for each function $f\in W_{2}^\alpha(\mathbb Z_h^d)$ we know a function $g\in l_2(\mathbb Z_h^d)$ such that where $\delta>0$. Based on this information, we want to recover (in the best possible way) the $\beta/2$th power ($0\leqslant\beta<\alpha$) of the Laplace difference operator on the class $W_{2}^\alpha(\mathbb Z_h^d)$. In what follows we write $W$ instead of $W_{2}^\alpha(\mathbb Z_h^d)$ for short. We regard any map $m\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)$ as a recovery method and call the quantity
$$
\begin{equation*}
e(\beta,W,\delta,m)=\sup_{\substack{f\in W,\ g\in l_{2}(\mathbb Z^d_h)\\ \|f-g\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta}}\|\Delta_h^{\beta/2}f-m(g)\|_{l_{2}(\mathbb Z^d_h)}
\end{equation*}
\notag
$$
the error of this method. We are interested in the quantity
$$
\begin{equation*}
E(\beta,W,\delta)=\inf_{m\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)}e(\beta,W,\delta,m)
\end{equation*}
\notag
$$
(called the optimal recovery error) and in methods $\widehat m$ delivering the infimum (called optimal recovery methods).
§ 2. Statement of the main resultLet $0<\beta<\alpha$ and $\delta>0$. We introduce the notation
$$
\begin{equation*}
\lambda_1=\frac{\alpha-\beta}{\alpha}\, \delta^{-2\beta/\alpha}\quad\text{and} \quad \lambda_2=\frac{\beta}{\alpha}\, \delta^{2(\alpha-\beta)/\alpha}
\end{equation*}
\notag
$$
for $\delta\geqslant (h^2/(4d))^{\alpha/2}$, and
$$
\begin{equation*}
\lambda_1=\biggl(\frac{4d}{h^2}\biggr)^\beta\quad\text{and} \quad \lambda_2=0
\end{equation*}
\notag
$$
for $\delta< (h^2/(4d))^{\alpha/2}$. Theorem. Let $\delta> 0$ and $0<\beta<\alpha$. Then the following hold: (1)
$$
\begin{equation*}
E(\beta,W,\delta)=\begin{cases} \delta^{(\alpha-\beta)/\alpha}, & \delta\geqslant\biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \\ \biggl(\dfrac{4d}{h^2}\biggr)^{\beta/2}\delta, &\delta<\biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}; \end{cases}
\end{equation*}
\notag
$$
(2) if $\delta\geqslant (h^2/(4d))^{\alpha/2}$, then the method $\widehat m_\omega\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)$ acting on Fourier images as
$$
\begin{equation*}
F\widehat m_\omega(g)(\xi)=\omega(\xi) F[g](\xi) \quad\textit{for a.a. } \xi\in \mathbb T^d_h
\end{equation*}
\notag
$$
is optimal for each function $\omega(\cdot)\in L_\infty(\mathbb T_h^d)$ such that
$$
\begin{equation}
\frac{|\omega(\xi)|^2}{\lambda_1}+ \frac{|(\psi(\xi))^{\beta/2}-\omega(\xi)|^2} {\lambda_2|\psi(\xi)|^{\alpha}}\leqslant 1;
\end{equation}
\tag{2.1}
$$
(3) if $\delta< (h^2/(4d))^{\alpha/2}$, then the method $\widehat m\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)$ acting as is optimal.The problem under consideration here is a difference analogue of the optimal recovery problem for fractional powers of the ordinary Laplace operator on $\mathbb R^d$ (see [1] and [2]). Topics related to the optimal recovery of linear functionals and operators on classes of sets from inaccurate information about elements in these sets have actively been developed since the 1960s. An approach to optimal recovery problems based on methods of extremum theory and convex duality was developed at V. M. Tikhomirov’s seminar “Approximation theory and the theory of extremal problems” at Moscow State University. We note here several works, namely, [3]–[8], presenting results based on this approach. To a certain extent the activity related to optimal recovery problems was summarized in [9]. We also note that the optimal recovery problem for difference analogues of derivatives was solved in [10]. § 3. Proof of the theoremFirst we prove the following estimate for the optimal recovery error:
$$
\begin{equation}
E(\beta,W,\delta)\geqslant\sup_{\substack{f\in W,\ \|f\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta}}\|\Delta^{\beta/2}f\|_{l_{2}(\mathbb Z^d_h)}.
\end{equation}
\tag{3.1}
$$
In fact, let $f_0 \in W$ and $\|f_0\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta$. It is obvious that the function $-f_0$ also satisfies these relations; hence
$$
\begin{equation}
\begin{aligned} \, 2\|\Delta^{\beta/2}f_0\|_{l_{2}(\mathbb Z^d_h)} &=\|\Delta^{\beta/2}f_0-m(0)-(\Delta^{\beta/2}(-f_0)-m(0)\|_{l_{2}(\mathbb Z^d_h)} \nonumber \\ &\leqslant 2\sup_{\substack{f\in W\\ \|f\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta}} \|\Delta^{\beta/2}f-m(0)\|_{l_{2}(\mathbb Z^d_h)} \nonumber \\ &\leqslant 2\sup_{\substack{f\in W,\ g\in l_{2}(\mathbb Z^d_h)\\ \|f-g)\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta}} \|\Delta^{\beta/2}f-m(g)\|_{l_{2}(\mathbb Z^d_h)}=2e(\beta,W,\delta,m) \end{aligned}
\end{equation}
\tag{3.2}
$$
for any $m\colon l_{2}(\mathbb Z^d_h)\to l_{2}(\mathbb Z^d_h)$. Taking the supremum over all functions $f$ such that $f\in W$ and $\|f\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta$ on the left-hand side of (3.2), we arrive at the inequality
$$
\begin{equation*}
\sup_{\substack{f\in W\\ \|f\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta}} \|\Delta^{\beta/2}f\|_{l_{2}(\mathbb Z^d_h)}\leqslant e(\beta,W,\delta,m).
\end{equation*}
\notag
$$
The method $m$ can be arbitrary. Thus, taking the infimum over all methods $m$ on the right-hand side, we obtain (3.1). The quantity on the right-hand side of (3.1) is the value of the extremal problem
$$
\begin{equation}
\|\Delta^{\beta/2}f\|_{l_{2}(\mathbb Z^d_h)}\to\max, \qquad \|f\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta, \quad \|\Delta^{\alpha/2}f\|_{l_{2}(\mathbb Z^d_h)}\leqslant 1,
\end{equation}
\tag{3.3}
$$
that is, the supremum of the functional in question under the given restrictions. According to the definition of a fractional power of the Laplace difference operator and Plancherel’s theorem, the squared value of problem (3.3) is equal to the value of the problem
$$
\begin{equation}
\begin{gathered} \, \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|\psi(\xi)|^{\beta}|F[f](\xi)|^2\,d\xi\to\max, \\ \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|F[f](\xi)|^2\,d\xi\leqslant\delta^2,\qquad \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|\psi(\xi)|^{\alpha}|F[f](\xi)|^2\,d\xi\leqslant 1. \end{gathered}
\end{equation}
\tag{3.4}
$$
We can regard it as a problem on the set of finite positive measures of the form on the $\sigma$-algebra $\Sigma$ of Lebesgue-measurable subsets of $\mathbb T^d_h$, for $f$ ranging over all admissible functions. However, it is convenient to consider an extended version of it, when all finite positive measures on $\Sigma$ are taken into account, namely,
$$
\begin{equation}
\begin{gathered} \, \int_{\mathbb T^d_h}|\psi(\xi)|^{\beta}\,d\mu(\xi)\to\max, \\ \int_{\mathbb T^d_h}d\mu(\xi)\leqslant\delta^2, \quad \int_{\mathbb T^d_h}|\psi(\xi)|^{\alpha}\,d\mu(\xi)\leqslant 1, \qquad d\mu(\cdot)\geqslant 0. \end{gathered}
\end{equation}
\tag{3.5}
$$
It is obvious that the value of this problem is at least the value of (3.4). Problem (3.5) is a convex problem on the linear space of all finite measures on $\Sigma$. The Karush–Kuhn–Tucker theorem (see [11]) gives necessary and sufficient conditions of maximum for problems of this type. Using this theorem we can find a solution of (3.5). This is the Dirac $\delta$-function at some point. Then we can construct a sequence of admissible functions $\varphi_n$ for problem (3.4) whose Fourier transforms approximate this $\delta$-function. Clearly, the value of the functional in problem (3.4) at any function $\varphi_n$ is at most the value of the problem itself. Passing to the limit as $n\to\infty$, we derive a lower estimate for the value of problem (3.4) and thus of problem (3.3) too. Thus, by (3.1) we find a lower estimate for the optimal recovery error. This estimate turns out to be sharp. Omitting rather routine constructions related to the application of the Karush–Kuhn–Tucker theorem, we immediately produce a sequence of functions yielding the required lower estimate for the optimal recovery error. For $\delta\geqslant (h^2/(4d))^{\alpha/2}$ set
$$
\begin{equation*}
\xi^*=(\xi_1^*,\dots,\xi_d^*)\in \mathbb{T}_h^d, \quad\text{where } \xi_j^*=\frac{2}{h}\arcsin\frac{h}{2\sqrt{d}\delta^{1/\alpha}}, \quad j=1,\dots,d.
\end{equation*}
\notag
$$
For $\delta< (h^2/(4d))^{\alpha/2}$ set
$$
\begin{equation*}
\xi^*=(\xi_1^*,\dots,\xi_d^*)\in \mathbb{T}^d, \quad\text{where } \xi_j^*=\frac{\pi}{h}, \quad j=1,\dots,d.
\end{equation*}
\notag
$$
For each $n\in\mathbb N$ let $\Box_{n}$ denote the cube formed by the vectors $\xi=(\xi_1,\dots,\xi_d)\,{\in}\, \mathbb T^d_h$ such that $\xi^*_{j}-1/n\leqslant\xi_j\leqslant\xi_{j}^*$, $j=1,\dots,d$. It is clear that $\Box_{n}\subset\mathbb T^d_h$ for sufficiently large $n$. We consider a sequence of functions $\varphi_n\in l_{2}(\mathbb Z^d_h)$, $n\in\mathbb N$, whose Fourier transforms have the form
$$
\begin{equation*}
F[\varphi_n](\xi) = \begin{cases} \delta(2\pi n)^{d/2}, &\xi \in \Box_{n}, \\ 0, & \xi \notin \Box_{n}. \end{cases}
\end{equation*}
\notag
$$
Clearly, $F[\varphi_n](\cdot)\in L_{2}(\mathbb T^d_h)$; therefore, the functions $\varphi_n$ are well defined. We show that they are admissible in problem (3.4). By the definition of $\varphi_n$
$$
\begin{equation*}
\frac1{(2\pi)^d}\int_{\mathbb T^d_h}|F[\varphi_n](\xi)|^2\,d\xi=\frac1{(2\pi)^d}\int_{\Box_n}|\delta(2\pi n)^{d/2}|^2\,d\xi= \delta^2 n^d\int_{\Box_n}d\xi=\delta^2.
\end{equation*}
\notag
$$
Furthermore, for all $\xi\in\Box_{n}$, in view of the expressions for $\xi_j^*$, $j=1,\dots,d$, we have
$$
\begin{equation}
\begin{aligned} \, \notag |\psi(\xi)| &\leqslant \frac{1}{h^2}\sum_{j=1}^d|1-e^{i\xi_jh}|^2= \frac{1}{h^2}\sum_{j=1}^d((1-\cos\xi_jh)^2+\sin^2\xi_jh) \\ &=\frac{1}{h^2}\sum_{j=1}^d4\sin^2\frac{\xi_jh}{2} \leqslant\frac{1}{h^2}\sum_{j=1}^d4\sin^2\frac{\xi_j^*h}{2} =\begin{cases} \delta^{-2/\alpha},&\delta\geqslant \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \\ \dfrac{4d}{h^2}, & \delta< \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \end{cases} \end{aligned}
\end{equation}
\tag{3.6}
$$
which implies the relations
$$
\begin{equation*}
\begin{aligned} \, &\frac1{(2\pi)^d}\int_{\mathbb T^d_h}|\psi(\xi)|^{\alpha}|F[\varphi_n](\xi)|^2\,d\xi= \frac1{(2\pi)^d}\int_{\Box_n}|\psi(\xi)|^{\alpha} (\delta(2\pi n)^{d/2})^2\,d\xi \\ &\qquad = \delta^2 n^d\int_{\Box_n}|\psi(\xi)|^{\alpha}\,d\xi\leqslant \begin{cases} \displaystyle n^d\int_{\Box_n}\,d\xi= 1, &\delta\geqslant \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \\ \displaystyle \biggl(\frac{4d}{h^2}\biggr)^\alpha\delta^2 n^d\int_{\Box_n}\,d\xi<1, & \delta<\biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
Hence the sequence $\varphi_n$ is admissible in problem (3.4). Now we estimate the value of the maximized functional in (3.4) at $\varphi_n$. First we note that if $a_i\geqslant0$, $i=1,\dots,d$, then it is obvious that
$$
\begin{equation*}
a_1+a_2+\dots+a_d\geqslant d\min\{a_1,a_2,\dots,a_d\};
\end{equation*}
\notag
$$
this yields the inequality
$$
\begin{equation}
(a_1+a_2+\dots+a_d)^2\geqslant d^2\min\{a_1^2,a_2^2,\dots,a_d^2\},
\end{equation}
\tag{3.7}
$$
for numbers $a_i$ of the same sign. Let $\xi\in\Box_{n}\subset\mathbb T^d_h$. It is straightforward to see that for sufficiently large $n$ the expressions are of the same sign and the expressions are also of the same sign. Applying (3.7) to the sums
$$
\begin{equation*}
\sum_{j=1}^d((1-\cos\xi_jh)^2-\sin^2 \xi_jh)\quad\text{and} \quad \sum_{j=1}^d 2(1-\cos\xi_jh)\sin\xi_jh,
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation}
\begin{aligned} \, |\psi(\xi)|^2 &= \frac{1}{h^4}\biggl|\sum_{j=1}^d((1-\cos\xi_jh)^2-\sin^2 \xi_jh)-2i\sum_{j=1}^d(1-\cos\xi_jh)\sin\xi_jh\biggr|^2 \nonumber \\ &=\frac{1}{h^4}\biggl(\sum_{j=1}^d((1-\cos\xi_jh)^2-\sin^2 \xi_jh)\biggr)^2+\frac{1}{h^4}\biggl(\sum_{j=1}^d 2(1-\cos\xi_jh)\sin\xi_jh\biggr)^2 \nonumber \\ &\geqslant\frac{d^2}{h^4}\bigl(\min\bigl(((1-\cos\xi_1h)^2-\sin^2 \xi_1h)^2,\dots,((1-\cos\xi_dh)^2-\sin^2 \bigr)\xi_dh)^2) \nonumber \\ &\qquad + 4\min\bigl((1-\cos\xi_1h)^2\sin^2\xi_1h,\dots,(1-\cos\xi_dh)^2\sin^2\xi_dh\bigr)\bigr). \end{aligned}
\end{equation}
\tag{3.8}
$$
We let $K(\xi)$, $\xi\in \Box_n$, denote the function in brackets on the right-hand side of the inequality in (3.8). This function is obviously continuous. Then it follows from the mean value theorem for integrals that
$$
\begin{equation*}
\begin{aligned} \, \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|\psi(\xi)|^{\beta}|F[\varphi_n](\xi)|^2\,d\xi &=\frac1{(2\pi)^d}\int_{\Box_n} |\psi(\xi)|^{\beta}(\delta(2\pi n)^{d/2})^2\,d\xi \\ &\geqslant \frac{\delta^2 n^d d^\beta}{h^{2\beta}}\int_{\Box_n}(K(\xi))^{\beta/2}\,d\xi =\frac{\delta^2 d^\beta}{h^{2\beta}}(K(\xi_0))^{\beta/2}, \end{aligned}
\end{equation*}
\notag
$$
where $\xi_0\in\Box_n$. It is obvious that $K(\xi_0)$ tends to the quantity
$$
\begin{equation*}
K(\xi^*)=(1-\cos\xi_\delta^*h)^2-\sin^2 \xi_\delta^*h)^2+ 4(1-\cos\xi_\delta^*h)^2\sin^2\xi_\delta^*h
\end{equation*}
\notag
$$
as $n\to\infty$, where $\xi_\delta^*$ is equal to (any of) the (equal) coordinates of the vector $\xi^*$ defined above for $\delta$ under consideration. Thus,
$$
\begin{equation*}
\frac{\delta^2 d^\beta}{h^{2\beta}}(K(\xi^*))^{\beta/2} = \frac{\delta^2d^\beta}{h^{2\beta}}\biggl(4\sin^2\frac{\xi_\delta^*h}{2}\biggr)^{\beta} =\begin{cases} \delta^{2(\alpha-\beta)/\alpha}, &\delta\geqslant \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \\ \biggl(\dfrac{4d}{h^2}\biggr)^\beta\delta^2, & \delta< \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}. \end{cases}
\end{equation*}
\notag
$$
Hence the value of problem (3.4) is at least the quantity on the right-hand side of this equality; therefore, the optimal recovery error satisfies the estimate
$$
\begin{equation}
E(\beta,W,\delta)\geqslant \begin{cases} \delta^{(\alpha-\beta)/\alpha}, & \delta\geqslant\biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}, \\ \biggl(\dfrac{4d}{h^2}\biggr)^{\beta/2}\delta, &\delta< \biggl(\dfrac{h^2}{4d}\biggr)^{\alpha/2}. \end{cases}
\end{equation}
\tag{3.9}
$$
Now we prove the upper estimate for the optimal recovery error and the optimality of the recovery methods indicated in the theorem. First consider the case when $\delta\geqslant (h^2/(4d))^{\alpha/2}$. Assume that the numbers $\lambda_1$ and $\lambda_2$ defined before the formulation of the theorem correspond to this case. We prove that the set of functions $\omega(\cdot)$ satisfying (2.1) is nonempty. We define a function $h(\cdot)$ on the interval $[0, 4d/h^2]$ by We can easily verify that $h(\cdot)$ is nonnegative on this interval.The absolute value of $\psi(\cdot)$ takes values in $[0, 4d/h^2]$. In fact,
$$
\begin{equation*}
\begin{aligned} \, 0&\leqslant |\psi(\xi)|\leqslant \frac{1}{h^2}\sum_{j=1}^d|1-e^{i\xi_jh}|^2= \frac{1}{h^2}\sum_{j=1}^d((1-\cos\xi_jh)^2+\sin^2\xi_jh) \\ &= \frac{1}{h^2}\sum_{j=1}^d 4\sin^2\frac{\xi_jh}{2}\leqslant \frac{1}{h^2}\sum_{j=1}^d 4\sin^2\frac{\pi}{2}=\frac{4d}{h^2}. \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
-|\psi(\xi)|^\beta +\lambda_1 +\lambda_2 |\psi(\xi)|^\alpha\geqslant0 \quad \forall\,\xi\in \mathbb T^d_h.
\end{equation*}
\notag
$$
Extracting a full square we can verify directly that (2.1) is equivalent to the relation
$$
\begin{equation*}
| \omega(\xi) - \frac {\lambda_{1}(\psi(\xi))^{\beta/2}}{\lambda_{1}+ \lambda_{2}| \psi(\xi)|^{\alpha}}|\leqslant \frac {\sqrt{ \lambda_{1}\lambda_{2}}|\psi(\xi)|^{\alpha/2}}{\lambda_{1}+ \lambda_{2}|\psi(\xi)|^{\alpha}}\sqrt{\lambda_{1}+\lambda_{2}|\psi(\xi)|^{\alpha}-|\psi(\xi)|^{\beta}}
\end{equation*}
\notag
$$
for almost all $\xi\in \mathbb T^d_h$; in this case, as shown above, the expression under the root sign is nonnegative. It obviously follows that the set of functions $\omega(\cdot)$ satisfying (2.1) is nonempty. Now we turn directly to an upper estimate for the optimal recovery error and the optimality of the methods indicated in the theorem. Assume that a function $\omega(\cdot)$ satisfies (2.1). We estimate the error of the method $\widehat m_\omega$. By definition this method equals the value of the problem
$$
\begin{equation*}
\begin{gathered} \, \|\Delta^{\beta/2}f-\widehat m_\omega(g) \|_{l_{2}(\mathbb Z^d_h)}\to\max, \\ \|f-g\|_{l_{2}(\mathbb Z^d_h)}\leqslant\delta, \qquad \|\Delta^{\alpha/2}f\|_{l_{2}(\mathbb Z^d_h)}\leqslant1, \qquad g\in l_{2}(\mathbb Z^d_h). \end{gathered}
\end{equation*}
\notag
$$
Switching to Fourier images, we derive from Plancherel’s theorem that the squared value of this problem is the value of the problem
$$
\begin{equation}
\begin{gathered} \, \notag \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|(\psi(\xi))^{\beta/2}F[f](\xi)-\omega(\xi)F[g](\xi)|^2\,d\xi\to\max, \\ \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|F[f](\xi)-F[g](\xi)|^2\,d\xi\leqslant \delta^2, \\ \notag \frac1{(2\pi)^d}\int_{\mathbb T^d_h}|\psi(\xi)|^{\alpha}|F[f](\xi)|^2\,d\xi\leqslant1, \qquad g\in l_{2}(\mathbb Z^d_h). \end{gathered}
\end{equation}
\tag{3.10}
$$
First let $\delta\geqslant (h^2/(4d))^{\alpha/2}$. We estimate the expression under the integral sign in the functional in (3.10). Assume that $\lambda_1$ and $\lambda_2$ are the numbers defined for this $\delta$ before the formulation of the theorem. By the Cauchy–Bunyakovsky–Schwarz inequality we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl|(\psi(\xi))^{\beta/2}F[f](\xi)-\omega(\xi)F[g](\xi)\bigr|^2 \\ &\ \ =\bigl|(\psi(\xi))^{\beta/2}F[f](\xi)-\omega(\xi)F[f](\xi) +\omega(\xi)F[f](\xi)-\omega(\xi)F[g](\xi)\bigr|^2 \\ &\ \ =|\omega(\xi)(F[f](\xi)-F[g](\xi))+F[f](\xi)((\psi(\xi))^{\beta/2}-\omega(\xi))|^2 \\ &\ \ =\biggl|\frac{\omega(\xi)}{\sqrt{\lambda_1}}\sqrt{\lambda_1}(F[f](\xi)-F[g](\xi))+ \frac{(\psi(\xi))^{\beta/2}-\omega(\xi)} {\sqrt{\lambda_2}((\psi(\xi))^{\alpha/2}} \sqrt{\lambda_2}(\psi(\xi))^{\alpha/2}F[f](\xi)\bigg|^2 \\ &\ \ \leqslant \biggl(\!\frac{|\omega(\xi)|^2}{\lambda_1}+\frac{|(\psi(\xi))^{\beta/2}- \omega(\xi)|^2}{\lambda_2|\psi(\xi)|^\alpha}\!\biggr) \bigl(\lambda_1|F[f](\xi)\,{-}\,F[g](\xi)|^2\,{+}\, \lambda_2|\psi(\xi)|^\alpha|F[f](\xi)|^2\bigr) \\ &\ \ \leqslant\bigl(\lambda_1|F[f](\xi)-F[g](\xi)|^2+ \lambda_2|\psi(\xi)|^\alpha|F[f](\xi)|^2\bigr). \end{aligned}
\end{equation*}
\notag
$$
Integrating this inequality, we conclude that the functional maximized in problem (3.10) does not exceed the quantity
$$
\begin{equation*}
\lambda_1\delta^2+\lambda_2=\delta^{2(\alpha-\beta)/\alpha};
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
e(\beta,W,\delta,\widehat m_\omega)\leqslant\delta^{(\alpha-\beta)/\alpha}.
\end{equation*}
\notag
$$
It follows from this estimate and (3.9) that
$$
\begin{equation*}
\delta^{(\alpha-\beta)/{\alpha}}\geqslant e(\beta,W,\delta,\widehat m_\omega)\geqslant E(\beta,W,\delta)\geqslant\delta^{(\alpha-\beta)/{\alpha}}.
\end{equation*}
\notag
$$
Therefore, if $\delta\geqslant (h^2/(4d))^{\alpha/2}$, then $\widehat m_{\omega}$ is an optimal method, and the required upper estimate for the optimal recovery error is obtained. Now let $\delta< (h^2/(4d))^{\alpha/2}$. We estimate the error of the method $\widehat m$. In this case, in view of (3.6) the expression under the integral sign in the functional considered in problem (3.10) is estimated as
$$
\begin{equation*}
\begin{aligned} \, &|(\psi(\xi))^{\beta/2}F[f](\xi)-(\psi(\xi))^{\beta/2}F[g](\xi)|^2 \\ &\qquad =|\psi(\xi)|^{\beta}|F[f](\xi)-F[g](\xi)|^2 \leqslant \biggl(\frac{4d}{h^2}\biggr)^\beta|F[f](\xi)-F[g](\xi)|^2, \end{aligned}
\end{equation*}
\notag
$$
which yields the inequalities
$$
\begin{equation*}
\begin{aligned} \, &\frac1{(2\pi)^d}\int_{\mathbb T^d_h}|(\psi(\xi))^{\beta/2}F[f](\xi)- (\psi(\xi))^{\beta/2}F[g](\xi)|^2\,d\xi \\ &\qquad \leqslant \frac1{(2\pi)^d}\int_{\mathbb T^d_h}\biggl(\frac{4d}{h^2}\biggr)^\beta |F[f](\xi)-F[g](\xi)|^2d\xi\leqslant\biggl(\frac{4d}{h^2}\biggr)^\beta\delta^2. \end{aligned}
\end{equation*}
\notag
$$
Consequently, taking account of estimate (3.9) for $\delta< (h^2/(4d))^{\alpha/2}$, we deduce that the method $\widehat m$ is optimal and the expression for the optimal recovery error indicated in the theorem is valid. The theorem is proved. Corollary. For all $f\in l_2(\mathbb Z_h^d)$ such that Proof. We prove (3.12). If $f=0$, then this inequality is obvious. Assume that a nonzero function $f\in l_2(\mathbb Z_h^d)$ is such that (3.11) is true (it follows from the definition of $\Delta_h^{\alpha/2}$ that $\Delta_h^{\alpha/2}f\ne0$). Clearly, the function $f_1=\|\Delta_h^{\alpha/2}f\|_{l_2(\mathbb Z_h^d)}^{-1}f$ is admissible in problem (3.3) for
$$
\begin{equation}
\delta=\frac{\|f\|_{l_2(\mathbb Z_h^d)}}{\|\Delta_h^{\alpha/2}f\|_{l_2(\mathbb Z_h^d)}}.
\end{equation}
\tag{3.14}
$$
The value at this function of the functional maximized in (3.3) is at most the value of the problem itself, which equals the optimal recovery error. Since $\delta\geqslant(h^2/(4d))^{\alpha/2}$ by (3.11), the value of this functional at $f_1$ is at most $\delta^{(\alpha-\beta)/\alpha}$, that is,
$$
\begin{equation*}
\frac{\|\Delta_h^{\beta/2}f\|_{l_2(\mathbb Z_h^d)}}{\|\Delta_h^{\alpha/2}f\|_{l_2(\mathbb Z_h^d)}}\leqslant\delta^{(\alpha-\beta)/\alpha}.
\end{equation*}
\notag
$$
Substituting the expression for $\delta$ from (3.14) into this estimate we obtain (3.12).
Inequality (3.13) can be proved similarly. The corollary is proved. 
Citation in format AMSBIB
\Bibitem{Siv25} |
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