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# Тезисы

## Оценка погрешности восстановления эллиптического дифференциального оператора на анизотропных классах Никольского-Бесова по информации о спектре

Балгимбаева Ш.А.

Институт математики, Казахстан, 050010, Алматы, ул.Пушкна, 125

1  стр. (принято к публикации)

Let us consider the problem of for elliptic differential operator with constant coefficients on non - isotropic Nikol'skii-Besov spaces $B_{p\,\theta}^\mathbf{s}(\mathbb{R}^n)$ using spectrum information (information on Fourier transform) in $L_q$ --- norm.

Namely, let $$\mathcal{L}:= \sum\limits_{|\alpha| \leq m}a_\alpha \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}...\partial x_n^{\alpha_n}}$$ be elliptic differential operator with constant coefficients; let $B_{p\, \theta}^\mathbf{s}(\mathbb{R}^n)$ be Nikol'skii-Besov space and $L_q$ be the space of measurable function on $\mathbb{R}^n$ with standard norm $\|\cdot\|_q$.

Denote Fourier transform of distribution $f \in S'(\mathbb{R}^n)$ by $\mathcal{F}(f)$. For $f \in S'(\mathbb{R}^n)$ denote restriction of $\mathcal{F}(f)$ on $\Omega_\sigma :=\{\xi: \|\xi\|_{\mathbf{a}} < \sigma\} \subset \mathbb{R}^n$ by $\mathcal{F}|_\Omega_\sigma}.$ We use $\mathcal{F}(f)|_{\Omega_\sigma}$ as information on function $f \in B_{p\,\theta}^\mathbf{s}(\mathbb{R}^n)$.

Then the problem of recovery is to estimate the quantity $$E(B_{p\, \theta}^\mathbf{s}(\mathbb{R}^n),\mathcal{L},\mathcal{F}|_{\Omega_\sigma}, L_q):= \inf\limits_{S}\sup\limits_{\|f|B_{p\,\theta}^\mathbf{s}(\mathbb{R}^n)\| \leq 1} \|\mathcal{L}f - \mathcal{L}S[\mathcal{F}(f)|_{\Omega_\sigma}]\|_q$$ where $\inf$ is taken over all linear methods $S: \mathcal{F}(B_{p\,\theta}^\mathbf{s})|_\Omega_\sigma}\rightarrow L_q$, and to find linear methods $\tilde{S}$ for which the order of the quantity is realized.

We obtained order exact estimate for the quantity $E(B_{p\,\theta}^\mathbf{s}(\mathbb{R}^n),\mathcal{L},\mathcal{F}|_{\Omega_\sigma}, L_q)$ and construct corresponding optimal linear method as action of the differentiation operator on special "partial"\, sum of the expansion of function $f$ with respect to Meyer-David system of orthonormal multivariate wavelets.

© 2004 Дизайн Лицея Информационных технологий №1533