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Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion

Mikhail A. Lifshits Saint Petersburg State University, St. Petersburg, Russia
Werner Linde Friedrich-Schiller University, Jena, Germany
Available Formats:
Electronic ISBN: 978-1-4704-0338-6
Product Code: MEMO/157/745.E
List Price: $59.00 MAA Member Price:$53.10
AMS Member Price: $35.40 Click above image for expanded view Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion Mikhail A. Lifshits Saint Petersburg State University, St. Petersburg, Russia Werner Linde Friedrich-Schiller University, Jena, Germany Available Formats:  Electronic ISBN: 978-1-4704-0338-6 Product Code: MEMO/157/745.E  List Price:$59.00 MAA Member Price: $53.10 AMS Member Price:$35.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1572002; 87 pp
MSC: Primary 47; Secondary 60;

We consider the Volterra integral operator $T_{\rho,\psi}:L_p(0,\infty)\to L_q(0,\infty)$ for $1\leq p,q\leq \infty$, defined by $(T_{\rho,\psi}f)(s) =\rho(s)\int_0^s \psi(t) f(t) dt$ and investigate its degree of compactness in terms of properties of the kernel functions $\rho$ and $\psi$. In particular, under certain optimal integrability conditions the entropy numbers $e_n(T_{\rho,\psi})$ satisfy $c_1\Vert{\rho\,\psi}\Vert_r\leq \liminf_{n\to\infty} n\, e_n(T_{\rho,\psi}) \leq \limsup_{n\to\infty} n\, e_n(T_{\rho,\psi})\leq c_2\Vert{\rho\,\psi}\Vert_r$ where $1/r = 1- 1/p +1/q >0$. We also obtain similar sharp estimates for the approximation numbers of $T_{\rho,\psi}$, thus extending former results due to Edmunds et al. and Evans et al.. The entropy estimates are applied to investigate the small ball behaviour of weighted Wiener processes $\rho\, W$ in the $L_q(0,\infty)$–norm, $1\leq q\leq \infty$. For example, if $\rho$ satisfies some weak monotonicity conditions at zero and infinity, then $\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2$.

Graduate students and research mathematicians interested in operator theory, probability theory, and stochastic processes.

• Chapters
• 1. Introduction
• 2. Main results
• 3. Scale transformations
• 4. Upper estimates for entropy numbers
• 5. Lower estimates for entropy numbers
• 6. Approximation numbers
• 7. Small ball behaviour of weighted Wiener processes
• Requests

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Volume: 1572002; 87 pp
MSC: Primary 47; Secondary 60;

We consider the Volterra integral operator $T_{\rho,\psi}:L_p(0,\infty)\to L_q(0,\infty)$ for $1\leq p,q\leq \infty$, defined by $(T_{\rho,\psi}f)(s) =\rho(s)\int_0^s \psi(t) f(t) dt$ and investigate its degree of compactness in terms of properties of the kernel functions $\rho$ and $\psi$. In particular, under certain optimal integrability conditions the entropy numbers $e_n(T_{\rho,\psi})$ satisfy $c_1\Vert{\rho\,\psi}\Vert_r\leq \liminf_{n\to\infty} n\, e_n(T_{\rho,\psi}) \leq \limsup_{n\to\infty} n\, e_n(T_{\rho,\psi})\leq c_2\Vert{\rho\,\psi}\Vert_r$ where $1/r = 1- 1/p +1/q >0$. We also obtain similar sharp estimates for the approximation numbers of $T_{\rho,\psi}$, thus extending former results due to Edmunds et al. and Evans et al.. The entropy estimates are applied to investigate the small ball behaviour of weighted Wiener processes $\rho\, W$ in the $L_q(0,\infty)$–norm, $1\leq q\leq \infty$. For example, if $\rho$ satisfies some weak monotonicity conditions at zero and infinity, then $\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2$.

Graduate students and research mathematicians interested in operator theory, probability theory, and stochastic processes.

• Chapters
• 1. Introduction
• 2. Main results
• 3. Scale transformations
• 4. Upper estimates for entropy numbers
• 5. Lower estimates for entropy numbers
• 6. Approximation numbers
• 7. Small ball behaviour of weighted Wiener processes
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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