Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
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
Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion
eBook 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
Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion
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
eBook 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 \(\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2\).

    Readership

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

  • Table of Contents
     
     
    • 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
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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 \(\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2\).

Readership

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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.