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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, and Werner Linde, Friedrich-Schiller University, Jena, Germany

Memoirs of the American Mathematical Society
2002; 87 pp; softcover
Volume: 157
ISBN-10: 0-8218-2791-X
ISBN-13: 978-0-8218-2791-8
List Price: US$59
Individual Members: US$35.40
Institutional Members: US$47.20
Order Code: MEMO/157/745
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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.

Table of Contents

  • Introduction
  • Main results
  • Scale transformations
  • Upper estimates for entropy numbers
  • Lower estimates for entropy numbers
  • Approximation numbers
  • Small ball behaviour of weighted Wiener processes
  • Appendix
  • Bibliography
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