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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
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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

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 $$\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2$$.

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