eBook ISBN:  9781470403386 
Product Code:  MEMO/157/745.E 
List Price:  $59.00 
MAA Member Price:  $53.10 
AMS Member Price:  $35.40 
eBook ISBN:  9781470403386 
Product Code:  MEMO/157/745.E 
List Price:  $59.00 
MAA Member Price:  $53.10 
AMS Member Price:  $35.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 157; 2002; 87 ppMSC: 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 \(\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2\).
ReadershipGraduate 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


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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.

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