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

by Lifshits, M. A.; Linde, Werner

• ISBN13: 9780821827918
• ISBN10: 082182791X
• Format: Paperback
• Copyright: 2002-05-01
• Publisher: Amer Mathematical Society

Summary

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

Table of Contents

Introduction	1	(3)
Main Results	4	(5)
Scale Transformations	9	(11)
Increasing Transformations	9	(4)
Decreasing Transformations	13	(1)
Examples	13	(4)
Transformations and Norms	17	(3)
Upper Estimates for Entropy Numbers	20	(20)
A General Bound Based on Partitions	20	(2)
Proof of Theorem 2.2 (1)	22	(3)
Proof of Parts (2) and (3) in Theorem 2.2	25	(3)
Entropy Estimates for Tρ,ψ	28	(4)
Proof of Theorem 2.3	32	(5)
Upper Bounds for Forward Integration Operators	37	(1)
Proof of Theorem 4.9	37	(3)
Lower Estimates for Entropy Numbers	40	(11)
A General Construction for Lower Estimates	40	(3)
Proof of Theorem 2.4	43	(3)
Lower Bounds for Operators Tρ,ψ and Sχ,η	46	(2)
Regular Kernel Functions	48	(3)
Approximation Numbers	51	(21)
Maz'ja-Rosin Theorem	51	(2)
Estimates for an (Tρ) on Finite Intervals	53	(6)
Estimates for an (Tρ) in the Case 1 < p < 2 < q < ∞	59	(6)
Approximation Numbers of Tρ (General Case)	65	(7)
Small Ball Behaviour of Weighed Wiener Processes	72	(13)
Gaussian Processes and Metric Entropy	74	(1)
Weighted Wiener Processes	74	(2)
Small Ball Estimates for Wiener Processes	76	(2)
Exact Small Ball Estimates	78	(7)
Appendix	85	(1)
Bibliography	86