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Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet. / Dunker, T.; Linde, W.; Kühn, T.; Lifshits, M. A.

In: Journal of Approximation Theory, Vol. 101, No. 1, 01.11.1999, p. 63-77.

Research output: Contribution to journal › Article › peer-review

Harvard

Dunker, T, Linde, W, Kühn, T & Lifshits, MA 1999, 'Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet', Journal of Approximation Theory, vol. 101, no. 1, pp. 63-77. https://doi.org/10.1006/jath.1999.3354

APA

Dunker, T., Linde, W., Kühn, T., & Lifshits, M. A. (1999). Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet. Journal of Approximation Theory, 101(1), 63-77. https://doi.org/10.1006/jath.1999.3354

Vancouver

Dunker T, Linde W, Kühn T, Lifshits MA. Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet. Journal of Approximation Theory. 1999 Nov 1;101(1):63-77. https://doi.org/10.1006/jath.1999.3354

Author

Dunker, T. ; Linde, W. ; Kühn, T. ; Lifshits, M. A. / Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet. In: Journal of Approximation Theory. 1999 ; Vol. 101, No. 1. pp. 63-77.

BibTeX

@article{26f599dbbe5a4b73bd4fd0485d58a6a7,
title = "Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet",
abstract = "Let Td:L2([0, 1]d)→C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k-1(logk)d-1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(-Cε-2logε2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.",
author = "T. Dunker and W. Linde and T. K{\"u}hn and Lifshits, {M. A.}",
year = "1999",
month = nov,
day = "1",
doi = "10.1006/jath.1999.3354",
language = "English",
volume = "101",
pages = "63--77",
journal = "Journal of Approximation Theory",
issn = "0021-9045",
publisher = "Elsevier",
number = "1",

}

RIS

TY - JOUR

T1 - Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

AU - Dunker, T.

AU - Linde, W.

AU - Kühn, T.

AU - Lifshits, M. A.

PY - 1999/11/1

Y1 - 1999/11/1

N2 - Let Td:L2([0, 1]d)→C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k-1(logk)d-1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(-Cε-2logε2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.

AB - Let Td:L2([0, 1]d)→C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k-1(logk)d-1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(-Cε-2logε2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.

UR - http://www.scopus.com/inward/record.url?scp=0008809676&partnerID=8YFLogxK

U2 - 10.1006/jath.1999.3354

DO - 10.1006/jath.1999.3354

M3 - Article

AN - SCOPUS:0008809676

VL - 101

SP - 63

EP - 77

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 1

ER -

ID: 37011736