Asymptotic behaviour of iterates of Volterra operators on L-p(0,1)

S.P. Eveson

doi:10.1007/s00020-003-1329-6

Asymptotic behaviour of iterates of Volterra operators on L-p(0,1)

S.P. Eveson

Mathematics

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

Abstract

Given k is an element of L-1(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V-k defined on L-P(0,1) by

(Vku) (t) = integral(0)(t) (t-s)u(s)ds,

and its iterates (V-k(n))n is an element of N. We construct some much simpler sequences which, as n -> infinity, are asymptotically equal in the operator norm to V-k(n). This leads to a simple asymptotic formula for parallel to V(k)(n)parallel to and to a simple `asymptotically extremal sequence'; that is, a sequence (u(n)) in L-P (0, 1) with parallel to u(n)parallel to(P) = 1 and parallel to V-k(n) u(n)parallel to similar to parallel to V(k)(n)parallel to as n -> infinity. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.

Original language	English
Pages (from-to)	331-341
Number of pages	11
Journal	Integral Equations and Operator Theory
Volume	53
Issue number	3
DOIs	https://doi.org/10.1007/s00020-003-1329-6
Publication status	Published - Nov 2005

Keywords

Volterra operators
POWERS
NORM

Access to Document

10.1007/s00020-003-1329-6

http://dx.doi.org/10.1007/s00020-003-1329-6

Cite this

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title = "Asymptotic behaviour of iterates of Volterra operators on L-p(0,1)",

abstract = "Given k is an element of L-1(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V-k defined on L-P(0,1) by(Vku) (t) = integral(0)(t) (t-s)u(s)ds,and its iterates (V-k(n))n is an element of N. We construct some much simpler sequences which, as n -> infinity, are asymptotically equal in the operator norm to V-k(n). This leads to a simple asymptotic formula for parallel to V(k)(n)parallel to and to a simple `asymptotically extremal sequence'; that is, a sequence (u(n)) in L-P (0, 1) with parallel to u(n)parallel to(P) = 1 and parallel to V-k(n) u(n)parallel to similar to parallel to V(k)(n)parallel to as n -> infinity. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.",

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author = "S.P. Eveson",

year = "2005",

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doi = "10.1007/s00020-003-1329-6",

language = "English",

volume = "53",

pages = "331--341",

journal = "Integral Equations and Operator Theory",

issn = "0378-620X",

publisher = "Birkhauser Verlag Basel",

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TY - JOUR

T1 - Asymptotic behaviour of iterates of Volterra operators on L-p(0,1)

AU - Eveson, S.P.

PY - 2005/11

Y1 - 2005/11

N2 - Given k is an element of L-1(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V-k defined on L-P(0,1) by(Vku) (t) = integral(0)(t) (t-s)u(s)ds,and its iterates (V-k(n))n is an element of N. We construct some much simpler sequences which, as n -> infinity, are asymptotically equal in the operator norm to V-k(n). This leads to a simple asymptotic formula for parallel to V(k)(n)parallel to and to a simple `asymptotically extremal sequence'; that is, a sequence (u(n)) in L-P (0, 1) with parallel to u(n)parallel to(P) = 1 and parallel to V-k(n) u(n)parallel to similar to parallel to V(k)(n)parallel to as n -> infinity. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.

AB - Given k is an element of L-1(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V-k defined on L-P(0,1) by(Vku) (t) = integral(0)(t) (t-s)u(s)ds,and its iterates (V-k(n))n is an element of N. We construct some much simpler sequences which, as n -> infinity, are asymptotically equal in the operator norm to V-k(n). This leads to a simple asymptotic formula for parallel to V(k)(n)parallel to and to a simple `asymptotically extremal sequence'; that is, a sequence (u(n)) in L-P (0, 1) with parallel to u(n)parallel to(P) = 1 and parallel to V-k(n) u(n)parallel to similar to parallel to V(k)(n)parallel to as n -> infinity. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.

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KW - POWERS

KW - NORM

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DO - 10.1007/s00020-003-1329-6

M3 - Article

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SP - 331

EP - 341

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

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ER -