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

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JournalIntegral Equations and Operator Theory
DatePublished - Nov 2005
Issue number3
Volume53
Number of pages11
Pages (from-to)331-341
Original languageEnglish

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.

    Research areas

  • Volterra operators, POWERS, NORM

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