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ESTIMATES FOR THE NORM OF THE nTH INDEFINITE INTEGRAL

Published online by Cambridge University Press:  01 September 1998

G. LITTLE  and
J. B. READE
G. LITTLE
Affiliation:
Mathematics Department, Manchester University, Manchester M13 9PL
J. B. READE
Affiliation:
Mathematics Department, Manchester University, Manchester M13 9PL
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Abstract

Let T be the Volterra operator on L2[0, 1]

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where f∈L2[0, 1], 0[les ]x[les ]1. It is well known that ∥n!Tn∥=O(1/n!). In a recent paper [1], D. Kershaw has proved that

formula here

a result which was first conjectured by Lao and Whitley in [2]. It is easy to prove that

formula here

For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves that

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by analysing the special positivity preserving properties of T*T. He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2).

Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 5 , September 1998 , pp. 539 - 542
Copyright
© The London Mathematical Society 1998

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