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ESTIMATES FOR THE NORM OF THE nTH INDEFINITE INTEGRAL

  • G. LITTLE (a1) and J. B. READE (a1)
    • Published online: 01 September 1998
Abstract

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

formula here

where fL2[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

formula here

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).

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Bulletin of the London Mathematical Society
  • ISSN: 0024-6093
  • EISSN: 1469-2120
  • URL: /core/journals/bulletin-of-the-london-mathematical-society
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