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New Approximations for Wiener Integrals, with Error Estimates

Published online by Cambridge University Press:  20 November 2018

Henry C. Finlayson
Henry C. Finlayson*
Affiliation:
University of Manitoba
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The principal theorem of this paper, a generalization of a theorem given by R. H. Cameron (2), provides a means of approximating certain Wiener integrals to any desired degree of accuracy by an (n + 1)-fold Riemann integral with sufficiently large n. The generalization is in the use of a general complete orthonormal set of functions, whereas Cameron's paper used only the odd harmonic set.

Let C′ be the class of real-valued functions x(t) defined on [0, 1] and such that x(0) = 0 and which are continuous except perhaps for one left continuous jump. Let C be the class of continuous members of C′.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 58 - 105
Copyright
Copyright © Canadian Mathematical Society 1967

References

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2. Cameron, R. H., A “Simpson's Rule” for the numerical evaluation of Wiener's integrals in function space, Duke Math. J., 18 (1951), 111–130.Google Scholar
3. Hille, Einar, Analytic function theory, vol. 1 (New York, 1959).Google Scholar
4. Hobson, E. W., The theory of functions of a real variable and the theory of Fourier's series, 3rd ed. (Cambridge, 1957).Google Scholar
5. Titchmarsh, E. C., The theory of functions, 2nd ed. (Oxford, 1950).Google Scholar