Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-18T14:28:09.418Z Has data issue: false hasContentIssue false
  • English
  • Français

The expansion of continuous functions in series of integrals of orthonormal functions

Published online by Cambridge University Press:  18 May 2009

H. C. Finlayson
H. C. Finlayson
Affiliation:
University of Manitoba, Manitoba, Canada
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with the following problem: Can an arbitrary continuous function on [0, 1], which vanishes at the origin, be represented in some sense as a series of constant multiples of indefinite integrals of a complete orthonormal set of functions on [0,1]? Four contexts in which this problem arises naturally will be given in the introduction and the remainder of the paper will be devoted to giving a partial answer to the specific problem formulated in one of these contexts.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 1 , March 1972 , pp. 29 - 38
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Banach, Stefan, Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
2.Burkill, J. C., The Lebesgue integral, Cambridge Tract No. 40 (Cambridge, 1961).Google Scholar
3.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.CrossRefGoogle Scholar
4.Ciesielski, Z., On Haar functions and on the Schauder basis of the space C<0, i>. (Russian summary, unbound insert), Bull. Acad. Polon. Sci. Sir. Sci. Math. Astronom. Phys. 7 (1959), 227–232, Math. Reviews 24A, 1599.Google Scholar
5.Finlayson, H. C., Vladimirov-type approximation of Wiener integrals of suitably dominated functionals, Trans. Amer. Math. Soc. 132, 2 (1968), 461–469.CrossRefGoogle Scholar
6.Finlayson, H. C., Approximation of Wiener integrals of functionals continuous in the uniform topology, Pacific J. Math. 34 (1970). 61–71.CrossRefGoogle Scholar
7.Ince, E. L., Ordinary differential equations (New York, 1956).Google Scholar
8.Kamke, E., Differentialgleichungen Lösungsmethoden und Losungen, 3rd edn. (New York, 1948).Google Scholar
9.Konheim, Alan G. and Miranker, Willard L., Numerical evaluation of Wiener integrals, Math. Comp. 21 (1967), 49–65.CrossRefGoogle Scholar
10.Szego, G., Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications, Vol. 23 (New York, 1939).Google Scholar
11.Vladimirov, V. S., The approximate evaluation of Wiener integrals, Uspeki Mat. Nauk 15 (1960), 4 (94), 129–135. English translation: Amer. Math. Soc. Transl.(2) 34 (1963), 405–412.Google Scholar
12.Paley, R. E. A. C., Wiener, N. and Zygmund, A., Notes on random functions, Math. Zeit. 37 (1933), 647–668.CrossRefGoogle Scholar
13.Zygmund, A., Trigonometric series, vol. I, 2nd edn. (Cambridge, 1959).Google Scholar