Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T14:17:08.754Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert spaces of generalized functions extending L2, (I)

Published online by Cambridge University Press:  09 April 2009

John Boris Miller
John Boris Miller
Affiliation:
Australian National University, Canberra, A.C.T.
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.

By using certain fractional integrals and derivatives it is possible to construct a continuum of Hilbert spaces within the space L2 (0, ∞); these are the spaces gλ of functions f(x) for which 1xλf(λ)(x) є L2(0, ∞), and they exhibit invariance properties under generalized Fourier transformations. They are described in (6) and (7).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 3 , August 1960 , pp. 281 - 298
Copyright
Copyright © Australian Mathematical Society 1960

References

[1]Bochner, S. and Chandrasekharan, K., Fourier transforms, Princeton, (1949).Google Scholar
[2]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, Cambridge (1934).Google Scholar
[3]Kober, H., On fractional integrals and derivatives, Quart. J. of Math. (Oxford) 11 (1940), 193–211.Google Scholar
[4]Love, E. R., A Banach space of distributions, (I) and (II), Journal London Math. Soc. 32 (1957) 483–98 and 33 (1958) 288–306.Google Scholar
[5]McShane, E. J., Integration, Princeton (1947).Google Scholar
[6]Miller, J. B., A continuum of Hilbert spaces in L2, Proc. London Math. Soc. 9 (1959) 208–26.Google Scholar
[7]Miller, J. B., A symmetrical convergence theory for general transforms, (II), Proc. London Math. Soc. 9 (1959) 451–64.Google Scholar
[8]Schwartz, L., Théorie des distributions, I, Paris (1957).Google Scholar
[9]Temple, G., Theories and applications of generalized functions, Journal London Math. Soc. 28 (1953) 134–48.Google Scholar
[10]Temple, G., The theory of generalized functions, Proc. Roy. Soc. A, 228 (1955) 175–90.Google Scholar
[11]Titchmarsh, E. C., Theory of Fourier integrals, Oxford (1948).Google Scholar
[12]Zaanen, A. C., Linear Analysis, Amsterdam (1956).Google Scholar