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Hilbert spaces of generalized functions extending L2, (I)

  • John Boris Miller (a1)

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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).

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References

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[1]Bochner, S. and Chandrasekharan, K., Fourier transforms, Princeton, (1949).
[2]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, Cambridge (1934).
[3]Kober, H., On fractional integrals and derivatives, Quart. J. of Math. (Oxford) 11 (1940), 193211.
[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.
[5]McShane, E. J., Integration, Princeton (1947).
[6]Miller, J. B., A continuum of Hilbert spaces in L2, Proc. London Math. Soc. 9 (1959) 208–26.
[7]Miller, J. B., A symmetrical convergence theory for general transforms, (II), Proc. London Math. Soc. 9 (1959) 451–64.
[8]Schwartz, L., Théorie des distributions, I, Paris (1957).
[9]Temple, G., Theories and applications of generalized functions, Journal London Math. Soc. 28 (1953) 134–48.
[10]Temple, G., The theory of generalized functions, Proc. Roy. Soc. A, 228 (1955) 175–90.
[11]Titchmarsh, E. C., Theory of Fourier integrals, Oxford (1948).
[12]Zaanen, A. C., Linear Analysis, Amsterdam (1956).
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