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Notes on special systems of orthogonal Functions (III): A System of orthogonal polynomials

  • G. H. Hardy (a1)

Generalities. Suppose that

is real and L2(0, ∞), that

is its Mellin transform, and that

Then K(1)(x) generates a “Watson transformation”, with a “Fourier theorem”

valid for every F(x) of L2(0, ∞).

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(1)Bateman, H.Some properties of a certain set of polynomials”. Tóhoku math. J. 37 (1933), 23–8.
(2)Bateman, H.The polynomial F n(x)”. Annals of math. 35 (1934), 767–75.
(3)Hardy, G. H.Notes on some points in the integral calculus (46–47)”. Messenger of math. 46 (1917), 175–82, and 47 (1918), 81–8.
(4)Titchmarsh, E. C.Theory of Fourier integrals (Oxford, 1937).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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