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The Integral Expansions of Arbitrary Functions connected with Integral Equations

Published online by Cambridge University Press:  24 October 2008

J. Hyslop
J. Hyslop
Affiliation:
St John's College.
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Extract

The following paper aims at a more general treatment than has hitherto been given, of the integral expansions of arbitrary functions, from the point of view of integral equation theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 2 , May 1924 , pp. 169 - 185
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Göttingen Dissertation, ad fin.

† Pp. 158–169.

* In this summation, if Δf(λ), Δ;ρ(λ) both vanish, the fraction is to be replaced by zero.

* Hahn, , §§ 8–13.Google Scholar

† Weyl, , p. 275.Google Scholar

* P. 294, Eqn. (20), and p. 301.

† For I cf. Weyl, , p. 293Google Scholar; for II, II (a) cf. pp. 294–5.

* Pp. 300–1.

† Carleman, , p. 25 and p. 75.Google Scholar

* See Weyl, , p. 276 and p. 286.Google Scholar

† For III see Carleman, , p. 102; for IV of. pp. 48–9 and p. 104.Google Scholar

* Cf. Carleman, , p. 47.Google Scholar

† It is assumed that every finite interval of the λ-axis is transformed into a finite interval of the ρ-axis, and ( –∞, ∞ ) into ( – ∞, ∞ ). The changes to be made, if this is not so, are slight, and do not affect the argument.

* Cf. Carleman, , pp. 82 sqq.Google Scholar

* Weyl, , Göltingen Dissertation, 1908.Google Scholar

* An application of the ordinary Fourier Theorem, cf. Weyl, , p. 315, and Plan-cherel.Google Scholar

† It is readily proved that g has, w.r.t. these functions, the characteristic property of ρ w.r.t. P (s, p).

* Acta Math, xxv (1902), p. 161.Google Scholar