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A System of Linear Equations with an Infinity of Unknowns

Published online by Cambridge University Press:  24 October 2008

E. C. Titchmarsh
E. C. Titchmarsh
Affiliation:
(Trinity College)
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Extract

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations is

the numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 3 , 20 September 1924 , pp. 282 - 286
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* For the ordinary theory of linear equations with an infinity of unknowns see Riesz, F., Systèmes d'équations linéaires à une infinité d'inconnues (Paris, 1913)Google Scholar.

† Hardy, G. H. and Titchmarsh, E. C., Proc. London Math. Soc. (2), 23 (1924), 1–26.Google Scholar

‡ See von Koch, H., Proceedings of the fifth International Congress of Mathematicians (Cambridge, 1913)Google Scholar.