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A Determinantal Expansion for a Class of Definite Integral

Part 4.

Published online by Cambridge University Press:  20 January 2009

L. R. Shenton
L. R. Shenton
Affiliation:
Department of Mathematics, College of Technology, Manchester, 1.
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We shall show in this part the relation of generalised C.F.'s to ordinary C.F.'s, in the main confining our attention to Stieltjes type fractions. Moreover we shall bring out the part played by Parseval's theorem in our development of the subject, and a property of extremal solutions of the Stieltjes moment problem given by M. Riesz.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 4 , June 1957 , pp. 153 - 166
Copyright
Copyright © Edinburgh Mathematical Society 1957

References

page 153 note 1 Riesz, M., “Sur le problème des moments,” Arkiv for matematik, astronomi och fysik, 16 (12), 1–21; 16 (19), 1–21; 17 (16) 1–52.Google Scholar

page 153 note 2 See, for example, Shohat, J. A. and Tamarkin, J. D., The Problem of Moments (American Mathematical Society Surveys No. 1, 1943).Google Scholar

page 155 note 1 Stieltjes, T. J., Oeucres Complètes, Vol. 2, pp. 505–506, 518–520.Google Scholar

page 155 note 2 Hardy, G. H., “On Stieltjes' ‘problème des moments,’” Messenger of Mathematics., 46 (1917), 175–182; 47 (1917), 81–88.Google Scholar

page 156 note 1 The following abbreviated notation for alternant types of determinants will be used throught:

where any functional symbol cannot be separated from its argument.

Thus

but | ω2r(z 1), ω2r+2(z 2) | is unambiguous. Similarly when the symbol of functionality is tied to its suffix we shall write

Thus

page 157 note 1 See Shohat, J., “On Stieltjea Continued Fractions,” American Journal of Math., LIV. (1932), 79–84.CrossRefGoogle Scholar

page 161 note 1 As a particular example suppose that by using (4) and an equivalence transformation we find the convergent expansion

Then by (30) with z 1 = i t = − z 2, t>0, we have a convergent expansion for

But the stieltjes C.F. for diverges by oscillation.

page 162 note 1 l.d.s. means limit of the decreasing sequence.

page 164 note 1 There is a similar identity for the diagonal determinants given in Part 3, 3 (b).