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Higher Monotonicity Properties of Certain Sturm-Liouville Functions. III

Published online by Cambridge University Press:  20 November 2018

Lee Lorch ,
M. E. Muldoon  and
Peter Szego
Lee Lorch
Affiliation:
York University, Toronto, Ontario
M. E. Muldoon
Affiliation:
(L. L. and M. E. M.) Ampex Corporation, Redwood City, California (P. S.)
Peter Szego
Affiliation:
(L. L. and M. E. M.) Ampex Corporation, Redwood City, California (P. S.)
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A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation

(1.1)

considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be

(1.2)

where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1238 - 1265
Copyright
Copyright © Canadian Mathematical Society 1970

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