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Higher Monotonicity Properties of Certain Sturm-Liouville Functions. IV

Published online by Cambridge University Press:  20 November 2018

Lee Lorch ,
Martin E. Muldoon  and
Peter Szego
Lee Lorch
Affiliation:
York University, Downsview, Ontario
Martin E. Muldoon
Affiliation:
York University, Downsview, Ontario
Peter Szego
Affiliation:
Ampex Corporation, Redwood City, California
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The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation

1.1

Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term

1.2

where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 349 - 368
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bihari, I., Oscillation and monotonity theorems concerning non-linear differential equations of the second order, Acta Math. Acad. Sci. Hungar. 9 (1958), 83–104.Google Scholar
2. Butlewski, Z., Sur les intégrales d'une équation différentielle du second ordre, Mathematica (Cluj) 12 (1936), 36–48.Google Scholar
3. Cheng, M.-T., The Gibbs phenomenon and Bochner's summation method (I), Duke Math. J. 17 (1950), 83–90.Google Scholar
4. Hartman, P., On differential equations and the function J»2 + FM 2, Amer. J. Math. 83 (1961), 154–188.Google Scholar
5. Lorch, L., Comparison of two formulations of Sonin's theorem and of their respective applications to Bessel functions, Studia Sci. Math. Hungar. 1 (1966), 141–145.Google Scholar
6. Lorch, L. and Moser, L., A remark on completely monotonie sequences, with an application to summability, Can. Math. Bull. 6 (1963), 171–173.Google Scholar
7. Lorch, L. and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions, Acta Math. 109 (1963), 55–73.Google Scholar
8. Lorch, L. and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 455–457.Google Scholar
9. Lorch, L., Muldoon, M. E., and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions. III, Can. J. Math. 22 (1970), 1238–1265.Google Scholar
10. Makai, E., On a monotonie property of certain Sturm-Liouville functions, Acta Math. Acad. Sci. Hungar. 3 (1952), 165–172.Google Scholar
11. Muldoon, M. E., Elementary remarks on multiply monotonic functions and sequences, Can. Math. Bull. 14 (1971), 69–72.Google Scholar
12. Pólya, G. and Szegő, G., Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, 2te Aufl., Die Grundlehren der mathematischen Wissenschaften, Bd. XX (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954).Google Scholar
13. Richard, U., Sulle successioni di valori stazionari delle soluzioni di equazioni differenziali lineari del 2° ordine, Univ. e Politecnico Torino. Rend. Sem. Mat. 9 (1950), 309–324.Google Scholar
14. Steinig, J., The real zeros of Struve's function, SIAM J. Math. Analysis 1 (1970), 365–375.Google Scholar
15. Szász, O., On the relative extrema of Bessel functions, Boll. Unione Matematica Italiana (3) 5 (1950), 225–229 (= Collected Mathematical Papers (University of Cincinnati, 1955), 1300-1304).Google Scholar
16. Szegő, G., Orthogonal polynomials, American Math. Soc. Colloquium Publications, 23, 3rd ed. (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
17. Veltkamp, G. W. and Brands, J. J. M., Problem 64-17. A property of real solutions to Bessel’ s equation, SIAM Rev. 8 (1966), 236–237.Google Scholar
18. Vosmanský, J., The monotonicity of extremants of integrals of the differential equation y” + q(t)y = 0, Arch. Math. (Brno) 2 (1966), 105–111.Google Scholar
19. Vosmanský, J., Monotonic properties of zeros and extremants of the differential equation y” + q﹛t)y = 0, Arch. Math. (Brno) 6 (1970/71), 37–74.Google Scholar
20. Watson, G. N., A treatise on the theory of Bessel functions, 2nd ed. (Cambridge University Press, London, 1944).Google Scholar
21. Wiman, A., Über eine Stabilitätsfrage in der Theorie der linear en Differentialgleichugen, Acta Math. 66 (1936), 121–145.Google Scholar