Interpolating functions associated with second-order differential equations
HTML articles powered by AMS MathViewer
- by William F. Trench PDF
- Proc. Amer. Math. Soc. 78 (1980), 253-258 Request permission
Abstract:
Functions are exhibited which interpolate the magnitude of a solution y of a linear, homogeneous, second-order differential equation at its critical points, $|y’|$ at the zeros of y, and $|\smallint _{{x_0}}^xy(t)h(t)\;dt|$ at the zeros of y. Except for a normalization condition, the interpolating functions are independent of the specific solution y. A theorem similar in its conclusions to the Sonin-Pólya-Butlewski theorem is presented and examples are given.References
- Otakar Bor vka, Linear differential transformations of the second order, The English Universities Press Ltd., London, 1971. Translated from the German by F. M. Arscott. MR 0463539
- Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
- Lee Lorch and Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions, Acta Math. 109 (1963), 55–73. MR 147695, DOI 10.1007/BF02391809
- Lee Lorch, M. E. Muldoon, and Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions. III, Canadian J. Math. 22 (1970), 1238–1265. MR 274845, DOI 10.4153/CJM-1970-142-1
- Lee Lorch, Martin E. Muldoon, and Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions. IV, Canadian J. Math. 24 (1972), 349–368. MR 298113, DOI 10.4153/CJM-1972-029-9 H. Milloux, Sur l’équation différentielle $x'' + xA(t) = 0$, Prace Mat.-Fiz. 41 (1933), 39-54. G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 253-258
- MSC: Primary 34C10; Secondary 33A40, 34B25
- DOI: https://doi.org/10.1090/S0002-9939-1980-0550507-9
- MathSciNet review: 550507