A minimal decay rate for solutions of stable 𝑛th order homogeneous differential equations with constant coefficients

Kammler, David W.

doi:10.1090/S0002-9939-1975-0369810-9

A minimal decay rate for solutions of stable $n$th order homogeneous differential equations with constant coefficients
HTML articles powered by AMS MathViewer

by David W. Kammler

Proc. Amer. Math. Soc. 48 (1975), 145-151

DOI: https://doi.org/10.1090/S0002-9939-1975-0369810-9

PDF | Request permission

Abstract:

In this paper we establish the existence of an envelope function (depending only on $n$ and $\alpha > 0$) which provides a pointwise bound on the size of any normalized solution $y$ of any homogeneous $n$th order differential equation with constant coefficients for which the roots of the corresponding characteristic polynomial have real parts which do not exceed $- \alpha$. An explicit representation for this envelope is obtained in the special case where these roots are further constrained to be real valued.

References

David W. Kammler, Existence of best approximations by sums of exponentials, J. Approximation Theory 9 (1973), 78–90. MR 344754, DOI 10.1016/0021-9045(73)90113-5
David W. Kammler, Chebyshev polynomials corresponding to a semi-infinite interval and an exponential weight factor, Math. Comp. 27 (1973), 633–637. MR 329204, DOI 10.1090/S0025-5718-1973-0329204-5
Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482
I. P. Natanson, Constructive function theory. Vol. I. Uniform approximation, Frederick Ungar Publishing Co., New York, 1964. Translated from the Russian by Alexis N. Obolensky. MR 0196340