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Some multipoint extensions of the Rayleigh–Ritz inequality

Published online by Cambridge University Press:  14 November 2011

Richard C. Brown  and
Ivo Vrkoč
Richard C. Brown
Affiliation:
Department of Mathematics, University of Alabama, University, Alabama 35486, U.S.A.
Ivo Vrkoč
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague, č.S.S.R.
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Synopsis

We obtain the best constants for multipoint extensions of the Rayleigh–Ritz inequality, and use our results to find good upper bounds for the constants involved in the standard spline approximation and interpolation error bound theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 157 - 174
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Ahlfors, L. V.. Complex Analysis, 2nd edn (New York: McGraw-Hill, 1966).Google Scholar
2Beesack, P. R.. Integral inequalities of the Wirtinger type. Duke Math. J. 25 (1958), 477–498.Google Scholar
3Brink, J.. Inequalities involving ∥f∥p and ∥f (n)∥q for f with n-zeros. Pacific J. Math. 42 (1972), 289–312.CrossRefGoogle Scholar
4Brown, R. C.. Duality theory for nth order differential operators under Stieltjes boundary conditions II; nonsmooth coefficients and nonsingular measures. Ann. Mat. Pura Appl. 105 (1975), 141–170.CrossRefGoogle Scholar
5Brown, R. C.. The operator theory of generalized boundary value problems. Canad. J. Math. 28 (1976), 486–512.Google Scholar
6Brown, R. C.. Wirtinger inequalities, Dirichlet functional inequalities and the spectral theory of linear operators and relations. Spectral Theory of Differential Operators, Knowles, I. W. and Lewis, R. T. (eds), pp. 69–79 (Amsterdam: North-Holland Publishing Company, 1981).Google Scholar
7Brown, R. C. and Krall, A. M.. Nth order differential systems under Stieltjes boundary conditions. Czechoslovak Math. J. 27 (102) (1977), 119–131.Google Scholar
8Fan, K., Taussky, O. and Todd, J.. Discrete analogues of inequalities of Wirtinger. Monatsh. Math. 59 (1955), 73–90.CrossRefGoogle Scholar
9Fink, A. M.. Conjugate inequalities for functions and their derivatives. SIAM J. Appl. Math. 5 (1974), 399–411.Google Scholar
10Fink, A. M.. Differential inequalities and disconjugacy. J. Math. Anal. Appl. 49 (1975), 758–771.CrossRefGoogle Scholar
11Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1966).Google Scholar
12Micchelli, Charles A.. Oscillation matrices and cardinal spline interpolation. Studies in Spline Functions and Approximation Theory, pp. 163–201 (New York: Academic Press, 1976).Google Scholar
13Mitrinović, D. S. and Vasić, P. M.. An integral inequality ascribed to Wirtinger and its variations and generalizations. Publ. Elektrotehn Fak. Univ. Beograd, Ser. Mat. Fiz. No. 247–273 (1969), 157–170.Google Scholar
14Schoenberg, I. J.. Cardinal Spline Interpolation (CBMS Series No. 12) (Philadelphia: SIAM, 1973).CrossRefGoogle Scholar
15Schultz, Martin H.. Spline Analysis (Englewood Cliffs, N.J.: Prentice Hall, 1973).Google Scholar