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Transactions of the American Mathematical Society

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An approximation theory for generalized Fredholm quadratic forms and integral-differential equations


Authors: J. Gregory and G. C. Lopez
Journal: Trans. Amer. Math. Soc. 222 (1976), 319-335
MSC: Primary 45J05; Secondary 34C10
DOI: https://doi.org/10.1090/S0002-9947-1976-0423024-8
MathSciNet review: 0423024
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Abstract: An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory.

The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas.

In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the ``Euler-Lagrange equations'' for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations.

In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.


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  • [1] J. Gregory, An approximation theory for elliptic quadratic forms on Hilbert spaces: Application to the eigenvalue problem for compact quadratic forms, Pacific J. Math. 37 (1970), 383-395. MR 46 # 2449. MR 0303311 (46:2449)
  • [2] -, A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space, Trans. Amer. Math. Soc. 157 (1971), 119-128. MR 43 # 3878. MR 0278147 (43:3878)
  • [3] -, An approximation theory for focal points and focal intervals, Proc. Amer. Math. Soc. 32 (1972), 477-483. MR 45 # 5847. MR 0296788 (45:5847)
  • [4] -, A theory of numerical approximation for elliptic forms associated with second order differential systems. Application to eigenvalue problems, J. Math. Anal. Appl. 38 (1972), 416-426. MR 48 # 1014. MR 0322652 (48:1014)
  • [5] J. Gregory and F. Richards, Numerical approximation for 2mth order differential systems via splines, Rocky Mountain J. Math. 5 (1975). MR 0388790 (52:9624)
  • [6] M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. 1 (1951), 525-581. MR 13, 759. MR 0046590 (13:759a)
  • [7] -, Calculus of variations and optimal control theory, Wiley, New York, 1966. MR 34 # 3390. MR 0203540 (34:3390)
  • [8] G. C. Lopez, Quadratic variational problems involving higher order ordinary derivates, Dissertation, University of California, Los Angeles, 1961.
  • [9] E. Y. Mikami, Focal points in a control problem, Pacific J. Math. 35 (1970), 473-485. MR 43 # 6800. MR 0281081 (43:6800)
  • [10] M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ., vol. 18, Amer. Math. Soc., Providence, R. I., 1934. MR 1451874 (98f:58070)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0423024-8
Keywords: Approximation theory, conjugate points, oscillations, calculus of variations, Fredholm integral differential equations, spline approximations
Article copyright: © Copyright 1976 American Mathematical Society

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