Applications of the theory of quadratic forms in Hilbert space to the calculus of variations

Hestenes, Magnus R.

doi:10.2140/pjm.1951.1.525

Cited by 178 publications

(117 citation statements)

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“…, jr ) in ¿p, and some n > 0. This inequality is the ellipticity condition of Hestenes in this setting [6].…”

mentioning

confidence: 87%

“…The main results are given in terms of inequalities involving nonnegative indices. In particular we show that the hypotheses for these inequalities are sufficiently general to include the "resolution spaces" of Hestenes [6] for focal point theories and "continuous" perturbations of coefficients of quadratic forms and integral-differential equations.…”

mentioning

confidence: 99%

“…The first level leads to a theory of quadratic forms with applications given by Hestenes [6] and Lopez [8]. The second level leads to an approximation theory for "level one" problems exempUfied by Theorems 11 to 15.…”

mentioning

confidence: 99%

“…A part of this section is found in [8]. This work is an "application" of the quadratic form theory of Hestenes [6].…”

mentioning

confidence: 99%

“…Let 8 denote a subspace of A such that x is in 8 if and only if For any arc x(r) in A set (6) Tßit) = Rllßit)x^it) + f*K%is, t)x«\s) ds for almost all t on a < t < b. Define the recursive relations (7a) vîit) = faT<>is)ds + cl…”

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confidence: 99%

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

Gregory¹,

López-Acedo²

1976

Trans. Amer. Math. Soc.

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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 integraldifferential 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.Received by the editors August 27, 1973 and, in revised form, April 8, 1975 AMS (MOS) subject classifications (1970). Primary 45J05, 49A30, 34C10.Key words and phrases. Approximation theory, conjugate points, oscillations, calculus of variations, Fredholm integral differential equations, spline approximations. 1. Introduction. The main purpose of this paper is to present an approximation theory of quadratic forms which is applicable to a very general class of quadratic forms and to linear selfadjoint operators of a generalized Fredholm type. That is 2wth order, integral-differential systems such as T»(r) = vß~l(t), or as the generalized system:

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“…, jr ) in ¿p, and some n > 0. This inequality is the ellipticity condition of Hestenes in this setting [6].…”

mentioning

confidence: 87%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…A part of this section is found in [8]. This work is an "application" of the quadratic form theory of Hestenes [6].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Gregory¹,

López-Acedo²

1976

Trans. Amer. Math. Soc.

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Mathematical Basis of a Numerical Method

Jung¹,

Sarkar²,

Zhang³

et al. 2010

Time and Frequency Domain Solutions of EM Problems

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Index Theorems for Polynomial Pencils

Kollár¹,

Bosák²

2013

Nonlinear Physical Systems

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We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretationand generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u = u(λ) of an operator pencil L(λ) satisfying L(λ)u(λ) = µ(λ)u(λ) evaluated at a characteristic value of L(λ) do not only generate an arbitrary chain of root vectors of L(λ) but the chain that carries an extra information.

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Applications of the theory of quadratic forms in Hilbert space to the calculus of variations

Cited by 178 publications

References 10 publications

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

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

Mathematical Basis of a Numerical Method

Index Theorems for Polynomial Pencils

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