An approximation theory for oscillations of differential equations

Author:
John Gregory

Journal:
Proc. Amer. Math. Soc. **40** (1973), 166-172

MSC:
Primary 34A45; Secondary 34C10

DOI:
https://doi.org/10.1090/S0002-9939-1973-0322248-0

MathSciNet review:
0322248

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Abstract: In an earlier paper we gave an approximation theory of focal points and focal intervals. The fundamental purpose of this paper is to show that the ideas and methods of that paper can be used to give an approximation theory for oscillation for linear selfadjoint differential (and integral-differential) equations. These methods follow from the theory of quadratic forms given by Hestenes.

In §1 we give the preliminaries needed for this paper. In §2 we define oscillation for quadratic control problems and discuss their connection with differential equations. In §3 we give our main approximation results relating the oscillation points for a given problem to the oscillation points of an approximating problem. The element belongs to a metric space . The main result is to show that the th oscillation point is a continuous function of . For completeness in §4 we present an example for fourth order differential equations where the approximation is by discrete problems. Thus oscillation points can be ``easily'' computed by numerical algorithms.

**[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**0303311 (46:2449)****[2]**-,*An approximation theory for focal points and focal intervals*, Proc. Amer. Math. Soc.**32**(1972), 477-483. MR**0296788 (45:5847)****[3]**M. R. Hestenes,*Applications of the theory of quadratic forms in Hilbert space in the calculus of variations*, Pacific J. Math.**1**(1951), 525-581. MR**13**, 759. MR**0046590 (13:759a)****[4]**E. Y. Mikami,*Focal points in a control problem*, Pacific J. Math.**35**(1970), 473-485. MR**43**#6800. MR**0281081 (43:6800)****[5]**I. J. Schoenberg,*Spline interpolation and the higher derivatives*, Proc. Nat. Acad. Sci. U.S.A.**51**(1964), 24-28. MR**28**#3278. MR**0160064 (28:3278)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0322248-0

Keywords:
Approximation theory,
oscillation,
quadratic forms,
Hilbert space,
conjugate points,
calculus of variations

Article copyright:
© Copyright 1973
American Mathematical Society