Random quadratic forms

Authors:
John Gregory and H. R. Hughes

Journal:
Trans. Amer. Math. Soc. **347** (1995), 709-717

MSC:
Primary 47B80; Secondary 34B24, 34C10, 34F05

DOI:
https://doi.org/10.1090/S0002-9947-1995-1254841-4

MathSciNet review:
1254841

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Abstract | References | Similar Articles | Additional Information

Abstract: The results of Boyce for random Sturm-Liouville problems are generalized to random quadratic forms. Order relationships are proved between the means of eigenvalues of a random quadratic form and the eigenvalues of an associated mean quadratic form. Finite-dimensional and infinite-dimensional examples that show these are the best possible results are given. Also included are some results for a general approximation theory for random quadratic forms.

**[1]**William E. Boyce,*Random eigenvalue problems*, Probabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, pp. 1–73. MR**0263171****[2]**William E. Boyce,*On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems*, Quart. Appl. Math.**38**(1980/81), no. 2, 241–245. MR**580882**, https://doi.org/10.1090/S0033-569X-1980-0580882-5**[3]**R. Courant and D. Hilbert,*Methods of mathematical physics. Vol. I*, Interscience Publishers, Inc., New York, N.Y., 1953. MR**0065391****[4]**John Gregory,*An approximation theory for elliptic quadratic forms on Hilbert spaces: Application to the eigenvalue problem for compact quadratic forms*, Pacific J. Math.**37**(1971), 383–395. MR**0303311****[5]**John Gregory,*Quadratic form theory and differential equations*, Mathematics in Science and Engineering, vol. 152, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**599362****[6]**Magnus 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**0046590****[7]**Kurt Kreith,*Lower estimates for zeros of stochastic Sturm-Liouville problems*, Proc. Amer. Math. Soc.**92**(1984), no. 4, 515–518. MR**760936**, https://doi.org/10.1090/S0002-9939-1984-0760936-X

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1254841-4

Keywords:
Sturm-Liouville problem,
random eigenvalues,
continuity of eigenvalues

Article copyright:
© Copyright 1995
American Mathematical Society