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Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions

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Abstract

A bundle of differential operators

$$\mathcal{L}(\lambda ),\lambda \in \mathbb{C}:\mathcal{L}(\lambda )y(x) = y^{(4)} (x) - 2\lambda ^2 y^{(2)} (x) + \lambda ^4 y(x),|x| \leqslant 1,y( \pm 1) = y\prime ( \pm 1) = 0,$$

is considered. In various function spaces, we establish the facts about the expansions of a pair of functionsf(x) andg(x) in the system {y k (x),iλ k y k (x)} ∞ k=1 , wherey k (x),k=1,2,..., are the eigenfunctions of the bundle\(\mathcal{L}\)(λ) corresponding to the eigenvalues λ k , with Im λ k >0.

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Authors and Affiliations

  1. Institute of Hydromechanics, Ukrainian Academy of Sciences, Kiev

    A. M. Gomilko

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  1. A. M. Gomilko
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1601–1612, December, 1993.

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Gomilko, A.M. Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions. Ukr Math J 45, 1801–1814 (1993). https://doi.org/10.1007/BF01061350

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  • DOI: https://doi.org/10.1007/BF01061350

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