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Sampling theorems associated with fourth- and higher-order self-adjoint eigenvalue problems. (English) Zbl 0808.34091

The paper is concerned with sampling series associated with regular selfadjoint boundary eigenvalue problems for ordinary differential equations of order \(n = 2k\) on a finite interval. The main theorem is a generalization of a result for the second-order Sturm-Liouville case representing the corresponding sampling series as a Lagrange interpolation series. The proof contains an explicit construction of the related kernel using ideas of the general theory of boundary eigenvalue problems for \(n\)-th order differential equations. The result is applied to one regular and one singular fourth-order eigenvalue problem. Finally, the question of reduction of \(n\)-th order problems to lower-order problems is discussed. In particular, the theorem obtained for the regular fourth-order problem is deduced from Kramer’s generalized sampling theorem for second-order eigenvalue problems.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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