Abstract
The paper deals with the Sturm-Liouville operator
generated in the space L 2 = L 2[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev spaceW p1 [0, 1] for some integer p ≥ 0 and satisfy the conditions q (k)(0) = q (k)(1) = 0 for 0 ≤ k ≤ s − 1, where s ≤ p. Let the functions Q and S be defined by the equalities
and let q n , Q n , and S n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system \( \{ e^{2\pi inx} \} _{ - \infty }^\infty \). Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2. Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition
holds.
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Original Russian Text © A. A. Shkalikov, O. A. Veliev, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 5, pp. 671–686.
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Shkalikov, A.A., Veliev, O.A. On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems. Math Notes 85, 647–660 (2009). https://doi.org/10.1134/S0001434609050058
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DOI: https://doi.org/10.1134/S0001434609050058