Geodesic
Expansion in root functions of strongly irregular pencil of differential operators of the second order with multiple characteristics
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 165-174.

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We consider the quadratic strongly irregular pencil of ordinary second order differential operators with constant coefficients and with a multiple root of the characteristic equation. The amounts of double expansions in biorthogonal Fourier series in the derived chains of such pencils and a necessary and sufficient condition for convergence of these expansions to the expanded vector-valued function are found. This necessary and sufficient condition is a differential equation relating the components of the expanded vector function. At the same time some conditions of smoothness on the components of the expanded vector-valued function and requirements of the vanishing of its components and some of their derivatives at the ends of the main segment are imposed. 
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V. S. Rykhlov. Expansion in root functions of strongly irregular pencil of differential operators of the second order with multiple characteristics. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 165-174. http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a6/

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