derived chains

Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients and Splitting Boundary Conditions

Authors: Rykhlov V. S.

In the space of square summable functions on the main segment [0,1], the class of polynomial pencils of ordinary differential operators of the n-th order is considered. The coefficients of the differential expression are assumed to be constants. The boundary conditions are assumed to be splitting and two-point at the ends 0 and 1 (l of boundary conditions is taken only at the point 0, and the remaining n − l is taken at the point 1). The differential expression and the boundary forms are assumed to be homogeneous, that is, they contain only main parts.

Expansion in Root Functions of Strongly Irregular Pencil of Differential Operators of the Second Order with Multiple Characteristics

Authors: Rykhlov V. S.

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.