bending strip; ribs; the Fadle-Papkovich functions; biorthogonal functions.

We study the basic properties (in particular, the Lagrangian decomposition) Feature- Fadl Papkovich arising when solving boundary value problem for the half-plane containing a periodic set of ribs, which receive only the flexural deformation and the tensile compression. Lagrangian decomposition, in contrast to the expansions, which appear in the solution of boundary value problems of elasticity theory in the half, when the unknown expansion coefficients are determined from the expansions of two given at the end of the half-strip functions in series of two systems of functions Fadl-Papkovich, called the expansion of only one function for whatever or a system of functions. In this sense, the Lagrange series play the same role as trigonometric series play in decisions Filon-Ribiere [1]. Fadl-type functions Papkovich depends on the boundary conditions on the long sides of the half-strip. Examples Lagrange expansions, Function-Fadl Papkovich arising in the solution of a boundary value problem can be found in the papers [2-7]. There are some common methods and approaches in the study of basic properties of systems functions Fadl- Papkovich. However, in each case having its own specific characteristics that are unique to a given boundary value problem and its corresponding functions Fadl-Papkovich. Lagrange features expansions arising when solving the boundary value problem, the subject of this article. Two types of decomposition. Decomposition (depending on a certain parameter, the whole of this parameter) generators [8] functions, ie functions generating any system of functions Fadl-Papkovich when the parameter runs through the set of eigenvalues ??of the boundary value problem and decomposition using finite parts of biorthogonal functions. In the first case, the whole generating function continues as a whole is a segment - end of the half-strip in the whole infinite straight line, biorthogonal functions not explicitly written out, and the desired expansion coefficients of the Lagrange series are determined directly from the equation for determining the bi-orthogonal functions. In the second case biorthogonal functions are written out explicitly. They are defined in the segment - the half-strip end, have a simple form, but most importantly, they can be used to build not only the expansion of analytic functions, but actually, all functions for which there is a Fourier integral. In order to construct a Lagrangian decomposition, defined on the interval - end of the half-strip - function, you must first this function in any way out of this segment to continue. The way this is done will depend on the continuation of the expansion coefficients in the Lagrange series. Thus, the Lagrange decomposition is not unique. Nonuniqueness Lagrange expansions caused by complex-systems functions Fadl-Papkovich - one of the most important properties of these systems functions.

Semenova Irina Alexandrovna, Postgraduate student, Departament of Mathematical Analysis, I.Yakovlev Chuvash State Pedagogical University, Cheboksary, Russia