theory of elasticity, half-strip, Fadle-Papkovich functions, biorthogonal systems of functions, Lagrange expansions, analytical solutions.

We consider the Lagrange expansions on the Fadle-Papkovich functions. The functions arise from solution of the first fundamental boundary value problem of elasticity theory in a halfstrip {Π+ : x ≥ 0, |y| ≤ h}. Independently of a homogeneous boundary conditions type on the long sides of a half-strip, there are always two representations for Fadle- Papkovich functions. Both of representations are considered in this article. Their equivalence for given, physically natural classes of expended functions are shown. Functions that are The biorthogonal system functions are constructed, and then the Lagrange expansions on the Fadle-Papkovich functions are given. The Lagrange expansions are called the expansions of only one function for any one system of Fadle-Papkovich functions, unlike expansions that arise in solving of the boundary value problem. The Lagrange expansions are the analogues of expansions into series on trigonometric systems of functions, and their role in solving of boundary value problems is the same as the role of trigonometric series in Filon-Ribiere [1] expansions. The Lagrange expansions was considered earlier, e.g., in [2-15], as much as it is required to solve a boundary value problem. The aim of this article is detailed study of the Lagrange expansions. The Fadle-Papkovich functions exactly satisfy to zero boundary conditions on the longitudinal sides of the half-strip, but they are more complicated: a complex-valued, not orthogonal and do not form classical basis in the segment (end face of the half-strip), where expanding functions are given. But it is possible to construct (defined on the Riemannian surface of logarithm) the biorthogonal systems of functions, and then get explicit expressions (in the form of a simple Fourier integrals of the boundary functions) for the required expansion coefficients in the same scheme as in the classical solutions of Filon-Ribiere [16]. The essence of the approach is a new concept of functions basis on an interval, which is a generalization of the classical understanding of the basis on the segment. In terms of work [17, 18], a classical basis can be interpreted as a basis in the complex plane. While the Fadle-Papkovich functions form the basis on the Riemannian surface of logarithm. Moreover, in the particular case when the Fadle-Papkovich functions degenerate into trigonometric systems of functions, the basis on the Riemannian surface becomes to classical basis on a segment. The basis of the corresponding theory is the Borel transform in the class of quasi-entire functions of exponential type (classical basis is based on the theory of entire functions of exponential type and Paley-Wiener theorem [19]). The class of quasi-entire functions of exponential type and the Borel transform in this class were first introduce in 1935 [20]. In article [21] the properties of this transform were studied, as much as this is necessary for solving boundary value problems of elasticity theory in a half-strip

Menshova Irina Vladimirovna e-mail: menshovairina@yandex.ru, Candidate of Phys. & Math., Senior Researcher, Laboratory of Geodynamics, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow, Russia.