Table of contents

Strong asymptotic formulae for the Hermite-Padé polynomials for systems of Markov-type functions in the Nikishin class are obtained. The proof is based on the properties of certain rational functions on Riemann surfaces associated with the supports of the measures generating the Nikishin system.

The well-known conjecture of Mishchenko-Fomenko on the integrability of finite-dimensional Lie algebras is proved in the class of soluble Lie algebras.

A study is made of functions on the lattice generated by the integer powers of , . A -analogue of the Fourier transformation is constructed based on the Jackson integral in the space of generalized functions on the lattice.

For an exponential series with positive exponents making up a sequence of positive step Mandelbrojt's estimates of the length of a strip in which this series can be continued are improved. On the way, an estimate of the Leont'ev condensation index is obtained which is best possible in the class of sequences of fixed step and fixed upper density.

Two-gap solutions of the Boussinesq equation are considered. It is shown that for almost every Riemann surface of genus covering the elliptic surface it is possible to construct an elliptic (in ) two-gap solution of the Boussinesq equation. The existence of third- and fourth-order differential operators with elliptic "two-gap" potentials having an arbitrary number of poles is also established. An example is given.