Table of contents

Invariant lattices in the irreducible module associated with a complex permutation representation of the group , are studied. The submodule structure of a permutation module of this group over a finite field is described.

A method of geometric models is proposed and applied to the study of the spectral properties of the classical transformations . It is proved that the class of ergodic transformations under consideration with absolutely continuous and mixing components contains no transformation with a non-simple spectrum. A criterion for the ergodicity of the transformations is obtained in terms of the geometric models. The multiplicity function of the spectrum of is determined for any when is the golden section.

The Fourier series of a function of real variables is said to be -convergent at a point for if there exists the limit

over all indices such that for all and . An example of a continuous function of variables with modulus of continuity

is constructed such that the Fourier series of  with respect to the trigonometric system -diverges everywhere for an arbitrary fixed .

For a subspace of a Banach space the quasiorthogonal set is the set of all such that 0 is one of the best approximation elements of in . The properties of the sets are studied; several criteria in terms of these sets for to be a Hilbert space are established; in particular, generalizations of the well-known theorems of Rudin-Smith-Singer and Kakutani are proved.

A -ring is said to be -periodic if its Adams operators satisfy the relation for each . The quotient by the radical of the free periodic -ring generated by one element is described. Using this description, the order of a finite group is shown to divide the group's exponent to the power equal to the dimension of an arbitrary faithful complex representation.

A new approach to the study of asymptotic distributions of eigenvalues and singular values of Toeplitz matrices is presented; this approach involves a matrix criterion for equidistribution. Theorems of Szego type for generating functions in are obtained.

It is proved that in any finitely generated algebra of finite signature over an arbitrary field or commutative associative Noetherian ring satisfying the Capelli identities of some order there exists a largest nilpotent ideal.

The distribution of the spectrum of the operator in the complex plane is studied. The operator is the closure in of the operator originally defined on smooth functions satisfying the condition , where , , , is a model operator in a Hilbert space and . Conditions (criteria) in terms of the parameter ensuring that the eigenfunctions of the operator make up a complete system, a minimal system, or a (Riesz) basis in the Hilbert space are obtained.

The q-analogues of modified Bessel functions and Macdonald functions were defined in the previous paper of the authors as general solutions of certain second-order difference equations. Several representations of these functions based on the Jackson integral are presented.

The study of the widths of the unit ball in the Hardy space in weighted spaces is carried out. Sharp lower estimates of these widths in terms of the capacity of the support of the measure are obtained. The precise values of the widths are calculated for Blaschke lemniscates. For the measures , where is plane Lebesgue measure and is a positive continuous weight, an asymptotic formula is found.