Vol 104, No 3-4 (2018)

The spaces dual to spaces of holomorphic functions of given growth on Carathéodory domains are described by using the Cauchy transform of functionals. A pseudoanalytic extension of such transforms to the whole plane is constructed, which makes it possible to remove convexity constrains and consider spaces determined by weights of general form, rather than only by those whose dependence on the distance from a point of the domain to its boundary is one-dimensional.

The Dirichlet problem for a class of properly elliptic sixth-order equations in the unit disk is considered. Formulas for determining the defect numbers of this problem are obtained. Linearly independent solutions of the homogeneous problem and conditions for the solvability of the inhomogeneous problem are given explicitly.

In this paper, we investigate spectral expansion for the asymptotically spectral differential operators generated in L₂^m (−∞,∞) by ordinary differential expressions of arbitrary order with periodic matrix-valued coefficients.

A lattice graph with 2–3 reachability constraints is considered. The graph’s vertices are the points with integer nonnegative coordinates in the plane. Each vertex has two outgoing edges, one entering its immediate right neighbor and the other entering its immediate upper neighbor. The admissible paths for 2–3 reachability are those in which the numbers of edges in all but the last inclusion-maximal straight-line segments are divisible by 2 for horizontal segments and by 3 for vertical segments. A formula for the number of 2–3 paths from a vertex to a vertex is obtained. A random walk process on the 2–3 paths in the lattice graph is considered. It is proved that this process can locally be reduced to a Markov process on subgraphs determined by the type of the initial vertex. Formulas for the probabilities of vertex-to-vertex transitions along 2–3 paths are obtained.

An integral representation and embedding theorems for functions in multianisotropic Sobolev spaces are proved. Unlike in previous works, the general case where the characteristic Newton polyhedron in ℝⁿ has an arbitrary number of vertices is considered.

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given.

The Riesz potentials L^af, 0 < α < ∞, are considered in the framework of a grand Lebesgue space L_a^p),θ, 1 < p < ∞, θ > 0, on Rn with grandizers a ∈ L¹(ℝⁿ), which are understood in the case α ≥ n/p in terms of distributions on test functions in the Lizorkin space. The images under I^α of functions in a subspace of the grand space which satisfy the so-called vanishing condition is studied. Under certain assumptions on the grandizer, this image is described in terms of the convergence of truncated hypersingular integrals of order α in this subspace.

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted L^p spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces L^p, p≠2.

A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.

Given a smooth embedding of manifolds and a Fourier integral operator on the ambient manifold, the trace of this operator on the submanifold (i.e., its composition with the boundary and coboundary operators, which is an operator on the submanifold) is considered. Conditions under which such a trace is also a Fourier integral operator are determined, and its amplitude in canonical local coordinates is calculated. The results are applied to quantized canonical transformations.

The aim of this paper is to obtain some common fixed point results on an S-metric space. For this purpose, we prove new generalized common fixed point theorems using the notions of weak commutation, compatibility, and orbital continuity for two and four continuous self-mappings. Our main results generalize the known common fixed point theorems on metric spaces.