Vol 104, No 5-6 (2018)

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ⊂ Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.

New estimates of the sums of sine series with monotone coefficients of special classes in terms of the Salem majorant are obtained. The asymptotic sharpness of the obtained estimates for sequences of coefficients from the classes under consideration is proved.

We show that the global dimension of a broad class of radical Banach algebras of power series is at least 3 and obtain applications to cohomology groups.

A singularly perturbed boundary-value problem for a nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems with discontinuous advective and reactive terms is considered. The existence of contrast structures in problems of this type is proved, and an asymptotic approximation of the solution with an internal transition layer of arbitrary order of accuracy is obtained.

Connections associated with the Grassmann-like manifold of centered planes in the multidimensional projective space are studied. A geometric interpretation of these connections in terms of maps and translations of equipping planes is given. An intrinsic analog of Norden’s strong normalization of the manifold under consideration is constructed.

We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a spectral problem for the harmonic oscillator Schrödinger equation. The second approach is based on a reduction of the finite-difference equation for the Hermite polynomials to a pseudodifferential equation. Associated with each of the approaches are Lagrangian manifolds that give the asymptotics of Hermite polynomials via the Maslov canonical operator.

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

Let 1 ≤ 2l ≤ m < d. A vector x ∈ ℤ^d is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Z^d from the vector y = Ax ∈ R^m is considered. In the case m = 2l, we obtain necessary conditions and sufficient conditions on the numbers m, d, and k ensuring the existence of an integer matrix A all of whose elements do not exceed k in absolute value which makes it possible to reconstruct l-sparse vectors in ℤ^d. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.

Two approaches to systems of linear partial differential equations are considered: the traditional approach based on the generalized Fourier transform and the semigroup approach, under which the system is considered as a particular case of an operator-differential equation. For these systems, the semigroup classification and the Gelfand–Shilov classification are compared.

An example of a Markov function f = const + \(\hat \sigma \) such that the three functions f, f², and f³ constitute a Nikishin systemis given. It is conjectured that there exists aMarkov function f such that, for each n ∈ N, the system of f, f²,..., fⁿ is a Nikishin system.

The norm on the sum of Lorentz spaces endowed with norms equal to the products of the classical norm by some numbers is exactly calculated. The obtained result makes it possible to prove an extrapolation theorem for collections of Lorentz, Lebesgue, and Marcinkiewicz spaces with a sharp constant.

In this paper, we develop a new technique for the asymmetric approximation of discrete functions arising in seasonal customer demand extrapolation. We adapt the technique for two different settings, the so-called pull and push models. Our main goal here is to find effectively extrapolations minimizing the loss. For bothmodels, we discuss several features related to sampling, approximation, and extrapolation.

Some properties of the modular lambda function that are similar to those of the modular invariant functions are proved. An algorithm for constructing the minimal polynomial for the values of the lambda function at the points of imaginary quadratic fields is presented; the numbers conjugate to these values are given.

My paper contains an error. Namely, inequality (9) does not hold. For example, for k = 2,

I do not know whether the error in the argument can be corrected and whether the main result of the paper is valid.

I wish to express gratitude to O. L. Vinogradov for pointing out the error.