For averages over parallelepipeds of unitary actions of the groups R d \mathbb {R}^d and Z d {\mathbb {Z}^d} , a criterion for a power rate of convergence in norm with all possible exponents is obtained. The proof is based on the asymptotics of the integrals of product of cardinal sines.
Let S-k, 1 < k < m, be pairwise disjoint segments, S-k =[a(k), b(k)], 1 < pk < infinity. Suppose that functions f(k) are defined on Sk, fk belongs to C(S-k) and f(k) are belongs to L-k(P) (S-k). It is shown that for n = 1, 2, ... there are polynomials Pn, deg Pn < n, that approximate all functions f(k) in the metric of L-k(p) with weights tending to infinity near the points a(k), b(k).
. The Polyak and Brandenbursky invariants are applied to estimate the crossing number of (closed) braids and extend the previously known minimality criteria for diagrams of positive and alternating braids to homogeneous ones. In particular, it is proved that a diagram of a homogeneous braid is minimal if and only if this diagram is homogeneous. These results lay the groundwork for a potential solution to the recognition problem for homogeneous knots and links. The approach developed here is conceptually similar to recognizing alternating links on the basis of the Tait conjectures.
. A well-posed initial-boundary value problem is stated for the wave equation with variable propagation speed in a half-plane on a semi-infinite time interval. Certain conditions on the boundary data are indicated that ensure the existence and uniqueness of the solution as well as its stability under small changes of the boundary data in a certain function class. The problem statement is motivated by the need to justify the Poincare wavelet-based integral representation of the wave equation solution in terms of localized solutions, in particular, quasiphotons.
. The article provides a systematic presentation of an operator pattern that forms the basis of a certain approach to inverse problems in mathematical physics - the boundary control method. The pattern is based on the triangular factorization of operators. Not only solves it inverse problems but also provides functional models for an important class of symmetric semibounded operators. These models are constructed by using an evolutionary dynamical system determined by the operator.
