Vol 88, No 1 (2024)

Order estimates for the Kolmogorov widths of an intersection of Sobolev classeson a $d$-dimensional John domain and on the 1-dimensional torus are obtained.In particular, one Galeev's result is generalized.

In this paper, we establish the existence of a weak solution to theinitial boundary value problem for the motion equations ofviscoelastic incompressible fluid with constitutive lawcontaining high-order fractional derivatives and with memoryalong the trajectories of the velocity field. The proof is by approximation of the original problem by a sequence ofregularized problems followed by a passage to the limit based onappropriate a priori estimates. Methods of the theory of fractionalderivatives calculus and the theory of regular Lagrangian flows(generalization of the classical solution of ODE systems) are used.

The paper proposes an approach for construction of families of optimal methodsfor the recovery of linear operators from inaccurately given information.The proposed method of construction is then applied to recover derivatives from inaccurately specified otherderivatives in the multidimensional case and to recover solutions of the heat equationfrom inaccurately specified temperature distributions at some time instants.

Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complexplane $\mathbb C$. The following problem dates back toF. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin,J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$,for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, thereexists an entire function $f\neq 0$ of exponential typethat vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis?The classical Malliavin–Rubel theorem of the early 1960s completelysolves this problem for “positive” $\mathrm Z$ and $\mathrm W$ (which lie onlyon the positive semiaxis). Several generalizations of this criterion wereestablished by the author of the present paper in the late 1980s for “complex”$\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separatedby angles from the imaginary axis, with some advances in the 2020s.In this paper, we solve more involved problems in a more general subharmonicframework for distributions of masses on $\mathbb C$. All the previouslymentioned results can be obtained from the main results of this paperin a much stronger form (even for the initial formulation for distributionsof points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type).Some results of the present paper are closely relatedto the famous Beurling–Malliavin theorems on the radius of completeness and a multiplier.