Ukrainian Mathematical Journal最新文献

We study the representation of real numbers by Perron series (P-representation) given by

where r_n, p_n ∈ ℕ, p_n+1 ≥ r_n + 1, and its transcoding ((overline{P })-representation)

where g_n = p_n − r_n−1. We establish the properties of (overline{P })-representations typical of almost all numbers with respect to the Lebesgue measure (normal properties of the representations of numbers). We also examine the conditions of existence of the frequency of a digit i in the (overline{P })-representation of a number ({x=Delta }_{{g}_{1}{g}_{2}dots {g}_{2}dots }^{overline{P} }) defined by the equality

where ({N}_{i}^{overline{P} }left(x,kright)) denotes the amount of numbers n such that g_n = i and n ≤ k. In particular, we establish conditions under which the frequency ({nu }_{i}^{overline{P} }left(xright)) exists and is constant for almost all x ∈ (0; 1]. In addition, we also determine the conditions under which the digits in (overline{P })-representations are encountered finitely or infinitely many times for almost all numbers from (0; 1].

We prove and discuss some new weak-type (1,1) inequalities for the maximal operators of T means with respect to the Vilenkin system generated by monotonic coefficients. We also apply the accumulated results to prove that these T means are almost everywhere convergent. As applications, we present both some well-known and new results.

We consider a Schrödinger equation (i{partial }_{t}^{rho }uleft(x,tright)-{u}_{xx}left(x,tright)=pleft(tright)qleft(xright)+fleft(x,tright),0<tle T,0<rho <1,) with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u(x, t), a time-dependent factor p(t) of the source function is also unknown. To solve this inverse problem, we use an additional condition B[u(∙, t)] =ψ(t) with an arbitrary bounded linear functional B. The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method makes it possible to study a similar problem by taking, instead of d²/dx², an arbitrary elliptic differential operator A(x,D) with compact inverse.

Suppose that p(z) = 1 + zϕ″(z)/ϕ′(z), where ϕ(z) is a locally univalent analytic function in the unit disk D with ϕ(0) = ϕ′(1) − 1 = 0. We establish the lower and upper bounds for the best constants σ₀ and σ₁ such that ({e}^{{-sigma }_{0}/2}<left|pleft(zright)right|<{e}^{{sigma }_{0}/2}) and |p(w)/p(z)| < ({e}^{{sigma }_{1}}) for z, w ∈ D, respectively, imply the univalence of ϕ(z) in D.

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for 2π-periodic functions f that can be represented in the form of generalized Poisson integrals of functions φ from the space L_p, 1 ≤ p ≤ ∞. In these inequalities, the moduli of deviations of the interpolation Lagrange polynomials (left|fleft(xright)-{widetilde{S}}_{n-1}left(f;xright)right|) for every x ∈ ℝ are expressed via the best approximations ({E}_{n}{left(varphi right)}_{{L}_{p}}) of the functions φ by trigonometric polynomials in the L_p-metrics. We also deduce asymptotic equalities for the exact upper bounds of pointwise approximations of the generalized Poisson integrals of functions that belong to the unit balls in the spaces L_p, 1 ≤ p ≤ ∞, by interpolating trigonometric polynomials on the classes ({C}_{beta ,p}^{alpha ,r}).