Vol 31, No 153 (2026)

In the paper, we obtain new exact inequalities that relate the best approximations by trigonometric polynomials of differentiable 2π-periodic square summable functions to integrals containing generalized moduli of continuity of higher orders of the r-th derivative of the functions in the metric of the space L2[0;2π]: Let N be the set of natural numbers, Z+=N∪{0}. For arbitrary m∈N; r∈Z+; t>0; the classes of 2π-periodic functions are defined: W_m^(r)(t,ω)⊂L_2^(r), for which the average value of the generalized modulus of continuity of the r-th derivative of a function is bounded from above by the majorant ω; and W_m^(r)(t)⊂L_2^(r), for which the average value of the smoothness characteristic m(f^(r); t) is bounded from above by unity. We consider the extremal problem of calculating the exact upper bounds of best polynomial approximations of the class W_m^(r)(t,ω). The exact results obtained in the paper are also applied to calculating the value of some known n-widths of the class W_m^(r)(t); when obtaining upper bounds. Lower bounds for these same widths can be obtained using the theorem on the width of a ball.

This paper studies the compactness of a linear integral operator of the form (Ku)(t) = ∫Ω k(t, s)u(s) ds with kernel (operator function) k: Ω² → L(X, Y), where Ω is a connected compact set in R^ℓ, X, Y are separable real Banach spaces such that Y does not contain c0, L(X, Y) is the space of bounded linear operators from X to Y, and the integral is understood in the Pettis sense. Criteria are established for the action and compactness of the operator K from the space C(X) of continuous functions u: Ω → X to the space C(Y). It is also proved that the compactness of the operator K: C(X) → C(Y) implies the action and compactness of K from L₁(X) to C(Y). The results generalize a well-known criterion for the compactness of the operator K in the space of continuous scalar functions. We also generalize a well-known sufficient condition for the action and compactness of an integral operator in C[a, b] in the form of a condition for the boundedness of the kernel k and its continuity everywhere except for a finite number of continuous 'discontinuity curves' of the kernel in the square [a, b]². This condition is of interest for application to integral operators with variable limits of integration, in particular, to Volterra integral operators. For two cases relevant for applications: when for some t₀ ∈ Ω k(t₀, s) = 0 holds almost everywhere on Ω (called the zero-condition in the paper), and when dim Y < ∞, the conditions necessary and sufficient for the action and compactness of K are formulated exclusively in terms of the kernel k of the operator, and are integral analogues of the conditions of uniform boundedness and equidistant continuity from the Arzelà–Ascoli theorem.

The article is devoted to the estimates of the moduli of Taylor coefficients on classes of holomorphic functions that do not vanish and are bounded in the unit disk. Let us denote the set of all holomorphic functions that are bounded from below by modulus in the unit disk by E. Let t ≥ 0 be a parameter. The sharp estimates of the moduli of all Taylor coefficients have been obtained for subclasses Et of the class E that contain mappings from E normalized at the origin. We consider the class B consisting of holomorphic functions that are bounded from above by modulus and do not vanish in the unit disk, as well as its subclasses Bt which consist of functions from the class B normalized at the origin. The classes E and Et are related to the classes B and Bt through involution. An intermediate position between the two aforementioned classes E and B is occupied by the set of holomorphic functions Aζ,r, 0 < ζ < r, that are bounded from above and from below by modulus. The classes Aζ,r can be divided into subclasses Aζ,r,t of functions from class Aζ,r normalized at the origin. We fix ζ and r. The study of the class Aζ,r can be reduced to the study of the class A1,r/ζ which is a subclass of the class E, or to the study of the class Aζ/r,1 which is a subclass of the class B. Besides, A1,r/ζ → E and Aζ/r,1 → B as r → 1. A majorizing function with real coefficients has been constructed for the classes A1,r,t, enabling us using the theory of subordination to obtain the sharp estimates of the moduli of the first two Taylor coefficients for each class A1,r,t and also for the class A1,r.