moduli of smoothness
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Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.

The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube [0,1]q, q>1. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.

"This note focuses on a sequence of linear positive operators of integral type in the sense of Kantorovich. The construction is based on a class of discrete operators representing a new variant of Jain operators. By our statements, we prove that the integral family turns out to be useful in approximating continuous signals de ned on unbounded intervals. The main tools in obtaining these results are moduli of smoothness of rst and second order, K-functional and Bohman- Korovkin criterion."

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

The aim of the paper is twofold: we introduce new positive linear operators acting on continuous functions defined on a simplex and then estimate differences involving them and/or other known operators. The estimates are given in terms of moduli of smoothness and K -functionals. Several applications and examples illustrate the general results.

AbstractThe aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences $u_{m} $ u m and $v_{m}$ v m of functions. We prove that the new operators provide better weighted uniform approximation over $[0,\infty )$ [ 0 , ∞ ) . In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ. Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.

The problem of estimating the moduli of smoothness of functions from Lq in terms of their moduli of smoothness from Lp is well known. The first stage in the estimation of moduli of smoothness was the study of the properties of functions from the Lipschitz classes and obtaining the corresponding embeddings in the works of Titchmarsh, Hardy, Littlewood, Nikol’skii. The classical Hardy-Littlewood embedding for Lipschitz spaces can be obtained as a consequence of the Ulyanov’s inequality for the moduli of continuity of a function of one variable. In the works of Ulyanov, the modulus of smoothness of natural order was considered. The introduction of fractional moduli of smoothness made it possible in the works of Potapov, Simonov, Tikhonov to strengthen the Ulyanov’s inequality. Later, the same authors were able to generalize Ulyanov’s inequality to functions of two variables, obtaining estimates for mixed moduli of smoothness. The sharpness of these inequalities was proved in the case when 1 < 𝑝 < 𝑞 < ∞ or 1 = 𝑝 < 𝑞 = ∞. In this article, we study mixed moduli of smoothness of fractional orders of a function of two variables. Inequalities are obtained that refine the previously known estimates of the Ulyanov type inequalities between mixed moduli of smoothness in the metrics Lp and Lq for values 1 < 𝑝 < 𝑞 = ∞. The accuracy of the obtained estimates is investigated. The relationship between these and previously known estimates has been studied.