Free interpolation for the Zygmund class

We introduce interpolation sets for the Zygmund class 𝒵 {\mathcal{Z}} in the unit disc of the complex plane. This space lies between the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the class of order α = 1 {\alpha=1} , whose interpolation sets are given in a different way. We prove that the interpolation sets for 𝒵 {\mathcal{Z}} are interpolation sets for the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the latter are interpolation sets for a space slightly larger than 𝒵 {\mathcal{Z}} .

Approximation Theorem for New Modification of q -Bernstein Operators on (0,1)

In this work, we extend the works of F. Usta and construct new modified q -Bernstein operators using the second central moment of the q -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s K -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

Absolute convergence of Fourier integrals and Lipschitz classes

The necessary and sufficient conditions, in terms of Fourier transforms $\hat{f}$ of functions $f \in L^1(\mathbb{R})$, are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$ and $h^{\omega}(\mathbb{R})$.

INEQUALITIES REFINING RELATIONSHIPS BETWEEN MIXED MODULES OF SMOOTHNESS IN METRICS OF $L_p$ AND $L_{\infty}$

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.

Approximation by sampling-type nonlinear discrete operators in φ-variation

In the present paper, our purpose is to obtain a nonlinear approximation by using convergence in ?-variation. Angeloni and Vinti prove some approximation results considering linear sampling-type discrete operators. These types of operators have close relationships with generalized sampling series. By improving Angeloni and Vinti?s one, we aim to get a nonlinear approximation which is also generalized by means of summability process. We also evaluate the rate of approximation under appropriate Lipschitz classes of ?-absolutely continuous functions. Finally, we give some examples of kernels, which fulfill our kernel assumptions.