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 OF FUNCTIONS BY GAUSS-WEIERSTRASS INTEGRALS

When considering various schemes and algorithms for game problems of dynamics, researchers often have to deal with solutions of partial differential equations. A special place among the latter is occupied by the so-called equations of elliptic type (according to the corresponding classification), with the help of which natural and social processes can be described most fully and qualitatively. Moreover, the mathematical apparatus of partial differential equations of elliptic type makes it possible to get into the environment of deterministic phenomena and thus makes it possible to foresee their future. This fact undoubtedly increases the significance of the above type of equations among others in the sense of their application to mathematical modeling. At the same time, one of the most important concepts in applied mathematics is the concept of the modulus of continuity. The term "modulus of continuity" and its definition were introduced by Henri Lebesgue at the beginning of the last century in order to study various properties of continuous functions. Using the concept of the modulus of continuity and its properties, it is possible to investigate the belonging of the object under study to a certain class of functions: Hölder, Lipschitz, Zygmund, etc. This undoubtedly makes it possible to approximate functions of various kinds of operators most effectively. In this paper, using the example of the Gauss-Weierstrass integral as a solution to the corresponding differential equation of elliptic type, we study its rate of convergence in terms of the modulus of continuity of the second order to the function by which it was actually constructed. Namely, the boundary properties of the Gauss-Weierstrass integral were studied as a linear positive operator that realizes its best approximation on functions from the Zygmund class. The results obtained in this article can further be used to solve many problems in applied mathematics.

Approximation of function using generalized Zygmund class

In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and $g^{\prime }$ g ′ , where $g^{\prime }$ g ′ is a derived function of a 2π-periodic function g, in the generalized Zygmund class $X_{z}^{(\eta )}$ X z ( η ) , $z\geq 1$ z ≥ 1 , using matrix-Cesàro $(TC^{\delta })$ ( T C δ ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.

On the equivalence between weak BMO and the space of derivatives of the Zygmund class

In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy space H 1 {H}^{1} strictly contains the special atom space.

On Approximation of Signals in the Generalized Zygmund Class Using (E,r)(N,qn) Mean of Conjugate Derived Fourier Series

In the present article, we have established a result on degree of approximation of function in the generalized Zygmund class Zl(m),(l ≥ 1) by (E,r)(N,qn)- mean of conjugate derived Fourier series.

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form\mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dxwith non-autonomous integrand {F(x,\xi)} which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map {x\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class.

Approximation of Signals in the Weighted Zygmund Class Via Euler-hausdorff Product Summability Mean of Fourier Series

Approximation of functions of Lipschitz and zygmund classes have been considered by various researchers under different summability means. In the proposed paper, we have studied an estimation of the order of convergence of Fourier series in the weighted Zygmund class <em>W(Z_r^(ω))</em> by using Euler-Hausdorff product summability mean and accordingly established some (presumably new) results. Moreover, the results obtained here are the generalization of several known results.