Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Relations between the orthogonal matrix polynomials on [a, b], Dyukarev-Stieltjes parameters, and Schur complements

We obtain explicit interrelations between new Dyukarev-Stieltjes matrix parameters and orthogonal matrix polynomials on a finite interval [a, b], as well as the Schur complements of the block Hankel matrices constructed through the moments of the truncated Hausdorff matrix moment (THMM) problem in the nondegenerate case. Extremal solutions of the THMM problem are described with the help of matrix continued fractions.

The degree of approximation of functions, and their conjugates, belonging to several general Lipschitz classes, by Hausdorff matrix means of their Fourier series and conjugate series of their Fourier series

In this paper Hausdorff matrix approximations are obtained for a function and its conjugate belonging to any one of several generalized Lipschitz classes.

Bounded Domains of Generalized Riesz Methods with the Hahn Property

In 2002 Bennett et al. started the investigation to which extent sequence spaces are determined by the sequences of 0s and 1s that they contain. In this relation they defined three types of Hahn properties for sequence spaces: the Hahn property, separable Hahn property, and matrix Hahn property. In general all these three properties are pairwise distinct. If a sequence spaceEis solid and(0,1ℕ∩E)β=Eβ=ℓ1then the two last properties coincide. We will show that even on these additional assumptions the separable Hahn property and the Hahn property still do not coincide. However if we assumeEto be the bounded summability domain of a regular Riesz matrixRpor a regular nonnegative Hausdorff matrixHp, then this assumption alone guarantees thatEhas the Hahn property. For any (infinite) matrixAthe Hahn property of its bounded summability domain is related to the strongly nonatomic property of the densitydAdefined byA. We will find a simple necessary and sufficient condition for the densitydAdefined by the generalized Riesz matrixRp,mto be strongly nonatomic. This condition appears also to be sufficient for the bounded summability domain ofRp,mto have the Hahn property.

A Bauer-Hausdorff Matrix Inequality

We present a biorthogonal process for two subspaces ofℂn. Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality.