### A modification of the Whittaker-Kotelnikov-Shannon sampling series.

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We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.We establish direct and converse theorems for generalized parameter dependent Bernstein-type operators. The direct estimate is given using a K-functional and the inverse result is a strong converse inequality of type A, in the terminology of [2].

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.The paper presents a method of relating two K-functionals by means of a continuous linear transform of the function. In particular, a characterization of various weighted K-functionals by unweighted fixed-step moduli of smoothness is derived. This is applied in estimating the rate of convergence of several approximation processes.Partially supported by grant No. 103/2007 of the National Science Fund of the Sofia University....

Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for ${L}_{p}\left(T\right)$ and ${L}_{p}[-1,1]$ for 0 < p < ∞ using the moduli of smoothness ${\omega}^{r}{(f,t)}_{p}$ and ${\omega}_{\phi}^{r}{(f,t)}_{p}$ respectively.

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of appropriately defined Kfunctional. The differential operator in the K-functional is defined by means of strong limit and described explicitly in terms of its Fourier coefficients. The description is simple and independent of the homogeneous Banach space. Saturation of such...