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....

We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces $B_{2,\infty }^s\left(ℝ\right)$, $s*\left(f\right)=sups>0:f\in B_{2,\infty }^s\left(ℝ\right)$. The existing results on estimation of smoothness [K. Dziedziul, M. Kucharska and B. Wolnik, J. Nonparametric Statist. 23 (2011)] employ the Haar basis and are limited to the case 0 < s*(f) < 1/2. Using p-regular (p ≥ 1) spline wavelets with exponential decay we extend them to density functions with 0 < s*(f) < p+1/2. Applying the Franklin-Strömberg wavelet p = 1, we prove that the presented estimator...

Let $Y\subset l_{\infty }^{\left(n\right)}$ be a hyperplane and let $A\in ℒ\left(Y\right)$ be given. Denote $$\begin{array}{ccc}\hfill 𝒜=& \{L\in ℒ(l_{\infty }^{\left(n\right)},Y):L\mid Y=A\}\text{and}\hfill & \hfill {\lambda }_A=inf\{\parallel L\parallel :L\in 𝒜\}.\end{array}$$ In this paper the problem of calculating of the constant ${\lambda }_A$ is studied. We present a complete characterization of those $A\in ℒ\left(Y\right)$ for which ${\lambda }_A=\parallel A\parallel$. Next we consider the case ${\lambda }_A>\parallel A\parallel$. Finally some computer examples will be presented.