The present paper is a continuation of [5, 7] where a Fredholm theory for approximation sequences is proposed and some of its properties and consequences are studied. Here this theory is specified to the class of fractal approximation methods. The main result is a formula for the so-called α-number of an approximation sequence (Aₙ) which is the analogue of the kernel dimension of a Fredholm operator.

Extensions from $H^1\left({\Omega }_P\right)$ into $H^1\left(\Omega \right)$ (where ${\Omega }_P\subset \Omega$) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary $\partial \Omega$ of $\Omega$. The corresponding extension operator is linear and bounded.

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval $[-a,a]$, $a\in (0,1)$, have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general...

We study the complexity of Banach space valued integration in the randomized setting. We are concerned with r times continuously differentiable functions on the d-dimensional unit cube Q, with values in a Banach space X, and investigate the relation of the optimal convergence rate to the geometry of X. It turns out that the nth minimal errors are bounded by $c{n}^{-r/d-1+1/p}$ if and only if X is of equal norm type p.

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.

We present a detailed proof of the density of the set $C^{\infty }\left(\overline{\Omega }\right)\cap V$ in the space of test functions $V\subset H^1\left(\Omega \right)$ that vanish on some part of the boundary $\partial \Omega$ of a bounded domain $\Omega$.