Metric Fourier approximation of set-valued functions of bounded variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.Comment: 26 pages, 1 figur

ON THE BEST APPROXIMATION OF THE INFINITESIMAL GENERATOR OF A CONTRACTION SEMIGROUP IN A HILBERT SPACE

Let $A$ be the infinitesimal generator  of a strongly continuous contraction semigroup in a  Hilbert space $H$.   We give an upper estimate for the best approximation of the operator $A$ by bounded linear operators with a prescribed norm  in the space $H$ on the class $Q_2 = \{x\in \mathcal{D}(A^2) : \|A^2 x\| \leq 1\}$,  where $\mathcal D(A^2)$ denotes the domain of $A^2$

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

AbstractWe prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional