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Siciak’s extremal function via Bernstein and Markov constants for compact sets in ℂ N

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set E ⊂ ℂ N . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function Φ E . Moreover, we show that one of these extremal-like functions is equal to Φ E if E is a nonpluripolar set with l i m n → ∞ M ₙ ( E ) 1 / n = 1 where M ₙ ( E ) : = s u p | | | g r a d P | | | E / | | P | | E , the supremum is taken over all polynomials P of N variables of total...

Sigma order continuity and best approximation in L ϱ -spaces

Shelby J. Kilmer, Wojciech M. Kozƚowski, Grzegorz Lewicki (1991)

Commentationes Mathematicae Universitatis Carolinae

In this paper we give a characterization of σ -order continuity of modular function spaces L ϱ in terms of the existence of best approximants by elements of order closed sublattices of L ϱ . We consider separately the case of Musielak–Orlicz spaces generated by non- σ -finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.

Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions

Heiner Gonska, Radu Păltănea (2010)

Czechoslovak Mathematical Journal

We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S. N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.

Smooth and analytic solutions for analytic linear systems.

F. Acquistapace, F. Broglia, A. Tognoli (1996)

Revista Matemática de la Universidad Complutense de Madrid

We give some approximation theorems in the Whitney topology for a general class of analytic fiber bundles. This leads to a classification theorem which generalizes the classical ones.

Smooth approximation of data with applications to interpolating and smoothing

Segeth, Karel (2013)

Programs and Algorithms of Numerical Mathematics

In the paper, we are concerned with some computational aspects of smooth approximation of data. This approach to approximation employs a (possibly infinite) linear combinations of smooth functions with coefficients obtained as the solution of a variational problem, where constraints represent the conditions of interpolating or smoothing. Some 1D numerical examples are presented.

Smooth approximation spaces based on a periodic system

Segeth, Karel (2015)

Programs and Algorithms of Numerical Mathematics

A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients obtained as the unique solution of a minimization problem. While the minimization guarantees the smoothness of the approximant and its derivatives, the constraints represent the interpolating or smoothing conditions at nodes. In the contribution, a special attention is paid to the periodic basis system exp ( - k x ) ....

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