The properties of shift invariant operators are proved: It is shown that Q has polynomial order r iff r is the rate of convergence of . A weak saturation theorem is given. If f is replaced by in the weak saturation formula the asymptotics of the expression is calculated. Moreover, bootstrap approximation is introduced.
The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set . 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 . Moreover, we show that one of these extremal-like functions is equal to if E is a nonpluripolar set with where , the supremum is taken over all polynomials P of N variables of total...
In this paper we give a characterization of -order continuity of modular function spaces in terms of the existence of best approximants by elements of order closed sublattices of . We consider separately the case of Musielak–Orlicz spaces generated by non--finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.
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.
We obtain simultaneous approximation equivalence theorem for Szász-Mirakian quasi-interpolants.
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.
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.