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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.
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 ....
Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family involves the concept of cumulant transformation and a standard homogenization procedure.
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