Approximation theorems for the iterated Boolean sums of Bernstein operators

On the Convergence and Iterates of q-Bernstein Polynomials

“…Micchelli [11] introduced certain linear combinations of the Bernstein polynomials. These linear combinations, which may be regarded as Boolean sums, are discussed in [5,17,18]. They proved that the iterated Boolean sum of (1.1) converges to the interpolating polynomial of f of degree n at equally spaced points on ½0; 1: Wenz [18] also obtained similar results for the Bernstein-Schoenberg and Sablonni!…”

On the Limits of (Linear Combinations of) Iterates of Linear Operators

“…The conditions (9) are slightly different from the usual ones for spline knots where t i <t i+k , i= &k+1, ..., n&1, is required. However, our condition is not really stronger, since for t i =t i+k&1 the intervals [t 0 , t i ] with knot sequence t &k+1 , ..., t i+k&1 and [t i+k&1 , t n ] with knot sequence t i , ..., t n+k&1 can be considered separately.…”

“…After 1973, iterated Boolean sums of univariate Bernstein operators were studied by other authors (see [9] for an overview). In 1995 Sevy [17] found that the limit of iterated Boolean sums of Bernstein polynomials is the interpolation polynomial with respect to the nodes (iÂn, f (iÂn)), i=0, ..., n, and that the limit of iterated Boolean sums of Bernstein Durrmeyer operators is the least squares polynomial with respect to the…”

Estimates for Iterates of Positive Linear Operators Preserving Linear Functions

“…Note that (U n − Id) r = ⊕ r U n − Id, where ⊕ r stands for iterated Boolean sum. In case of Bernstein operators, such quantities where considered, for example, in the very interesting paper by Gonska & Zhou [26], where many historical references can also be found.…”