On the rate of approximation of functions by the Bernstein polynomials

On partial derivatives of multivariate Bernstein polynomials

“…See [3], and some generalisations in [12]. The following similar result holds true for the derivatives.…”

“…It turned out that this observation about an approximation along with derivatives holds true for multivariate functions as well; however, the authors were unable to find an exact reference and this was the reason for writing the present paper. In particular, this property is very useful in one of the proofs of See [3], and some generalisations in [12]. The following similar result holds true for the derivatives.…”

“…For the proof by induction of the relations (12) and (13), let us note that the basis of induction (k = 0) coincides with the definition of the polynomial B n (f ; x); as for the inductive step, for the sake of clarity of the following bulky computations let us first differentiate once the function B n (f ; x). we have,…”

Approximation by modified Kantorovich–Stancu operators

“…The following inequality where is a constant independent of m , can be found in [16] for , and in [5] for .…”

Estimating functional dependences based on time series with the use of classes of regression functions of infinite capacity

“…Let us consider several examples of the estimates of the approximation of functions by polynomials [5][6][7], which can be used to estimate the functional dependence with the results of Theorem 1 taken into account.…”