“…See [3], and some generalisations in [12]. The following similar result holds true for the derivatives.…”
Section: The Case D =mentioning
confidence: 56%
“…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.…”
Section: Introductionmentioning
confidence: 56%
“…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,…”
It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.
“…See [3], and some generalisations in [12]. The following similar result holds true for the derivatives.…”
Section: The Case D =mentioning
confidence: 56%
“…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.…”
Section: Introductionmentioning
confidence: 56%
“…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,…”
It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.
In the present paper, we study a new kind of Kantorovich–Stancu type operators. For this modified form, we discuss a uniform convergence estimate. Some Voronovskaja-type theorems are given.
“…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.…”
Section: Uniform Convergence In the Case Of Regression Functions Apprmentioning
The problem of estimating the functional dependence of time series on the time index is considered in the case of short data samples. It is proved that empirical risk functional uniformly converges to the theoretical one in the case where regression functions can be approximated by finite-degree polynomials. An example of estimating the functional dependence for a class of trigonometric functions is presented.
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