Neural network interpolation operators optimized by Lagrange polynomial.

In this paper, we introduce a new type of interpolation operators by using Lagrange polynomials of degree r, which can be regarded as feedforward neural networks with four layers. The approximation rate of the new operators can be estimated by the (r+1)-th modulus of smoothness of the objective functions. By adding some smooth assumptions on the activation function, we establish two important inequalities of the derivatives of the operators. With these two inequalities, by using the K-functional and Berens-Lorentz lemma in approximation theory, we establish the converse theorem of approximation. We also give the Voronovskaja-type asymptotic estimation of the operators for smooth functions. Furthermore, we extend our operators to the multivariate case, and investigate their approximation properties for multivariate functions. Finally, some numerical examples are given to demonstrate the validity of the theoretical results obtained and the superiority of the operators.

Pointwise approximation by a Durrmeyer variant of Bernstein-Stancu operators.

In the present paper, we introduce a kind of Durrmeyer variant of Bernstein-Stancu operators, and we obtain the direct and converse results of approximation by the operators.