ON POINTWISE APPROXIMATION OF FUNCTIONS OF SEVERAL VARIABLES BY SINGULAR INTEGRALS

We prove a theorem on weighted pointwise convergence of \ multidimensional integral operators with radial kernels to generating function of several variables, which are in general non-integrable in $n$-dimensional Euclidean space $E_{n}$ in the sense of Lebesgue$.$ Main result holds at almost every point of $E_{n}.$

Asymptotic Expansion for Neural Network Operators of the Kantorovich Type and High Order of Approximation

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the density functions generated by suitable sigmoidal functions. Furthermore, to improve the rate of convergence, we consider finite linear combinations of the above neural network type operators, and also in the latter case, we obtain a Voronovskaja type theorem. Finally, concrete examples of sigmoidal activation functions have been deeply discussed, together with the case of rectified linear unit (ReLu) activation function, very used in connection with deep neural networks.

The best one-sided approximation of classes $W^r H^{\omega}$ with regard to point location on the interval

The estimation of the best one-sided pointwise approximation to the classes $W^r H^{\omega}$ with the remainder better than one of the previous estimation is obtained.

Rational Approximation on Exponential Meshes

Error estimates of pointwise approximation, that are not possible by polynomials, are obtained by simple rational operators based on exponential-type meshes, improving previous results. Rational curves deduced from such operators are analyzed by Discrete Fourier Transform and a CAGD modeling technique for Shepard-type curves by truncated DFT and the PIA algorithm is developed.

Pointwise approximation of modified conjugate functions by matrix operators of their Fourier series with the use of some parameters

We extend and generalize the results of Xh. Z. Krasniqi [Acta Comment. Univ.Tartu. Math. 17 (2013), 89-101] and the authors [Acta Comment. Univ. Tartu.Math. 13 (2009), 11-24], [Proc. Estonian Acad. Sci. 2018, 67, 1, 50--60] aswell the jont paper with M. Kubiak [Journal of Inequalities and Applications(2018) 2018:92]. We consider the modified conjugate function $\widetilde{f}%_{r}$ for $2\pi /\rho $--periodic function $f$ . Moreover, the measure ofapproximations depends on \textbf{\ }$\mathbf{\rho }$\textbf{ - }differencesof the entries of matrices defined the method of summability.

Pointwise approximation of 2 π/r-periodic functions by matrix operators of their Fourier series with r-differences of the entries

We extend the results of Xh. Z. Krasniqi [Acta Comment. Univ. Tartu. Math., 2013, 17, 89-101] and the authors [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] to the case of 2 π/r-periodic functions. Moreover, as a measure of approximation r-differences of the entries are used.