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Mathematical Foundations of Computing

November 2021 , Volume 4 , Issue 4

Special issue on approximation by linear and nonlinear operators with applications. Part I

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Preface: Special issue on approximation by linear and nonlinear operators with applications. Part I
Danilo Costarelli
2021, 4(4): i-ii doi: 10.3934/mfc.2021028 +[Abstract](5177) +[HTML](63) +[PDF](78.89KB)
Abstract:
Iyengar-Hilfer fractional inequalities
George A. Anastassiou
2021, 4(4): 221-252 doi: 10.3934/mfc.2021004 +[Abstract](574) +[HTML](293) +[PDF](465.71KB)
Abstract:

Here we present Iyengar type integral inequalities. At the univariate level they involve \begin{document}$ \psi $\end{document}-Hilfer left and right fractional derivatives. At the multivariate level they involve Hilfer left and right fractional derivatives, and they deal with radial and non-radial functions on the ball and spherical shell. All estimates are with respect to norms \begin{document}$ \left \Vert \cdot \right \Vert _{p} $\end{document}, \begin{document}$ 1\leq p\leq \infty $\end{document}. At the end we provide an application.

Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters
Giorgio Gnecco and Andrea Bacigalupo
2021, 4(4): 253-269 doi: 10.3934/mfc.2021014 +[Abstract](338) +[HTML](178) +[PDF](3462.42KB)
Abstract:

In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.

On multidimensional Urysohn type generalized sampling operators
Harun Karsli
2021, 4(4): 271-280 doi: 10.3934/mfc.2021015 +[Abstract](436) +[HTML](174) +[PDF](384.29KB)
Abstract:

The concern of this study is to construction of a multidimensional version of Urysohn type generalized sampling operators, whose one dimensional case defined and investigated by the author in [28] and [27]. In details, as a continuation of the studies of the author, the paper centers around to investigation of some approximation and asymptotic properties of the aforementioned linear multidimensional Urysohn type generalized sampling operators.

Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $
Antonio G. García
2021, 4(4): 281-297 doi: 10.3934/mfc.2021019 +[Abstract](271) +[HTML](104) +[PDF](440.56KB)
Abstract:

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define \begin{document}$ \Lambda $\end{document}-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice \begin{document}$ \Lambda $\end{document} in \begin{document}$ \mathbb{R}^{2d} $\end{document}. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.

A note on convergence results for varying interval valued multisubmeasures
Anca Croitoru, Alina GavriluŢ, Alina Iosif and Anna Rita Sambucini
2021, 4(4): 299-310 doi: 10.3934/mfc.2021020 +[Abstract](247) +[HTML](123) +[PDF](387.0KB)
Abstract:

Some limit theorems are presented for Riemann-Lebesgue integrals where the functions \begin{document}$ G_n $\end{document} and the measures \begin{document}$ M_n $\end{document} are interval valued and the convergence for the multisubmeasures is setwise. In particular sufficient conditions in order to obtain \begin{document}$ \int G_n dM_n \to \int G dM $\end{document} are given.

A numerical comparative study of generalized Bernstein-Kantorovich operators
Uğur Kadak and Faruk Özger
2021, 4(4): 311-332 doi: 10.3934/mfc.2021021 +[Abstract](262) +[HTML](205) +[PDF](725.02KB)
Abstract:

In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as \begin{document}$ \lambda $\end{document}-Bernstein, \begin{document}$ \lambda $\end{document}-Bernstein-Kantorovich, \begin{document}$ \alpha $\end{document}-Bernstein and \begin{document}$ \alpha $\end{document}-Bernstein-Kantorovich operators.

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