arXiv subjects

Danilo Costarelli

Publications and source records attributed to Danilo Costarelli.

28 records · Page 2Linked to original sources

arXiv2021

Quantitative estimates for nonlinear sampling Kantorovich operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of continuity in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $L^{p}$-spaces, $1\leq p<\infty $, and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming $f$ in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $L^p$-case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

math.FA↗

arXiv2020

On the convergence properties of Durrmeyer-Sampling Type Operators in Orlicz spaces

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces $L^φ(\mathbb{R})$. From the latter result, the convergence in $L^p(\mathbb{R})$-space, $L^α\log^βL$, and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.

math.FA↗

arXiv2020

Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators

In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces $L^φ$. The results here proved, extend those given by Costarelli and Vinti in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in $L^φ$ and use it to provide an upper bound for the approximation error of the above operators. Finally, examples of sigmoidal activation functions have been considered and studied in details.

math.FA↗

arXiv2020

Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications

In this paper, convergence results in a multivariate setting have been proved for a family of neural network operators of the max-product type. In particular, the coefficients expressed by Kantorovich type means allow to treat the theory in the general frame of the Orlicz spaces, which includes as particular case the $L^p$-spaces. Examples of sigmoidal activation functions are discussed, for the above operators in different cases of Orlicz spaces. Finally, concrete applications to real world cases have been presented in both uni-variate and multivariate settings. In particular, the case of reconstruction and enhancement of biomedical (vascular) image has been discussed in details.

math.FA↗

arXiv2019

Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing

In this paper we study the problem of the convergence in variation for the generalized sampling series based upon averaged-type kernels in the multidimensional setting. As a crucial tool, we introduce a family of operators of sampling-Kantorovich type for which we prove convergence in L^p on a subspace of L^p(R^N): therefore we obtain the convergence in variation for the multidimensional generalized sampling series by means of a relation between the partial derivatives of such operators acting on an absolutely continuous function f and the sampling-Kantorovich type operators acting on the partial derivatives of f. Applications to digital image processing are also furnished.

math.FA↗

arXiv2017

A characterization of the convergence in variation for the generalized sampling series

In this paper, we study the convergence in variation for the generalized sampling operators based upon averaged-type kernels and we obtain a characterization of absolutely continuous functions. This result is proved exploiting a relation between the first derivative of the above operator acting on $f$ and the sampling Kantorovich series of f'. By such approach, also a variation detracting-type property is established. Finally, examples of averaged kernels are provided, such as the central B-splines of order $n$ (duration limited functions) or other families of kernels generated by the Fejer and the Bochner-Riesz kernels (bandlimited functions).

math.FA↗

arXiv2017

Detection of thermal bridges from thermographic images for the analysis of buildings energy performance

In this paper, we develop a procedure for the detection of the contours of thermal bridges from thermographic images, in order to study the energetic performance of buildings. Two main steps of the above method are: the enhancement of the thermographic images by an optimized version of the mathematical algorithm for digital image processing based on the theory of sampling Kantorovich operators, and the application of a suitable thresholding based on the analysis of the histogram of the enhanced thermographic images. Finally, an accuracy improvement of the parameter that defines the thermal bridge is obtained.

math.NA↗

arXiv2015

Approximation of discontinuous signals by sampling Kantorovich series

In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous signals are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Further, also the problem of the linear prediction by sampling values from the past is analyzed. At the end, the role of duration-limited kernels in the previous approximation processes has been treated, and several examples have been provided.

math.FA↗

arXiv2014

Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces

In this paper, the problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of the uniform approximation for uniformly continuous and bounded functions/signals belonging to Lipschitz classes and the case of the modular approximation for functions in Orlicz spaces are considered. In the latter context, Lipschitz classes of Zygmund-type which take into account of the modular functional involved are introduced. Applications to Lp(R^n), interpolation and exponential spaces can be deduced from the general theory formulated in the setting of Orlicz spaces. The special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fejer's and B-spline kernels have been studied in details.

math.FA↗

arXiv2014

Applications of sampling Kantorovich operators to thermographic images for seismic engineering

In this paper, we present some applications of the multivariate sampling Kantorovich operators $S_w$ to seismic engineering. The mathematical theory of these operators, both in the space of continuous functions and in Orlicz spaces, show how it is possible to approximate/reconstruct multivariate signals, such as images. In particular, to obtain applications for thermographic images a mathematical algorithm is developed using MATLAB and matrix calculus. The setting of Orlicz spaces is important since allow us to reconstruct not necessarily continuous signals by means of $S_w$. The reconstruction of thermographic images of buildings by our sampling Kantorovich algorithm allow us to obtain models for the simulation of the behavior of structures under seismic action. We analyze a real world case study in term of structural analysis and we compare the behavior of the building under seismic action using various models.

cs.CV↗