1,721,017 research outputs found
A general method to study the convergence of nonlinear operators in Orlicz spaces
No full textWe continue the work started in a previous article and introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group. We analyze the behavior of such nets and detect the fairest assumption, which are needed for the nets to converge with respect to the uniform convergence and in the setting of Orlicz spaces. As a consequence, we give results of convergence in this frame, study some important special cases, and provide graphical representations
Convergence Results for Nonlinear Sampling Kantorovich Operators in Modular Spaces
No full textIn the present paper, convergence in modular spaces is investigated for a class of nonlinear discrete operators, namely the nonlinear multivariate sampling Kantorovich operators. The convergence results in the Musielak-Orlicz spaces, in the weighted Orlicz spaces, and in the Orlicz spaces follow as particular cases. Even more, spaces of functions equipped by modulars without an integral representation are presented and discussed
Estimations for the convex modular of the aliasing error of nonlinear sampling Kantorovich operators
No full textMay 15, January 30, February 26, Abstract. In this paper, we establish quantitative estimates for the nonlinear sampling Kantorovich operators in the general setting of modular spaces L rho. To achieve this, we consider a notion of modulus of smoothness based on the convex modular functional rho, which defines the space. The approach proposed is new in the sense that, in the literature, theorems for the order of approximation in L rho are mainly qualitative, i.e., are proved considering functions belonging to Lipschitz classes; here the estimates are achieved for every function belonging to the whole L rho. To show the effectiveness of the achieved results, several particular cases of modular spaces are presented in detail
Quantitative Estimates for Nonlinear Sampling Kantorovich Operators
No full textIn this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in Lp-spaces, 1 ≤ p< ∞, 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 Lp-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
Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
No full textIn the present work, we establish a quantitative estimate for the perturbed sampling Kantorovich operators in Orlicz spaces, in terms of the modulus of smoothness, defined by means of its modular functional. From the obtained result, we also deduce the qualitative order of approximation, by considering functions in suitable Lipschitz classes. This allows us to apply the above results in certain Orlicz spaces of particular interest, such as the interpolation spaces, the exponential spaces and the L-p -spaces, 1 <= p < + infinity. In particular, in the latter case, we also provide an estimate established using a direct proof based on certain properties of the L-p -modulus of smoothness, which are not valid in the general case of Orlicz spaces. The possibility of using a direct approach allows us to improve the estimate that can be deduced as a consequence of the one achieved in Orlicz spaces. In the final part of the article, we furnish some estimates and the corresponding qualitative order of approximation in the space of uniformly continuous and bounded functions
Approximation results in Sobolev and fractional Sobolev spaces by sampling Kantorovich operators
No full textThe present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in "Sobolev-like settings". More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions defining the above sampling type series. As a consequence, certain open problems related to the convergence in variation for the SK operators have been solved. Then, we considered the above operators in a fractional-type setting. It is well-known that, in the literature, several notions of fractional Sobolev spaces are available, such as, the Gagliardo Sobolev spaces (GSs) defined by means of the Gagliardo semi-norm, or the weak Riemann-Liouville Sobolev spaces (wRLSs) defined by the weak (left and right) Riemann-Liouville fractional derivatives and so on. Here, in order to face the above convergence problem, we introduced a new definition of fractional Sobolev spaces, that we called the tight fractional Sobolev spaces (tfSs) and generated as the intersection of the GSs and the symmetric Sobolev spaces (i.e., that given by the intersection of the left and the right wRLSs). In the latter setting, we obtain one of the main results of the paper, that is a convergence theorem for the SK operators with respect to a suitable norm on tfSs
Convergence of a Class of Generalized Sampling Kantorovich Operators Perturbed by Multiplicative Noise
No full textIn this paper a new family of sampling type series is introduced. From the mathematical point of view, the present definition generalizes the notion of the well-known sampling Kantorovich operators, in fact providing a weighted version of the original family of operators by functions gk,w, k∈ Z, w > 0, called noise functions. From the application point of view, this situation represents the reconstruction problem of signals perturbed by linear or nonlinear multiplicative noise sources. In this respect, approximation results have been established in various contexts. First, pointwise and uniform approximation theorems have been proved. Then, convergence theorems have been derived in the general setting of Orlicz spaces. The latter context allows us to deduce, in particular, an Lp-convergence theorem. Finally, the concept of delta convergent sequence is introduced and also used in order to prove that the above family of sampling type operators extend the well-known generalized sampling series of P.L. Butzer
On the Regularization by Durrmeyer-Sampling Type Operators in Lp-Spaces via a Distributional Approach
No full textIn this paper, we are concerned with the study of the regularization properties of Durrmeyer-sampling type operators Dw phi,psi in Lp-spaces, with 1 <= p <=+infinity. In order to reach the above results, we mainly use tools belonging to distribution theory and Fourier analysis. Here, we show how the regularization process performed by the operators is strongly influenced by the regularity of the discrete kernel phi. We investigate the classical case of continuous kernels, the more general case of kernels in Sobolev spaces, as well as the remarkable case of bandlimited kernels, i.e., belonging to Bernstein classes. In the latter case, we also establish a closed form for the distributional Fourier transform of the above operators applied to bandlimited functions. Finally, the main results presented herein will be also applied to specific instances of bandlimited kernels, such as de la Vall & eacute;e Poussin and Bochner-Riesz kernels
A Simplified Model for Estimating Household Air Pollution in Challenging Contexts: A Case Study from Ghana
Full text linkAlmost three billion people rely primarily on inefficient and polluting cooking systems worldwide. Household air pollution is a direct consequence of this practice, and it is annually associated with millions of premature deaths and diseases, mainly in low- and lower-middle-income countries. The use of improved cookstoves often represents an appropriate solution to reduce such health risks. However, in the distribution of such units, it can be necessary to prioritize the beneficiaries. Thus, in this study, we conducted field research involving five rural villages in the Northern part of Ghana, where using three-stone fires or rural stoves was common. Concentrations of PM2.5, PM10, and carbon monoxide (CO) were measured indoors and outdoors. Considering each field mission lasted less than 24 h, assumptions were made so as to calculate the average pollutant concentrations in 24 h through a new, simplified equation that combined efficiency and cost-savings by shortening field assessments. The obtained values were compared with international guidelines. The results showed that PM2.5 and PM10 limits were overstepped in two villages, which should thus be prioritized. However, further research will be necessary to strengthen and validate our proposed equation, which must be seen as a starting point
Convergence of sampling Kantorovich operators in modular spaces with applications
No full textIn the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the Lp and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented
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