L. Angeloni, N. Çetin, D. Costarelli, A. R. Sambucini, G. Vinti
In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.
{"title":"Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces","authors":"L. Angeloni, N. Çetin, D. Costarelli, A. R. Sambucini, G. Vinti","doi":"10.33205/CMA.876890","DOIUrl":"https://doi.org/10.33205/CMA.876890","url":null,"abstract":"In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48360133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nystrom methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.
{"title":"Some numerical applications of generalized Bernstein operators","authors":"D. Occorsio, M. Russo, W. Themistoclakis","doi":"10.33205/CMA.868272","DOIUrl":"https://doi.org/10.33205/CMA.868272","url":null,"abstract":"In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nystrom methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41487043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1142/9789811221644_0001
{"title":"Metric Spaces and Limits for Sequences","authors":"","doi":"10.1142/9789811221644_0001","DOIUrl":"https://doi.org/10.1142/9789811221644_0001","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86547024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesaro means of the Fourier series of the multi-dimensional function $fin L_1(log L)^{d-1}(mathbb{T}^d)supset L_p(mathbb{T}^d) (1
{"title":"Unrestricted Ces`aro summability of $d$-dimensional Fourier series and Lebesgue points","authors":"F. Weisz","doi":"10.33205/CMA.859583","DOIUrl":"https://doi.org/10.33205/CMA.859583","url":null,"abstract":"We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesaro means of the Fourier series of the multi-dimensional function $fin L_1(log L)^{d-1}(mathbb{T}^d)supset L_p(mathbb{T}^d) (1<p<infty)$ converge to $f$ at each strong Lebesgue point.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47943513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1007/978-1-4615-9990-6_15
M. Protter, C. B. Morrey
{"title":"Functions on Metric Spaces","authors":"M. Protter, C. B. Morrey","doi":"10.1007/978-1-4615-9990-6_15","DOIUrl":"https://doi.org/10.1007/978-1-4615-9990-6_15","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89079901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1002/9781118096864.ch13
John Quigg
Theorem 3. Let I be an interval, and let (fn) be a sequence of differentiable functions from I to R. Suppose that the sequence (f ′ n) of derivatives converges uniformly, and that there exists c ∈ I such that the sequence (fn(c)) of values converges. Then (fn) converges pointwise, lim fn is differentiable, and ( lim n→∞ fn )′ = lim n→∞ f ′ n. Theorem 4. Let A ⊂ R, let ∑∞ n=1 fn be a uniformly convergent series of functions from A to R, and let t ∈ A. If each fn is continuous at t, then so is ∑∞ n=1 fn. Theorem 5. Let ∑∞ n=1 fn be a uniformly convergent series of functions from [a, b] to R. If each fn is integrable, then so is ∑∞ n=1 fn, and ∫ b
{"title":"Uniform Convergence","authors":"John Quigg","doi":"10.1002/9781118096864.ch13","DOIUrl":"https://doi.org/10.1002/9781118096864.ch13","url":null,"abstract":"Theorem 3. Let I be an interval, and let (fn) be a sequence of differentiable functions from I to R. Suppose that the sequence (f ′ n) of derivatives converges uniformly, and that there exists c ∈ I such that the sequence (fn(c)) of values converges. Then (fn) converges pointwise, lim fn is differentiable, and ( lim n→∞ fn )′ = lim n→∞ f ′ n. Theorem 4. Let A ⊂ R, let ∑∞ n=1 fn be a uniformly convergent series of functions from A to R, and let t ∈ A. If each fn is continuous at t, then so is ∑∞ n=1 fn. Theorem 5. Let ∑∞ n=1 fn be a uniformly convergent series of functions from [a, b] to R. If each fn is integrable, then so is ∑∞ n=1 fn, and ∫ b","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88082002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.
{"title":"On the Korovkin-type approximation of set-valued continuous functions","authors":"M. Campiti","doi":"10.33205/CMA.863145","DOIUrl":"https://doi.org/10.33205/CMA.863145","url":null,"abstract":"This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43836838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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