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":" ","pages":""},"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":"10 1","pages":""},"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":"18 1","pages":""},"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":" ","pages":""},"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}
We give a proof of H older continuity for bounded local weak solutions to the equation ut =sum_{i=1}^N (|u_{x_i}|^{p_i−2} u_{x_i} )_{x_i} , in Ω × [0, T], with Ω ⊂⊂ R^N under the condition 2 < pi < p(1 + 2/N) for each i = 1, .., N, being p the harmonic mean of the pi's, via recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack estimates within the proper intrinsic geometry.
对于方程ut =sum_{i=1}^N (|u_{x_i}|^{p_i−2}u_{x_i})_{x_i}，在Ω x [0, T]中，在2 < pi < p(1 + 2/N)的条件下，对于每一个i=1，给出了一个有界局部弱解的H老连续性的证明，其中Ω≡R^N通过最近发现的内禀哈纳克估计，N是π的调和平均值p。此外，我们在适当的固有几何范围内建立了这些哈纳克估计的等价形式。
{"title":"On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation.","authors":"Simone Ciani, V. Vespri","doi":"10.33205/CMA.824336","DOIUrl":"https://doi.org/10.33205/CMA.824336","url":null,"abstract":"We give a proof of H older continuity for bounded local weak solutions to the equation ut =sum_{i=1}^N (|u_{x_i}|^{p_i−2} u_{x_i} )_{x_i} , in Ω × [0, T], with Ω ⊂⊂ R^N under the condition 2 < pi < p(1 + 2/N) for each i = 1, .., N, being p the harmonic mean of the pi's, via recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack estimates within the proper intrinsic geometry.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"34 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41262399","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}
Heun functions are important for many applications in Mathematics, Physics and in thus in interdisciplinary phenomena modelling. They satisfy second order differential equations and are usually represented by power series. Closed forms and simpler polynomial representations are useful. Therefore, we study and derive closed forms for several families of Heun functions related to classical entropies. By comparing two expressions of the same Heun function, we get several combinatorial identities generalizing some classical ones.
{"title":"Heun equations and combinatorial identities","authors":"Adina Bărar, G. Mocanu, I. Raşa","doi":"10.33205/CMA.810478","DOIUrl":"https://doi.org/10.33205/CMA.810478","url":null,"abstract":"Heun functions are important for many applications in Mathematics, Physics and in thus in interdisciplinary phenomena modelling. They satisfy second order differential equations and are usually represented by power series. Closed forms and simpler polynomial representations are useful. Therefore, we study and derive closed forms for several families of Heun functions related to classical entropies. By comparing two expressions of the same Heun function, we get several combinatorial identities generalizing some classical ones.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48475861","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}
After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a emph{weak lower $A$-semi-frame} and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in cite{GB}. We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.
{"title":"Weak A-frames and weak A-semi-frames","authors":"J. Antoine, G. Bellomonte, C. Trapani","doi":"10.33205/CMA.835582","DOIUrl":"https://doi.org/10.33205/CMA.835582","url":null,"abstract":"After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a emph{weak lower $A$-semi-frame} and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in cite{GB}. We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44365728","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 recent times quantitative Voronovskaya type theorems have been presented in spaces of non-periodic continuous functions. In this work we proved similar results but for Fejer-Korovkin trigonometric operators. That is we measure the rate of convergence in the associated Voronovskaya type theotem. Recall that these operators provide the optimal rate in approximation by positive linear operators. For the proofs we present new inequalities related with trigonometric polynomials as well as with the convergence factor of the Fej'er-Korovkin operators. Our approach includes spaces of Lebesgue integrable functions.
{"title":"Quantitative Voronovskaya-type theorems for Fej'er-Korovkin operators","authors":"J. Bustamante, Lázaro Flores De Jesús","doi":"10.33205/cma.818715","DOIUrl":"https://doi.org/10.33205/cma.818715","url":null,"abstract":"In recent times quantitative Voronovskaya type theorems have been presented in spaces of non-periodic continuous functions. In this work we proved similar results but for Fejer-Korovkin trigonometric operators. That is we measure the rate of convergence in the associated Voronovskaya type theotem. Recall that these operators provide the optimal rate in approximation by positive linear operators. For the proofs we present new inequalities related with trigonometric polynomials as well as with the convergence factor of the Fej'er-Korovkin operators. Our approach includes spaces of Lebesgue integrable functions.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42961491","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 we extend the Ostrowski inequality to the integral with respect to arc-length by providing upper bounds for the quantity |f(v)l(γ)-∫_{γ}f(z)|dz|| under the assumptions that γ is a smooth path parametrized by z(t), t∈[a,b] with the length l(γ), u=z(a), v=z(x) with x∈(a,b) and w=z(b) while f is holomorphic in G, an open domain and γ⊂G. An application for circular paths is also given. Several applications for circular paths and for some special functions of interest such as the exponential functions are also provided.
{"title":"Ostrowski's Type Inequalities for the Complex Integral on Paths","authors":"S. Dragomir","doi":"10.33205/cma.798861","DOIUrl":"https://doi.org/10.33205/cma.798861","url":null,"abstract":"In this paper we extend the Ostrowski inequality to the integral with respect to arc-length by providing upper bounds for the quantity |f(v)l(γ)-∫_{γ}f(z)|dz|| under the assumptions that γ is a smooth path parametrized by z(t), t∈[a,b] with the length l(γ), u=z(a), v=z(x) with x∈(a,b) and w=z(b) while f is holomorphic in G, an open domain and γ⊂G. An application for circular paths is also given. Several applications for circular paths and for some special functions of interest such as the exponential functions are also provided.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42753985","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 the present paper, we consider certain classes of bi-univalent Bazilevic functions with bounded boundary rotation involving Salăgeăn linear operator to obtain the estimates of their second and third coefficients. Further, certain special cases are also indicated. Some interesting remarks about the results presented here are also discussed. . .
{"title":"Certain Class of Bi-Bazilevic Functions with Bounded Boundary Rotation Involving Salăgeăn Operator","authors":"M. Aouf, T. Seoudy","doi":"10.33205/cma.781936","DOIUrl":"https://doi.org/10.33205/cma.781936","url":null,"abstract":"In the present paper, we consider certain classes of bi-univalent Bazilevic functions with bounded boundary rotation involving Salăgeăn linear operator to obtain the estimates of their second and third coefficients. Further, certain special cases are also indicated. Some interesting remarks about the results presented here are also discussed. . .","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41693207","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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