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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $mathbb{R}$ of fractional orders $ 2 < alpha leq 3 $. These estimate the size of first and second derivatives of a composition with a Banach space valued function over $mathbb{R}$. We give applications when $α = 2.5$.
{"title":"Abstract Generalized Fractional Landau inequalities over R","authors":"G. Anastassiou","doi":"10.33205/CMA.764161","DOIUrl":"https://doi.org/10.33205/CMA.764161","url":null,"abstract":"We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $mathbb{R}$ of fractional orders $ 2 < alpha leq 3 $. These estimate the size of first and second derivatives of a composition with a Banach space valued function over $mathbb{R}$. We give applications when $α = 2.5$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48991874","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}
It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds. ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds
{"title":"The A-integral and restricted Riesz transform","authors":"R. Aliev, Khanim I. Nebiyeva","doi":"10.33205/cma.728156","DOIUrl":"https://doi.org/10.33205/cma.728156","url":null,"abstract":"It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds. ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46431542","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}
Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable. This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.
{"title":"Ulam stability in real inner-product spaces","authors":"Bianca Moșneguțu, A. Mǎdutǎ","doi":"10.33205/cma.758854","DOIUrl":"https://doi.org/10.33205/cma.758854","url":null,"abstract":"Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable. This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48370366","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}
Let $sigma_n$ denotes the classical Fej'er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $(I-sigma_n)^r(f)$ in terms of the modulus of continuity of order $r$ (with specific constants) in all $mathbb{L}^p$ spaces $1leq p leq infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $(I-sigma_n)^r(f)$.
{"title":"Strong converse inequalities and quantitative Voronovskaya-type theorems for trigonometric Fej'er sums","authors":"J. Bustamante, Lázaro Flores De Jesús","doi":"10.33205/cma.653843","DOIUrl":"https://doi.org/10.33205/cma.653843","url":null,"abstract":"Let $sigma_n$ denotes the classical Fej'er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $(I-sigma_n)^r(f)$ in terms of the modulus of continuity of order $r$ (with specific constants) in all $mathbb{L}^p$ spaces $1leq p leq infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $(I-sigma_n)^r(f)$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48490943","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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