Pub Date : 2021-06-07DOI: 10.14321/realanalexch.47.2.1645501301
S. Mahanta, S. Ray
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.
{"title":"A Generalised Continuous Primitive Integral and Some of its Applications","authors":"S. Mahanta, S. Ray","doi":"10.14321/realanalexch.47.2.1645501301","DOIUrl":"https://doi.org/10.14321/realanalexch.47.2.1645501301","url":null,"abstract":"Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44894971","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 : 2020-12-01DOI: 10.14321/realanalexch.48.1.1606864896
D. Cruz-Uribe, S. Rodney
We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $prightarrowinfty$ is the $L^infty$ norm to the scale of Orlicz spaces.
{"title":"A Note on the Limit of Orlicz Norms","authors":"D. Cruz-Uribe, S. Rodney","doi":"10.14321/realanalexch.48.1.1606864896","DOIUrl":"https://doi.org/10.14321/realanalexch.48.1.1606864896","url":null,"abstract":"We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $prightarrowinfty$ is the $L^infty$ norm to the scale of Orlicz spaces.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46673069","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 : 2020-12-01DOI: 10.14321/REALANALEXCH.46.2.0441
D. Hruška
We prove a stronger version of a conjecture stated in a paper from 2017 by J. M. Ash and S. Catoiu concerning relations between various notions of the Lipschitz property and differentiability in the Euclidean plane. We also provide an improved version of the main result of that paper.
我们证明了J. M. Ash和S. Catoiu在2017年的一篇论文中提出的一个猜想的更强版本，该猜想涉及欧几里得平面上Lipschitz性质的各种概念与可微性之间的关系。我们还提供了该论文主要结果的改进版本。
{"title":"A NOTE ON DIRECTIONAL LIPSCHITZ CONTINUITY IN THE EUCLIDEAN PLANE","authors":"D. Hruška","doi":"10.14321/REALANALEXCH.46.2.0441","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0441","url":null,"abstract":"We prove a stronger version of a conjecture stated in a paper from 2017 by J. M. Ash and S. Catoiu concerning relations between various notions of the Lipschitz property and differentiability in the Euclidean plane. We also provide an improved version of the main result of that paper.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66983862","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0401
D. Sokołowski
{"title":"Stability of (n)-th Order Flett’s and Sahoo-Riedel’s Points","authors":"D. Sokołowski","doi":"10.14321/REALANALEXCH.45.2.0401","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0401","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66984003","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0411
J. Foran, J. Kardos
{"title":"Characterizing the Coordinate Functions of Space Filling Curves","authors":"J. Foran, J. Kardos","doi":"10.14321/REALANALEXCH.45.2.0411","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0411","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45826032","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0375
T. H. Steele
{"title":"The Dynamics of a Typical Measurable Function are Determined on a Zero Measure Set","authors":"T. H. Steele","doi":"10.14321/REALANALEXCH.45.2.0375","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0375","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44254718","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0387
Soon-Mo Jung, Doyun Nam
{"title":"On the Density of the Thinnest Covering of (mathbb{R}^n)","authors":"Soon-Mo Jung, Doyun Nam","doi":"10.14321/REALANALEXCH.45.2.0387","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0387","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45630067","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0465
G. Lewicki, M. Prophet, William E. Wood
{"title":"A Note on the Existence of Real Two-dimensional Symmetric Subspaces of (L^p[-1,1])","authors":"G. Lewicki, M. Prophet, William E. Wood","doi":"10.14321/REALANALEXCH.45.2.0465","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0465","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42514649","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0481
Otgonbayar Uuye
{"title":"A Generalization of the Riemann-Lebesgue Theorem for Riemann Integrability","authors":"Otgonbayar Uuye","doi":"10.14321/REALANALEXCH.45.2.0481","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0481","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66984154","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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