Pub Date : 2020-05-01DOI: 10.14321/realanalexch.45.1.0173
Reinwand
{"title":"Types of Convergence Which Preserve Continuity","authors":"Reinwand","doi":"10.14321/realanalexch.45.1.0173","DOIUrl":"https://doi.org/10.14321/realanalexch.45.1.0173","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"45 1","pages":"173-204"},"PeriodicalIF":0.2,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45147801","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 : 2020-03-22DOI: 10.14321/realanalexch.46.1.0121
T. Munoz-Darias, A. Karlovich, E. Shargorodsky
Let $BV_p[0,1]$, $1le p
设$BV_p[0,1]$, $1le p
{"title":"Multiplication is an open bilinear mapping in the Banach algebra of functions of bounded Wiener $p$-variation","authors":"T. Munoz-Darias, A. Karlovich, E. Shargorodsky","doi":"10.14321/realanalexch.46.1.0121","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0121","url":null,"abstract":"Let $BV_p[0,1]$, $1le p<infty$, be the Banach algebra of functions of bounded $p$-variation in the sense of Wiener. Recently, Kowalczyk and Turowska cite{KT19} proved that the multiplication in $BV_1[0,1]$ is an open bilinear mapping. We extend this result for all values of $pin[1,infty)$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44527986","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 : 2019-12-27DOI: 10.14321/REALANALEXCH.45.2.0453
T. K'atay
We show that a typical Besicovitch set $B$ has intersections of measure zero with every line not contained in it. Moreover, every line in $B$ intersects the union of all the other lines in $B$ in a set of measure zero.
{"title":"The Intersection of Typical Besicovitch Sets with Lines","authors":"T. K'atay","doi":"10.14321/REALANALEXCH.45.2.0453","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0453","url":null,"abstract":"We show that a typical Besicovitch set $B$ has intersections of measure zero with every line not contained in it. Moreover, every line in $B$ intersects the union of all the other lines in $B$ in a set of measure zero.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163851","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 : 2019-12-08DOI: 10.14321/REALANALEXCH.34.2.0347
A. Mirotin, E. Mirotin
The main purpose of this work is to ascertain when arithmetic operations with periodic functions whose domains may not coincide with the whole real line preserve periodicity.
{"title":"On Sums and Products of Periodic Functions","authors":"A. Mirotin, E. Mirotin","doi":"10.14321/REALANALEXCH.34.2.0347","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.34.2.0347","url":null,"abstract":"The main purpose of this work is to ascertain when arithmetic operations with periodic functions whose domains may not coincide with the whole real line preserve periodicity.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"34 1","pages":"347-358"},"PeriodicalIF":0.2,"publicationDate":"2019-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45519800","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 : 2019-12-08DOI: 10.14321/realanalexch.46.1.0149
Don'at Nagy
The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of $mathbb{R}$. If we consider a compact subset $Ksubseteq[0,1]$ and a power series $f$ which is absolutely convergent on $[0,1]$, then we may use these operations and the natural topology of the space of compact sets to substitute the compact set $K$ into the power series $f$. Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set $Ksubseteq [0,1]$ into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.
{"title":"Substituting the typical compact sets into a power series","authors":"Don'at Nagy","doi":"10.14321/realanalexch.46.1.0149","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0149","url":null,"abstract":"The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of $mathbb{R}$. If we consider a compact subset $Ksubseteq[0,1]$ and a power series $f$ which is absolutely convergent on $[0,1]$, then we may use these operations and the natural topology of the space of compact sets to substitute the compact set $K$ into the power series $f$. Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set $Ksubseteq [0,1]$ into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47758264","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 : 2019-11-23DOI: 10.14321/realanalexch.47.1.1619498208
T. Persson
I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer.
{"title":"A Mass Transference Principle and Sets with Large Intersections","authors":"T. Persson","doi":"10.14321/realanalexch.47.1.1619498208","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1619498208","url":null,"abstract":"I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42997770","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}
Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $mathsf{X}_{Omega}^{(l)}$ is the multiplicative subshift derived from the shift space $Omega$ with given $l > 1$. We show that $mathsf{X}_{Omega}^{(l)}$ is (topologically) transitive/mixing if and only if $Omega$ is extensible/mixing. After introducing $l$-directional mixing property, we derive the equivalence between $l$-directional mixing property of $mathsf{X}_{Omega}^{(l)}$ and weakly mixing property of $Omega$.
{"title":"Topologically Mixing Properties of Multiplicative Integer Systems","authors":"Jung-Chao Ban, Chih-Hung Chang, Wen-Guei Hu, Guan-Yu Lai, Yu-Liang Wu","doi":"10.14321/realanalexch.47.1.1614278701","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1614278701","url":null,"abstract":"Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $mathsf{X}_{Omega}^{(l)}$ is the multiplicative subshift derived from the shift space $Omega$ with given $l > 1$. We show that $mathsf{X}_{Omega}^{(l)}$ is (topologically) transitive/mixing if and only if $Omega$ is extensible/mixing. After introducing $l$-directional mixing property, we derive the equivalence between $l$-directional mixing property of $mathsf{X}_{Omega}^{(l)}$ and weakly mixing property of $Omega$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42116169","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 : 2019-11-15DOI: 10.14321/REALANALEXCH.45.2.0439
A. Shaikh, B. Datta
The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.
{"title":"Generalized almost statistical convergence","authors":"A. Shaikh, B. Datta","doi":"10.14321/REALANALEXCH.45.2.0439","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0439","url":null,"abstract":"The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43419645","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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