Pub Date : 2024-07-05DOI: 10.1134/s0001434624030027
R. M. Gadzhimirzaev
Abstract
We study the convergence of Fourier series in the polynomial system ({m_{n,N}^{alpha,r}(x)}) orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of (fin W^r_{l^p_{rho_N}(Omega_delta)}) in this system converges to (f) pointwise on the grid (Omega_delta) as (pge2). In addition, we study the approximation properties of partial sums of Fourier series in the system ({m_{n,N}^{0,r}(x)}).
{"title":"Convergence of the Fourier Series in Meixner–Sobolev Polynomials and Approximation Properties of Its Partial Sums","authors":"R. M. Gadzhimirzaev","doi":"10.1134/s0001434624030027","DOIUrl":"https://doi.org/10.1134/s0001434624030027","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the convergence of Fourier series in the polynomial system <span>({m_{n,N}^{alpha,r}(x)})</span> orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of <span>(fin W^r_{l^p_{rho_N}(Omega_delta)})</span> in this system converges to <span>(f)</span> pointwise on the grid <span>(Omega_delta)</span> as <span>(pge2)</span>. In addition, we study the approximation properties of partial sums of Fourier series in the system <span>({m_{n,N}^{0,r}(x)})</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Trigonometric Polynomials with Frequencies in the Set of Cubes","authors":"M. R. Gabdullin, S. V. Konyagin","doi":"10.1134/s0001434624030052","DOIUrl":"https://doi.org/10.1134/s0001434624030052","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove that for any <span>(varepsilon>0)</span> and any trigonometric polynomial <span>(f)</span> with frequencies in the set <span>({n^3: N leq nleq N+N^{2/3-varepsilon}})</span> one has </p><span>$$|f|_4 ll varepsilon^{-1/4}|f|_2$$</span><p> with implied constant being absolute. We also show that the set <span>({n^3: Nleq nleq N+(0.5N)^{1/2}})</span> is a Sidon set. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030131
A. O. Chebotarenko
Abstract
We generalize the Khinchin singularity phenomenon for the problem in which, for a given irrational linear subspace, rational subspaces forming the least angle with the given subspace are sought.
{"title":"On Angles between Linear Subspaces in $$mathbb R^4$$ and the Singularity","authors":"A. O. Chebotarenko","doi":"10.1134/s0001434624030131","DOIUrl":"https://doi.org/10.1134/s0001434624030131","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We generalize the Khinchin singularity phenomenon for the problem in which, for a given irrational linear subspace, rational subspaces forming the least angle with the given subspace are sought. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030180
A. R. Aliev, R. A. Aliev
Abstract
In this paper, conditions are found for the boundedness of the fractional maximal operator, the Riesz potential, and their commutators in Orlicz spaces.
摘要 本文为奥利奇空间中的分数最大算子、里兹势及其换元子的有界性找到了条件。
{"title":"On the Boundedness of the Fractional Maximal Operator, the Riesz Potential, and Their Commutators in Orlicz Spaces","authors":"A. R. Aliev, R. A. Aliev","doi":"10.1134/s0001434624030180","DOIUrl":"https://doi.org/10.1134/s0001434624030180","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, conditions are found for the boundedness of the fractional maximal operator, the Riesz potential, and their commutators in Orlicz spaces. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030362
K. S. Shklyaev
Abstract
It is proved that every connected boundedly compact locally Chebyshev set in a normed space is a Chebyshev set.
摘要 证明了规范空间中的每个有界紧凑局部切比雪夫集合都是切比雪夫集合。
{"title":"On Locally Chebyshev Sets","authors":"K. S. Shklyaev","doi":"10.1134/s0001434624030362","DOIUrl":"https://doi.org/10.1134/s0001434624030362","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> It is proved that every connected boundedly compact locally Chebyshev set in a normed space is a Chebyshev set. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030337
Q. H. Tran, B. Herrera
Abstract
This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the (n)-simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra.
{"title":"$$n$$ -Dimensional Generalizations of a Thébault Conjecture","authors":"Q. H. Tran, B. Herrera","doi":"10.1134/s0001434624030337","DOIUrl":"https://doi.org/10.1134/s0001434624030337","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the <span>(n)</span>-simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030076
E. A. Lebedeva
Abstract
We construct a Parseval wavelet frame with compact support for an arbitrary continuous (2pi)-periodic function (f), (f(0)=1), satisfying the inequality (|f(x)|^2+|f(x+pi)|^2le 1). The frame refinement mask uniformly approximates (f). The refining function has stable integer shifts.
{"title":"Approximation by Refinement Masks","authors":"E. A. Lebedeva","doi":"10.1134/s0001434624030076","DOIUrl":"https://doi.org/10.1134/s0001434624030076","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We construct a Parseval wavelet frame with compact support for an arbitrary continuous <span>(2pi)</span>-periodic function <span>(f)</span>, <span>(f(0)=1)</span>, satisfying the inequality <span>(|f(x)|^2+|f(x+pi)|^2le 1)</span>. The frame refinement mask uniformly approximates <span>(f)</span>. The refining function has stable integer shifts. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030258
Hongyan Guan, Jinze Gou
Abstract
This paper is the first to introduce a fixed point problem of integral type in a (b)-metric space. We study sufficient conditions for the existence and uniqueness of a common fixed point of contractive mappings of integral type. We also give two examples to support our results.
{"title":"Common Fixed Point Theorems for Contractive Mappings of Integral Type in $$b$$ -Metric Spaces","authors":"Hongyan Guan, Jinze Gou","doi":"10.1134/s0001434624030258","DOIUrl":"https://doi.org/10.1134/s0001434624030258","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> This paper is the first to introduce a fixed point problem of integral type in a <span>(b)</span>-metric space. We study sufficient conditions for the existence and uniqueness of a common fixed point of contractive mappings of integral type. We also give two examples to support our results. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030234
P. A. Borodin, A. M. Ershov
Abstract
In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space (L_2[-1,1]). In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov’s problem different from Komarov’s one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on ([-1,1]); (b) of the density in (L_2[-1,1]) of simple partial fractions with poles on the boundary of a given domain for which ([-1,1]) is an inner chord.
摘要 2014年,S. R. Nasyrov提出了这样一个问题:在复数空间(L_2[-1,1])中,极点在单位圆上的简单部分分数(复数多项式的对数导数)是否密集?2019 年,科马洛夫(M. A. Komarov)对这个问题做出了否定的回答。本文包含了对纳西洛夫问题的不同于科马洛夫问题的简单解答。本文得到了与以下问题相关的结果:(a) ([-1,1])上加权 Lebesgue 空间中单位圆上有极点的简单分式的密度;(b) (L_2[-1,1])中给定域边界上有极点的简单分式的密度,而 ([-1,1])是该域的内弦。
{"title":"S. R. Nasyrov’s Problem of Approximation by Simple Partial Fractions on an Interval","authors":"P. A. Borodin, A. M. Ershov","doi":"10.1134/s0001434624030234","DOIUrl":"https://doi.org/10.1134/s0001434624030234","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space <span>(L_2[-1,1])</span>. In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov’s problem different from Komarov’s one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on <span>([-1,1])</span>; (b) of the density in <span>(L_2[-1,1])</span> of simple partial fractions with poles on the boundary of a given domain for which <span>([-1,1])</span> is an inner chord. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-05DOI: 10.1134/s0001434624030118
V. D. Stepanov, E. P. Ushakova
Abstract
Upper and lower bounds are obtained for the approximation numbers of the two-dimensional rectangular Hardy operator on weighted Lebesgue spaces on (mathbb{R}_+^2).
{"title":"Approximation Numbers of the Two-Dimensional Rectangular Hardy Operator","authors":"V. D. Stepanov, E. P. Ushakova","doi":"10.1134/s0001434624030118","DOIUrl":"https://doi.org/10.1134/s0001434624030118","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Upper and lower bounds are obtained for the approximation numbers of the two-dimensional rectangular Hardy operator on weighted Lebesgue spaces on <span>(mathbb{R}_+^2)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}