Pub Date : 2017-08-01Epub Date: 2016-11-07DOI: 10.1016/j.trmi.2016.10.002
S. Basu , D. Sen
Bruckner proved that with exception of a set of first category, all other points of any second category set having Baire property in the Euclidean plane are points of directional linear categorical density of the set in almost all directions in the sense of category. In this article, we investigate this result of Bruckner in relation to sets not necessarily having Baire property and with respect to a more general definition of directional linear categorical density frammed after the pattern originally introduced by Wilczyński for linear categorical density.
{"title":"On a result of Bruckner relating to directional linear categorical density in Euclidean plane","authors":"S. Basu , D. Sen","doi":"10.1016/j.trmi.2016.10.002","DOIUrl":"10.1016/j.trmi.2016.10.002","url":null,"abstract":"<div><p>Bruckner proved that with exception of a set of first category, all other points of any second category set having Baire property in the Euclidean plane are points of directional linear categorical density of the set in almost all directions in the sense of category. In this article, we investigate this result of Bruckner in relation to sets not necessarily having Baire property and with respect to a more general definition of directional linear categorical density frammed after the pattern originally introduced by Wilczyński for linear categorical density.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 2","pages":"Pages 131-135"},"PeriodicalIF":0.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.10.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44379739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01Epub Date: 2017-04-01DOI: 10.1016/j.trmi.2017.03.001
Omar Dzagnidze
<div><p>It is well known that to each summable in the <span><math><mi>n</mi></math></span>-dimensional cube <span><math><msup><mrow><mrow><mo>[</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span> function <span><math><mi>f</mi></math></span> of variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> there corresponds one <span><math><mi>n</mi></math></span>-multiple trigonometric Fourier series <span><math><mi>S</mi><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></math></span> with constant coefficients.</p><p>In the present paper, with the function <span><math><mi>f</mi></math></span> we associate <span><math><mi>n</mi></math></span> one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></math></span>, with respect to variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively, with nonconstant coefficients and announce the preliminary results. In particular, if a continuous function <span><math><mi>f</mi></math></span> is differentiable at some point <span><math><mi>x</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span>, then all one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></math></span> converge at <span><math><mi>x</mi></math></span> to the value <span><math><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>.</p><p>For illustration we consider the well known example of Ch. Fefferman’s function <span><math><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math></span> whose double trigonometric Fourier series <span><math><mi>S</mi><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></math></span> diverges everywhere in the sense of Prinsheim. Namely, we establish the simultaneous convergence of the one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span> at almost all points <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><
{"title":"One-dimensional Fourier series of a function of many variables","authors":"Omar Dzagnidze","doi":"10.1016/j.trmi.2017.03.001","DOIUrl":"10.1016/j.trmi.2017.03.001","url":null,"abstract":"<div><p>It is well known that to each summable in the <span><math><mi>n</mi></math></span>-dimensional cube <span><math><msup><mrow><mrow><mo>[</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span> function <span><math><mi>f</mi></math></span> of variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> there corresponds one <span><math><mi>n</mi></math></span>-multiple trigonometric Fourier series <span><math><mi>S</mi><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></math></span> with constant coefficients.</p><p>In the present paper, with the function <span><math><mi>f</mi></math></span> we associate <span><math><mi>n</mi></math></span> one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></math></span>, with respect to variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively, with nonconstant coefficients and announce the preliminary results. In particular, if a continuous function <span><math><mi>f</mi></math></span> is differentiable at some point <span><math><mi>x</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span>, then all one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></math></span> converge at <span><math><mi>x</mi></math></span> to the value <span><math><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>.</p><p>For illustration we consider the well known example of Ch. Fefferman’s function <span><math><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math></span> whose double trigonometric Fourier series <span><math><mi>S</mi><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></math></span> diverges everywhere in the sense of Prinsheim. Namely, we establish the simultaneous convergence of the one-dimensional Fourier series <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>S</mi><msub><mrow><mrow><mo>[</mo><mi>F</mi><mo>]</mo></mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span> at almost all points <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 2","pages":"Pages 167-170"},"PeriodicalIF":0.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.03.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47727646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01Epub Date: 2017-03-18DOI: 10.1016/j.trmi.2017.01.001
Nawel Alaya , Moncef Dziri
In this work, we consider a generalized system of partial differential operators, we define the related Fourier transform and establish some harmonic analysis results. We also investigate a wide class of integral transforms of Riemann–Liouville type. In particular we give a good estimate of these integrals kernels, inversion formula and a Plancherel theorem for the dual.
{"title":"Harmonic analysis and integral transforms associated with a class of a system of partial differential operators","authors":"Nawel Alaya , Moncef Dziri","doi":"10.1016/j.trmi.2017.01.001","DOIUrl":"10.1016/j.trmi.2017.01.001","url":null,"abstract":"<div><p>In this work, we consider a generalized system of partial differential operators, we define the related Fourier transform and establish some harmonic analysis results. We also investigate a wide class of integral transforms of Riemann–Liouville type. In particular we give a good estimate of these integrals kernels, inversion formula and a Plancherel theorem for the dual.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 2","pages":"Pages 111-130"},"PeriodicalIF":0.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.01.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49042019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01Epub Date: 2017-01-30DOI: 10.1016/j.trmi.2017.01.002
Abdullah Shoaib , Muhammd Arshad , Tahair Rasham , Mujahid Abbas
The aim of this paper is to introduce the new concept of ordered complete dislocated quasi -metric space. The notion of dominated mappings is applied to approximate the unique solution of non linear functional equations. In this paper, we find the fixed point results for mappings satisfying the locally contractive conditions on a closed ball in an ordered complete dislocated quasi -metric space. Our results improve several well known classical results.
{"title":"Unique fixed point results on closed ball for dislocated quasi G-metric spaces","authors":"Abdullah Shoaib , Muhammd Arshad , Tahair Rasham , Mujahid Abbas","doi":"10.1016/j.trmi.2017.01.002","DOIUrl":"10.1016/j.trmi.2017.01.002","url":null,"abstract":"<div><p>The aim of this paper is to introduce the new concept of ordered complete dislocated quasi <span><math><mi>G</mi></math></span>-metric space. The notion of dominated mappings is applied to approximate the unique solution of non linear functional equations. In this paper, we find the fixed point results for mappings satisfying the locally contractive conditions on a closed ball in an ordered complete dislocated quasi <span><math><mi>G</mi></math></span>-metric space. Our results improve several well known classical results.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 2","pages":"Pages 221-230"},"PeriodicalIF":0.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.01.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48841761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01Epub Date: 2017-03-14DOI: 10.1016/j.trmi.2017.02.003
Duglas Ugulava
Jackson’s type theorem on approximation of square integrable functions is proved for functions defined on homogeneous spaces with a compact transitive transformation group actions. An example is proved which illustrates the theorem.
{"title":"Approximation in mean on homogeneous compact spaces","authors":"Duglas Ugulava","doi":"10.1016/j.trmi.2017.02.003","DOIUrl":"10.1016/j.trmi.2017.02.003","url":null,"abstract":"<div><p>Jackson’s type theorem on approximation of square integrable functions is proved for functions defined on homogeneous spaces with a compact transitive transformation group actions. An example is proved which illustrates the theorem.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 2","pages":"Pages 231-237"},"PeriodicalIF":0.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.02.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43178745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-01Epub Date: 2017-01-30DOI: 10.1016/j.trmi.2016.12.004
L. Ephremidze , W.H. Gerstacker , I. Spitkovsky
An elementary proof of Robinson’s Energy Delay Theorem on minimum-phase functions is provided. The situation in which the energy conservation property holds for an infinite number of lags is fully described.
{"title":"On Robinson’s Energy Delay Theorem","authors":"L. Ephremidze , W.H. Gerstacker , I. Spitkovsky","doi":"10.1016/j.trmi.2016.12.004","DOIUrl":"10.1016/j.trmi.2016.12.004","url":null,"abstract":"<div><p>An elementary proof of Robinson’s Energy Delay Theorem on minimum-phase functions is provided. The situation in which the energy conservation property holds for an infinite number of lags is fully described.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 1","pages":"Pages 16-23"},"PeriodicalIF":0.2,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.12.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44838067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-01Epub Date: 2017-02-15DOI: 10.1016/j.trmi.2017.01.003
S. Basu , D. Sen
Here we give abstract formulations of some generalized versions of the classical Vitali theorem on Lebesgue nonmeasurable sets which are due to Kharazishvili and Solecki.
{"title":"Abstract formulations of some theorems on nonmeasurable sets","authors":"S. Basu , D. Sen","doi":"10.1016/j.trmi.2017.01.003","DOIUrl":"10.1016/j.trmi.2017.01.003","url":null,"abstract":"<div><p>Here we give abstract formulations of some generalized versions of the classical Vitali theorem on Lebesgue nonmeasurable sets which are due to Kharazishvili and Solecki.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 1","pages":"Pages 10-15"},"PeriodicalIF":0.2,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.01.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41752856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-01Epub Date: 2016-11-16DOI: 10.1016/j.trmi.2016.10.004
Sayed Saber
In this paper we study the Sobolev regularity of the Bergman projection and the -Neumann operator on a certain pseudoconvex domain. We show that if is a domain with Lipschitz boundary, which is relatively compact in an -dimensional compact Kähler manifold and satisfies some “-pseudoconvexity” condition, the operators , and are regular in the Sobolev spaces for forms with values in a holomorphic vector bundle and for any , , , .
{"title":"Sobolev regularity of the Bergman projection on certain pseudoconvex domains","authors":"Sayed Saber","doi":"10.1016/j.trmi.2016.10.004","DOIUrl":"10.1016/j.trmi.2016.10.004","url":null,"abstract":"<div><p>In this paper we study the Sobolev regularity of the Bergman projection <span><math><mi>B</mi></math></span> and the <span><math><mover><mrow><mi>∂</mi></mrow><mo>¯</mo></mover></math></span>-Neumann operator <span><math><mi>N</mi></math></span> on a certain pseudoconvex domain. We show that if <span><math><mi>Ω</mi></math></span> is a domain with Lipschitz boundary, which is relatively compact in an <span><math><mi>n</mi></math></span>-dimensional compact Kähler manifold and satisfies some “<span><math><mo>log</mo><mspace></mspace><mi>δ</mi></math></span>-pseudoconvexity” condition, the operators <span><math><mi>B</mi></math></span>, <span><math><mi>N</mi></math></span> and <span><math><msup><mrow><mover><mrow><mi>∂</mi></mrow><mo>¯</mo></mover></mrow><mrow><mo>∗</mo></mrow></msup><mi>N</mi></math></span> are regular in the Sobolev spaces <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span> for forms with values in a holomorphic vector bundle <span><math><mi>E</mi></math></span> and for any <span><math><mi>k</mi><mo><</mo><mi>η</mi><mo>/</mo><mn>2</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>η</mi><mo><</mo><mn>1</mn></math></span>, <span><math><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mi>n</mi></math></span>, <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 1","pages":"Pages 90-102"},"PeriodicalIF":0.2,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.10.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42588369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-01Epub Date: 2017-01-27DOI: 10.1016/j.trmi.2016.12.005
Garnik Karapetyan, Mikael Arakelian
In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) [4], [5]. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) [6], [7], [8], [9], [10]. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) [11], [12]).
在完全正多面体具有任意数目的各向异性顶点的情况下，本文研究了三维情况下多各向异性Sobolev空间的函数的积分表示和嵌入定理。这项工作概括了Karapetyan(出版中)和Karapetyan(2016)中获得的结果。特别是，在Karapetyan (in press)中，二维情况得到了完全解决，在Karapetyan(2016)中，对于一个各向异性顶点的情况得到了类似的结果。这个问题的根源在于Sobolev(1938)和Sobolev(0000)[4]，[5]。许多作者都得到了相关的结果，如Besov et al.(1967)、Reshetnyak(1971)、Smith(1961)、Nikolsky(0000)和Il 'in(1967)[6]、[7]、[8]、[9]、[10]。专著(Besov, 1978)包含了对这个问题的概述。本文得到的结果是基于准齐次多项式对正则化的推广(见Uspenskii(1972)和Karapetyan(1990)[11]，[12])。
{"title":"Estimation of multianisotropic kernels and their application to the embedding theorems","authors":"Garnik Karapetyan, Mikael Arakelian","doi":"10.1016/j.trmi.2016.12.005","DOIUrl":"10.1016/j.trmi.2016.12.005","url":null,"abstract":"<div><p>In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) <span>[4]</span>, <span>[5]</span>. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) <span>[6]</span>, <span>[7]</span>, <span>[8]</span>, <span>[9]</span>, <span>[10]</span>. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) <span>[11]</span>, <span>[12]</span>).</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 1","pages":"Pages 48-56"},"PeriodicalIF":0.2,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.12.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55644192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Dirichlet generalized harmonic problem for finite right circular cylindrical domains is considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. It is shown that if a finite domain is bounded by several surfaces and the curves are placed in arbitrary form, then the generalized problem has a unique solution depending continuously on the data. The problem is considered for the simple case when the curves of discontinuity are circles with centers situated on the axis of the cylinder. An algorithm of numerical solution by a probabilistic method is given, which in its turn is based on a computer simulation of the Wiener process. A numerical example is considered to illustrate the effectiveness and simplicity of the proposed method.
{"title":"Investigation and numerical solution of some 3D internal Dirichlet generalized harmonic problems in finite domains","authors":"Mamuli Zakradze , Murman Kublashvili , Zaza Sanikidze , Nana Koblishvili","doi":"10.1016/j.trmi.2016.11.001","DOIUrl":"10.1016/j.trmi.2016.11.001","url":null,"abstract":"<div><p>A Dirichlet generalized harmonic problem for finite right circular cylindrical domains is considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. It is shown that if a finite domain is bounded by several surfaces and the curves are placed in arbitrary form, then the generalized problem has a unique solution depending continuously on the data. The problem is considered for the simple case when the curves of discontinuity are circles with centers situated on the axis of the cylinder. An algorithm of numerical solution by a probabilistic method is given, which in its turn is based on a computer simulation of the Wiener process. A numerical example is considered to illustrate the effectiveness and simplicity of the proposed method.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 1","pages":"Pages 103-110"},"PeriodicalIF":0.2,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2016.11.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41551985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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