Expositiones Mathematicae最新文献

{"title":"Inverse limits of CM points on certain Shimura varieties","authors":"Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin","doi":"10.1016/j.exmath.2026.125754","DOIUrl":"10.1016/j.exmath.2026.125754","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span> be a positive integer, and let <span><math><mrow><mi>D</mi><mo>≡</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mtext>mod</mtext><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span> be a negative integer. We define the sets <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> as subsets of the Shimura varieties <span><math><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span>, respectively, consisting of CM points of discriminant <span><math><mi>D</mi></math></span> that are primitive modulo <span><math><mi>N</mi></math></span>. By using the theory of definite form class groups, we show that the inverse limits <span><span><span><math><mrow><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span></span></span>naturally inherit group structures isomorphic to <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mo>/</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>∞</mi></mrow></msup><mo>)</mo></mrow><mo>/</mo><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, where <span><math><mrow><mi>K</mi><mo>=</mo><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mi>D</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>t</mi></math></span> is a transcendental number. These resu","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 2","pages":"Article 125754"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077601","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":"The Kaczmarz Algorithm in Hilbert C∗-modules","authors":"Daniel Alpay ,&nbsp;Chad Berner ,&nbsp;Eric S. Weber","doi":"10.1016/j.exmath.2025.125738","DOIUrl":"10.1016/j.exmath.2025.125738","url":null,"abstract":"<div><div>The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 2","pages":"Article 125738"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658446","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":"Special martingale transforms, queen beds, and the Ahlfors–Beurling operator","authors":"Oliver Dragičević ,&nbsp;Alexander Volberg","doi":"10.1016/j.exmath.2025.125714","DOIUrl":"10.1016/j.exmath.2025.125714","url":null,"abstract":"<div><div>In Dragičević et al. (2003, 2006) the Ahlfors–Beurling operator <span><math><mi>T</mi></math></span> and its powers <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> were represented as averages of two-dimensional martingale transforms. Our intention is to optimize the parameters of the averaging for the estimate of <span><math><msub><mrow><mo>‖</mo><mi>T</mi><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> from above. The second goal of ours is to give an estimate of <span><math><msub><mrow><mo>‖</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> from below.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125714"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395471","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":"The stochastic heat equation with fractional time and fractional time–space white noise","authors":"Rahma Yasmina Moulay Hachemi ,&nbsp;Bernt Øksendal","doi":"10.1016/j.exmath.2024.125625","DOIUrl":"10.1016/j.exmath.2024.125625","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time–space white noise, in the following two cases:&lt;/div&gt;&lt;div&gt;(i) With additive fractional time–space white noise&lt;/div&gt;&lt;div&gt;(ii) With multiplicative time–space Brownian noise&lt;/div&gt;&lt;div&gt;The fractional time derivative is interpreted as the Caputo derivative of order &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and we assume that the Hurst coefficient &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of the time–space fractional white noise is in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i).&lt;/div&gt;&lt;div&gt;In the multiplicative case (ii) we show that there is a unique solution in the Hida space &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; of stochastic distributions and we show that the solution coincides with the solution of an associated fractional stochastic Volterra equation. Then we give an explicit expression for the solution of this Volterra equation. See Appendix below or Di Nunno et al. (2008), Holden et al. (2010) and Øksendal (2023) for more information about Hida spaces and white noise theory.&lt;/div&gt;&lt;div&gt;A solution &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is called &lt;em&gt;mild&lt;/em&gt; if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. For both the additive noise case and the multiplicative noise case we show that if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; then the solution is mild if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, while if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; the solution is not mild for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research pap","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125625"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395462","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":"Bent Fuglede (8.10.1925–7.12.2023)","authors":"","doi":"10.1016/j.exmath.2025.125719","DOIUrl":"10.1016/j.exmath.2025.125719","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125719"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395457","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":"Frame vector group representations and amenability properties","authors":"Dorin Ervin Dutkay ,&nbsp;Catalin Georgescu ,&nbsp;Gabriel Picioroaga","doi":"10.1016/j.exmath.2025.125743","DOIUrl":"10.1016/j.exmath.2025.125743","url":null,"abstract":"<div><div>We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call <em>framenable</em>. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mrow><mtext>Aut</mtext><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mtext>Aut</mtext><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, all (countable) lattices of <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, the Baumslag–Solitar groups <span><math><mrow><mi>B</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow></math></span>, the braid groups <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and Thompson’s group <span><math><mi>F</mi></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125743"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694512","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":"The space of tracial states on a C∗-algebra","authors":"Bruce Blackadar ,&nbsp;Mikael Rørdam","doi":"10.1016/j.exmath.2024.125618","DOIUrl":"10.1016/j.exmath.2024.125618","url":null,"abstract":"<div><div>We give a simple and elementary proof that the tracial state space of a unital <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-algebra is a Choquet simplex, using the center-valued trace on a finite von Neumann algebra.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125618"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395335","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":"Generalizations of Weiss’ version of the Fuglede–Putnam theorem for unbounded operators","authors":"Mohammed Hichem Mortad","doi":"10.1016/j.exmath.2024.125606","DOIUrl":"10.1016/j.exmath.2024.125606","url":null,"abstract":"<div><div>The present paper discusses Weiss’ version of the Fuglede–Putnam theorem. We establish three generalizations of this version, each of which involves unbounded normal operators. The first generalization expands upon Weiss’ theorem by incorporating a single unbounded normal operator, while the second involves normal powers. The third generalization deals with two unbounded normal operators. These generalizations have several applications and are accompanied by counterexamples.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125606"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395460","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":"Fuglede’s conjecture, differential operators and unitary groups of local translations","authors":"Piyali Chakraborty ,&nbsp;Dorin Ervin Dutkay ,&nbsp;Palle E.T. Jorgensen","doi":"10.1016/j.exmath.2025.125662","DOIUrl":"10.1016/j.exmath.2025.125662","url":null,"abstract":"<div><div>The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125662"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395472","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":"Kernel-algorithms in frame-approximations","authors":"Palle E.T. Jorgensen ,&nbsp;Myung-Sin Song ,&nbsp;James Tian","doi":"10.1016/j.exmath.2024.125583","DOIUrl":"10.1016/j.exmath.2024.125583","url":null,"abstract":"<div><div>With view to applications, we present here general classes of non-orthogonal expansions in Hilbert space. The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our work includes a diverse list of optimization problems, new Karhunen–Loève transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. For this we also make use of specific reproducing kernel Hilbert spaces, kernel factorizations, and finite-dimensional approximations. Our projection algorithms are designed with view to maximum likelihood solutions, minimization of “cost” problems, identification of principal components, and data-dimension reduction.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125583"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141134319","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}