Pub Date : 2026-02-01Epub Date: 2025-01-25DOI: 10.1016/j.exmath.2025.125655
A. Bendikov , L. Saloff-Coste
We discuss the -analyticity (and lack thereof) for Gaussian symmetric Markov semigroups on compact connected groups in finite and infinite dimension. In the infinite dimensional case, a positive result gives some sufficient conditions for quantitative -differentiability.
{"title":"On the L1 differentiability of certain Markov semigroups","authors":"A. Bendikov , L. Saloff-Coste","doi":"10.1016/j.exmath.2025.125655","DOIUrl":"10.1016/j.exmath.2025.125655","url":null,"abstract":"<div><div>We discuss the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-analyticity (and lack thereof) for Gaussian symmetric Markov semigroups on compact connected groups in finite and infinite dimension. In the infinite dimensional case, a positive result gives some sufficient conditions for quantitative <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-differentiability.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125655"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395331","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 : 2026-02-01Epub Date: 2025-04-02DOI: 10.1016/j.exmath.2025.125683
Mohamed El Kadiri
We define the fine Laplacian of a finely superharmonic function on a regular (or more generally a Borel) finely open set , , as a Borel measure on . If is a Euclidean open set, then is the classical Laplacian of in the distributional sense. Moreover, if is finely of class on , then , where () are the fine partial derivatives of order 2 of .
{"title":"On the Laplacian of a finely superharmonic function","authors":"Mohamed El Kadiri","doi":"10.1016/j.exmath.2025.125683","DOIUrl":"10.1016/j.exmath.2025.125683","url":null,"abstract":"<div><div>We define the fine Laplacian <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi></mrow></math></span> of a finely superharmonic function <span><math><mi>u</mi></math></span> on a regular (or more generally a Borel) finely open set <span><math><mrow><mi>U</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, as a Borel measure on <span><math><mi>U</mi></math></span>. If <span><math><mi>U</mi></math></span> is a Euclidean open set, then <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi></mrow></math></span> is the classical Laplacian <span><math><mrow><mi>Δ</mi><mi>u</mi></mrow></math></span> of <span><math><mi>u</mi></math></span> in the distributional sense. Moreover, if <span><math><mi>u</mi></math></span> is finely of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> on <span><math><mi>U</mi></math></span>, then <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mfrac><mrow><msup><mrow><mi>∂</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mi>∂</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></math></span>, where <span><math><mfrac><mrow><msup><mrow><mi>∂</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mi>∂</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></math></span>\u0000 (<span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></math></span>) are the fine partial derivatives of order 2 of <span><math><mi>u</mi></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125683"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395334","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 : 2026-02-01Epub Date: 2025-02-18DOI: 10.1016/j.exmath.2025.125663
E.E. Berdysheva , M.D. Ramabulana , Sz.Gy. Révész
A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Turán extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.
{"title":"On extremal problems of Delsarte type for positive definite functions on LCA groups","authors":"E.E. Berdysheva , M.D. Ramabulana , Sz.Gy. Révész","doi":"10.1016/j.exmath.2025.125663","DOIUrl":"10.1016/j.exmath.2025.125663","url":null,"abstract":"<div><div>A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Turán extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125663"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395332","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 : 2026-02-01Epub Date: 2024-11-09DOI: 10.1016/j.exmath.2024.125629
Mihail N. Kolountzakis
In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.
{"title":"Orthogonal Fourier Analysis on domains","authors":"Mihail N. Kolountzakis","doi":"10.1016/j.exmath.2024.125629","DOIUrl":"10.1016/j.exmath.2024.125629","url":null,"abstract":"<div><div>In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125629"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395458","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 : 2026-02-01Epub Date: 2024-10-18DOI: 10.1016/j.exmath.2024.125622
Stephen J. Gardiner, Hermann Render
This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.
{"title":"Harmonic extension through cylindrical and conical surfaces","authors":"Stephen J. Gardiner, Hermann Render","doi":"10.1016/j.exmath.2024.125622","DOIUrl":"10.1016/j.exmath.2024.125622","url":null,"abstract":"<div><div>This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125622"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395459","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 : 2026-02-01Epub Date: 2024-12-07DOI: 10.1016/j.exmath.2024.125636
Gergely Kiss , Itay Londner , Máté Matolcsi , Gábor Somlai
The notion of weak tiling was a key ingredient in the proof of Fuglede’s spectral set conjecture for convex bodies, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised by Kolountzakis, Lev and Matolcsi, by giving an example of a set which tiles its complement weakly, but is neither spectral, nor a proper tile.
{"title":"A lonely weak tile","authors":"Gergely Kiss , Itay Londner , Máté Matolcsi , Gábor Somlai","doi":"10.1016/j.exmath.2024.125636","DOIUrl":"10.1016/j.exmath.2024.125636","url":null,"abstract":"<div><div>The notion of weak tiling was a key ingredient in the proof of Fuglede’s spectral set conjecture for convex bodies, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised by Kolountzakis, Lev and Matolcsi, by giving an example of a set <span><math><mi>T</mi></math></span> which tiles its complement weakly, but <span><math><mi>T</mi></math></span> is neither spectral, nor a proper tile.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125636"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395454","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 : 2026-02-01Epub Date: 2024-10-19DOI: 10.1016/j.exmath.2024.125620
Jaume de Dios Pont , Jan Grebík , Rachel Greenfeld , José Madrid
The periodic tiling conjecture asserts that if a region tiles by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in , and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.
{"title":"Periodicity and decidability of translational tilings by rational polygonal sets","authors":"Jaume de Dios Pont , Jan Grebík , Rachel Greenfeld , José Madrid","doi":"10.1016/j.exmath.2024.125620","DOIUrl":"10.1016/j.exmath.2024.125620","url":null,"abstract":"<div><div>The periodic tiling conjecture asserts that if a region <span><math><mrow><mi>Σ</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span> tiles <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> by translations then it admits at least one <em>fully periodic</em> tiling. This conjecture is known to hold in <span><math><mi>R</mi></math></span>, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for <em>polygonal sets</em>: bounded open sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that <em>any</em> translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125620"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395463","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 : 2026-02-01Epub Date: 2025-11-29DOI: 10.1016/j.exmath.2025.125745
Marcin Bownik , Ziemowit Rzeszotnik , Darrin Speegle
Larson’s problem Larson (2007, Problem 3) asks “Must the support of the Fourier transform of a wavelet contain a wavelet set?”. We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson’s problem and propose two new related problems.
{"title":"Are MSF wavelets minimally supported?","authors":"Marcin Bownik , Ziemowit Rzeszotnik , Darrin Speegle","doi":"10.1016/j.exmath.2025.125745","DOIUrl":"10.1016/j.exmath.2025.125745","url":null,"abstract":"<div><div>Larson’s problem Larson (2007, Problem 3) asks “Must the support of the Fourier transform of a wavelet contain a wavelet set?”. We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson’s problem and propose two new related problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125745"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145645748","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}
We establish upper and lower universal bounds for potentials of weighted designs on the sphere that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to -frame energy.
{"title":"Bounds on energy and potentials of discrete measures on the sphere","authors":"S.V. Borodachov , P.G. Boyvalenkov , P.D. Dragnev , D.P. Hardin , E.B. Saff , M.M. Stoyanova","doi":"10.1016/j.exmath.2025.125712","DOIUrl":"10.1016/j.exmath.2025.125712","url":null,"abstract":"<div><div>We establish upper and lower universal bounds for potentials of weighted designs on the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to <span><math><mi>p</mi></math></span>-frame energy.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125712"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395330","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 : 2026-02-01Epub Date: 2024-08-02DOI: 10.1016/j.exmath.2024.125601
Christian Berg
Let for . We prove that is a complete Bernstein function for and a Stieltjes function for . This answers a conjecture of David Bradley that is a Bernstein function when .
{"title":"A complete Bernstein function related to the fractal dimension of Pascal’s pyramid modulo a prime","authors":"Christian Berg","doi":"10.1016/j.exmath.2024.125601","DOIUrl":"10.1016/j.exmath.2024.125601","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mi>x</mi><mo>)</mo></mrow><mo>/</mo><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math></span>. We prove that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is a complete Bernstein function for <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn></mrow></math></span> and a Stieltjes function for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>r</mi></mrow></math></span>. This answers a conjecture of David Bradley that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is a Bernstein function when <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125601"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930603","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}
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