This paper studies the relations between extendability of different classes of Sobolev and BV functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak -Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected BV-extension sets are always also -extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.
{"title":"Closed BV-extension and W1,1-extension sets","authors":"Emanuele Caputo , Jesse Koivu , Danka Lučić , Tapio Rajala","doi":"10.1016/j.jfa.2025.111319","DOIUrl":"10.1016/j.jfa.2025.111319","url":null,"abstract":"<div><div>This paper studies the relations between extendability of different classes of Sobolev <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> and <em>BV</em> functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected <em>BV</em>-extension sets are always also <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111319"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to and to be suitably dominated.
We also prove that the solution that we find converges, as , to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in and therefore the usual regularity theory cannot be leveraged to our benefit in this framework.
The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as , every classical operator in divergence form.
{"title":"Nonlocal operators in divergence form and existence theory for integrable data","authors":"David Arcoya , Serena Dipierro , Edoardo Proietti Lippi , Caterina Sportelli , Enrico Valdinoci","doi":"10.1016/j.jfa.2025.111317","DOIUrl":"10.1016/j.jfa.2025.111317","url":null,"abstract":"<div><div>We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> and to be suitably dominated.</div><div>We also prove that the solution that we find converges, as <span><math><mi>s</mi><mo>↗</mo><mn>1</mn></math></span>, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> and therefore the usual regularity theory cannot be leveraged to our benefit in this framework.</div><div>The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as <span><math><mi>s</mi><mo>↗</mo><mn>1</mn></math></span>, every classical operator in divergence form.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111317"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2025-12-22DOI: 10.1016/j.jfa.2025.111321
Anh Xuan Do , Nguyen Lam , Guozhen Lu
Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension . Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in [27]. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincaré type inequality with Gaussian measure.
{"title":"Sharp stability of the Heisenberg Uncertainty Principle: Second-order and curl-free field cases","authors":"Anh Xuan Do , Nguyen Lam , Guozhen Lu","doi":"10.1016/j.jfa.2025.111321","DOIUrl":"10.1016/j.jfa.2025.111321","url":null,"abstract":"<div><div>Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>. Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in <span><span>[27]</span></span>. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincaré type inequality with Gaussian measure.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111321"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-09DOI: 10.1016/j.jfa.2026.111343
Camillo Brena
We construct two sequences of closed 4-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by 1, and volume bounded from below by , with different fundamental groups but with the same Gromov–Hausdorff limit. This provides a negative answer to the question posed in J. Pan (2025) [9].
构造了两个非负Ricci曲率的封闭四维流形序列，它们具有不同的基本群，但具有相同的Gromov-Hausdorff极限，其直径上界为1，体积下界为v>；0。这为J. Pan (2025) b[9]提出的问题提供了一个否定的答案。
{"title":"Instability of the fundamental group for non-collapsed Ricci-limits","authors":"Camillo Brena","doi":"10.1016/j.jfa.2026.111343","DOIUrl":"10.1016/j.jfa.2026.111343","url":null,"abstract":"<div><div>We construct two sequences of closed 4-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by 1, and volume bounded from below by <span><math><mi>v</mi><mo>></mo><mn>0</mn></math></span>, with different fundamental groups but with the same Gromov–Hausdorff limit. This provides a negative answer to the question posed in J. Pan (2025) <span><span>[9]</span></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111343"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-05DOI: 10.1016/j.jfa.2025.111336
Xilun Li, Yanan Ye
In this note, we construct a series of examples to show that various nonnegative curvature conditions, including Riemannian curvature and Bismut curvature, are not preserved by the generalized Ricci flow.
{"title":"Nonnegativity of curvature along generalized Ricci flow","authors":"Xilun Li, Yanan Ye","doi":"10.1016/j.jfa.2025.111336","DOIUrl":"10.1016/j.jfa.2025.111336","url":null,"abstract":"<div><div>In this note, we construct a series of examples to show that various nonnegative curvature conditions, including Riemannian curvature and Bismut curvature, are not preserved by the generalized Ricci flow.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111336"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145922958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2025-12-23DOI: 10.1016/j.jfa.2025.111320
Alexander Elgart , Abel Klein
It is shown that the infinite random Heisenberg XXZ spin- chain exhibits localization phenomena, such as spectral, eigenstate, and weak dynamical localization, in an arbitrary (but fixed) energy interval in a non-trivial region of the parameter space. This region depends only on the energy interval and includes weak interaction and strong disorder regimes. The crucial step in the argument is a proof that if the Green functions for the associated finite systems Hamiltonians exhibit certain (volume-dependent) decay properties in a fixed energy interval, then the infinite volume Green function decays in the same interval as well. The pertinent finite systems decay properties for the random XXZ spin chain had been previously verified by the authors.
{"title":"Localization phenomena in the random XXZ spin chain","authors":"Alexander Elgart , Abel Klein","doi":"10.1016/j.jfa.2025.111320","DOIUrl":"10.1016/j.jfa.2025.111320","url":null,"abstract":"<div><div>It is shown that the infinite random Heisenberg XXZ spin-<span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> chain exhibits localization phenomena, such as spectral, eigenstate, and weak dynamical localization, in an arbitrary (but fixed) energy interval in a non-trivial region of the parameter space. This region depends only on the energy interval and includes weak interaction and strong disorder regimes. The crucial step in the argument is a proof that if the Green functions for the associated finite systems Hamiltonians exhibit certain (volume-dependent) decay properties in a fixed energy interval, then the infinite volume Green function decays in the same interval as well. The pertinent finite systems decay properties for the random XXZ spin chain had been previously verified by the authors.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111320"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-05DOI: 10.1016/j.jfa.2025.111334
Miriam Gordin
We present vector-valued concentration inequalities for the biased measure on the discrete cube with an optimal dependence on the bias parameter and the Rademacher type of the target Banach space. These results allow us to obtain novel vector-valued concentration inequalities for the measure given by a product of Poisson distributions. We further obtain lower bounds on the average distortion with respect to the biased measure of embeddings of the hypercube into Banach spaces of nontrivial type which imply average non-embeddability.
{"title":"Vector-valued concentration inequalities on the biased discrete cube","authors":"Miriam Gordin","doi":"10.1016/j.jfa.2025.111334","DOIUrl":"10.1016/j.jfa.2025.111334","url":null,"abstract":"<div><div>We present vector-valued concentration inequalities for the biased measure on the discrete cube <span><math><msup><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> with an optimal dependence on the bias parameter and the Rademacher type of the target Banach space. These results allow us to obtain novel vector-valued concentration inequalities for the measure given by a product of Poisson distributions. We further obtain lower bounds on the average distortion with respect to the biased measure of embeddings of the hypercube into Banach spaces of nontrivial type which imply average non-embeddability.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111334"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145922959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-06DOI: 10.1016/j.jfa.2025.111335
Jonathan Ditlevsen , Quentin Labriet
In this article we study differential symmetry breaking operators between principal series representations induced from minimal parabolic subgroups for the pair . Using the source operator philosophy we construct such operators for generic induction parameters of the representations and establish that this approach yields all possible operators in this setting. We show that these differential operators occur as residues of a family of symmetry breaking operators that depends meromorphically on the parameters. Finally, in the case we classify and construct all differential symmetry breaking operators for any parameters, including the non-generic ones.
{"title":"Differential symmetry breaking operators for the pair (GLn+1(R),GLn(R))","authors":"Jonathan Ditlevsen , Quentin Labriet","doi":"10.1016/j.jfa.2025.111335","DOIUrl":"10.1016/j.jfa.2025.111335","url":null,"abstract":"<div><div>In this article we study differential symmetry breaking operators between principal series representations induced from minimal parabolic subgroups for the pair <span><math><mo>(</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span>. Using the source operator philosophy we construct such operators for generic induction parameters of the representations and establish that this approach yields all possible operators in this setting. We show that these differential operators occur as residues of a family of symmetry breaking operators that depends meromorphically on the parameters. Finally, in the <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> case we classify and construct all differential symmetry breaking operators for any parameters, including the non-generic ones.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111335"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145922960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2025-12-19DOI: 10.1016/j.jfa.2025.111324
Bartosz Malman
The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a space, the closure of analytic polynomials in a Lebesgue space corresponding to a measure of the form , with being the standard weighted area measure on , dm the Lebesgue measure on the unit circle , and w a general weight on . We characterize when of this form is a space of analytic functions on by computing the Thomson decomposition of the measure μ. The structure of the decomposition is expressed in terms of what we call the family of associated Beurling-Carleson sets. We characterize the cyclic singular inner functions in the analytic spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols which generate a de Branges-Rovnyak space with a dense subset of functions smooth on . The characterization is given explicitly in terms of the modulus of b on and the singular measure corresponding to the singular inner factor of b. Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained -optimization tools.
{"title":"Weighted Korenblum-Roberts theory","authors":"Bartosz Malman","doi":"10.1016/j.jfa.2025.111324","DOIUrl":"10.1016/j.jfa.2025.111324","url":null,"abstract":"<div><div>The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk <span><math><mi>D</mi></math></span> as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> space, the closure of analytic polynomials in a Lebesgue space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> corresponding to a measure of the form <span><math><mi>d</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>+</mo><mi>w</mi><mspace></mspace><mi>d</mi><mtext>m</mtext></math></span>, with <span><math><mi>d</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> being the standard weighted area measure on <span><math><mi>D</mi></math></span>, <em>dm</em> the Lebesgue measure on the unit circle <span><math><mi>T</mi></math></span>, and <em>w</em> a general weight on <span><math><mi>T</mi></math></span>. We characterize when <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> of this form is a space of analytic functions on <span><math><mi>D</mi></math></span> by computing the Thomson decomposition of the measure <em>μ</em>. The structure of the decomposition is expressed in terms of what we call the family of <em>associated Beurling-Carleson sets</em>. We characterize the cyclic singular inner functions in the analytic <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols <span><math><mi>b</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>D</mi></math></span> which generate a de Branges-Rovnyak space with a dense subset of functions smooth on <span><math><mi>T</mi></math></span>. The characterization is given explicitly in terms of the modulus of <em>b</em> on <span><math><mi>T</mi></math></span> and the singular measure corresponding to the singular inner factor of <em>b</em>. Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-optimization tools.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111324"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Schauder frames of discrete translates in L2(R)","authors":"Nir Lev , Anton Tselishchev","doi":"10.1016/j.jfa.2025.111318","DOIUrl":"10.1016/j.jfa.2025.111318","url":null,"abstract":"<div><div>We construct a uniformly discrete sequence <span><math><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo>}</mo><mo>⊂</mo><mi>R</mi></math></span> and functions <em>g</em> and <span><math><mo>{</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>}</mo></math></span> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, such that every <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> admits a series expansion<span><span><span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mo>〈</mo><mi>f</mi><mo>,</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>〉</mo><mspace></mspace><mi>g</mi><mo>(</mo><mi>x</mi><mo>−</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span></span></span> convergent in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> norm.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111318"},"PeriodicalIF":1.6,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145845559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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