Pub Date : 2024-11-08DOI: 10.1016/j.jfa.2024.110746
Luke D. Edholm , Yonatan Shelah
New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in . Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.
{"title":"The Leray transform: Distinguished measures, symmetries and polygamma inequalities","authors":"Luke D. Edholm , Yonatan Shelah","doi":"10.1016/j.jfa.2024.110746","DOIUrl":"10.1016/j.jfa.2024.110746","url":null,"abstract":"<div><div>New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110746"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-08DOI: 10.1016/j.jfa.2024.110745
Vicente Asensio , Enrique Jordá , Thomas Kalmes
<div><div>We characterize those pairs <span><math><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> of smooth mappings <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>C</mi><mo>,</mo><mi>φ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for which the corresponding weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>ψ</mi><mo>⋅</mo><mo>(</mo><mi>f</mi><mo>∘</mo><mi>φ</mi><mo>)</mo></math></span> acts continuously on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>ψ</mi><mo>,</mo><mi>φ</mi></math></span>. Among other things, as an application of our results we show that for a univariate polynomial <em>φ</em> with <span><math><mtext>deg</mtext><mo>(</mo><mi>φ</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> only depends on <em>φ</em> and that in this case power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> is equivalent to <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converging to 0 in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> as well as to the uniform mean ergodicity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for which neither the multiplication operator <span><math><mi>f</mi><mo>↦</mo><mi>ψ</mi><mi>f</mi></math></span> nor the composition
{"title":"Power boundedness and related properties for weighted composition operators on S(Rd)","authors":"Vicente Asensio , Enrique Jordá , Thomas Kalmes","doi":"10.1016/j.jfa.2024.110745","DOIUrl":"10.1016/j.jfa.2024.110745","url":null,"abstract":"<div><div>We characterize those pairs <span><math><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> of smooth mappings <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>C</mi><mo>,</mo><mi>φ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for which the corresponding weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>ψ</mi><mo>⋅</mo><mo>(</mo><mi>f</mi><mo>∘</mo><mi>φ</mi><mo>)</mo></math></span> acts continuously on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>ψ</mi><mo>,</mo><mi>φ</mi></math></span>. Among other things, as an application of our results we show that for a univariate polynomial <em>φ</em> with <span><math><mtext>deg</mtext><mo>(</mo><mi>φ</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> only depends on <em>φ</em> and that in this case power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> is equivalent to <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converging to 0 in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> as well as to the uniform mean ergodicity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for which neither the multiplication operator <span><math><mi>f</mi><mo>↦</mo><mi>ψ</mi><mi>f</mi></math></span> nor the composition","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110745"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-07DOI: 10.1016/j.jfa.2024.110743
Jean-Philippe Anker , Bartosz Trojan
We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.
在二面群的情况下，我们建立了邓克尔核的尖锐上下估计值。
{"title":"Optimal bounds for the Dunkl kernel in the dihedral case","authors":"Jean-Philippe Anker , Bartosz Trojan","doi":"10.1016/j.jfa.2024.110743","DOIUrl":"10.1016/j.jfa.2024.110743","url":null,"abstract":"<div><div>We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110743"},"PeriodicalIF":1.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659808","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 : 2024-11-07DOI: 10.1016/j.jfa.2024.110744
Thomas Tony
A closed connected oriented Riemannian manifold N with non-vanishing Euler characteristic, non-negative curvature operator and is area-rigid in the sense that any area non-increasing spin map with non-vanishing -degree and is a Riemannian submersion with . This is due to Goette and Semmelmann and generalizes a result by Llarull. In this article, we show area-rigidity for not necessarily orientable manifolds with respect to a larger class of maps by replacing the topological condition on the -degree by a less restrictive condition involving the so-called higher mapping degree. This includes fiber bundles over even dimensional spheres with enlargeable fibers, e.g. . We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost -harmonic sections. This also leads to a new proof of the fact that any closed connected spin manifold with non-negative scalar curvature and non-trivial Rosenberg index is Ricci flat.
{"title":"Scalar curvature rigidity and the higher mapping degree","authors":"Thomas Tony","doi":"10.1016/j.jfa.2024.110744","DOIUrl":"10.1016/j.jfa.2024.110744","url":null,"abstract":"<div><div>A closed connected oriented Riemannian manifold <em>N</em> with non-vanishing Euler characteristic, non-negative curvature operator and <span><math><mn>0</mn><mo><</mo><mn>2</mn><msub><mrow><mi>Ric</mi></mrow><mrow><mi>N</mi></mrow></msub><mo><</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> is area-rigid in the sense that any area non-increasing spin map <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> with non-vanishing <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-degree and <span><math><msub><mrow><mi>scal</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>≥</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span> is a Riemannian submersion with <span><math><msub><mrow><mi>scal</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>=</mo><msub><mrow><mi>scal</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span>. This is due to Goette and Semmelmann and generalizes a result by Llarull. In this article, we show area-rigidity for not necessarily orientable manifolds with respect to a larger class of maps <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> by replacing the topological condition on the <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-degree by a less restrictive condition involving the so-called higher mapping degree. This includes fiber bundles over even dimensional spheres with enlargeable fibers, e.g. <span><math><msub><mrow><mi>pr</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost <figure><img></figure>-harmonic sections. This also leads to a new proof of the fact that any closed connected spin manifold with non-negative scalar curvature and non-trivial Rosenberg index is Ricci flat.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110744"},"PeriodicalIF":1.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.jfa.2024.110740
Victor Wu
We introduce -algebras associated to directed graphs of groups. In particular, we associate a combinatorial -algebra to each row-finite directed graph of groups with no sources, and show that this -algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these -algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.
{"title":"C⁎-algebras associated to directed graphs of groups, and models of Kirchberg algebras","authors":"Victor Wu","doi":"10.1016/j.jfa.2024.110740","DOIUrl":"10.1016/j.jfa.2024.110740","url":null,"abstract":"<div><div>We introduce <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras associated to directed graphs of groups. In particular, we associate a combinatorial <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra to each row-finite directed graph of groups with no sources, and show that this <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110740"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.jfa.2024.110739
Joan Bosa , Eduard Vilalta
We introduce and study the notion of pureness for *-homomorphisms and, more generally, for cpc order-zero maps. After providing various important examples of pureness, we show our main result: Any composition of two pure maps factors through a pure object up to Cuntz equivalence. This is used to obtain several factorization results at the level of -algebras.
{"title":"Pure ⁎-homomorphisms","authors":"Joan Bosa , Eduard Vilalta","doi":"10.1016/j.jfa.2024.110739","DOIUrl":"10.1016/j.jfa.2024.110739","url":null,"abstract":"<div><div>We introduce and study the notion of pureness for *-homomorphisms and, more generally, for cpc order-zero maps. After providing various important examples of pureness, we show our main result: Any composition of two pure maps factors through a pure object up to Cuntz equivalence. This is used to obtain several factorization results at the level of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110739"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.jfa.2024.110733
Philipp Grohs , Lukas Liehr , Martin Rathmair
<div><div>Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function <em>f</em> from its spectrogram, i.e., the magnitudes of its short-time Fourier transform <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi></math></span> with window function <em>g</em>. While it is known that for appropriate windows, any function <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> can be reconstructed from the full spectrogram <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo></math></span>, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>∈</mo><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> is determined up to a global phase from<span><span><span><math><mrow><mo>(</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>)</mo></mrow></math></span></span></span> whenever <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> satisfies the density condition <span><math><mo>|</mo><mi>det</mi><mo></mo><mi>A</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≥</mo><mn>4</mn></math></span>. For real
{"title":"Multi-window STFT phase retrieval: Lattice uniqueness","authors":"Philipp Grohs , Lukas Liehr , Martin Rathmair","doi":"10.1016/j.jfa.2024.110733","DOIUrl":"10.1016/j.jfa.2024.110733","url":null,"abstract":"<div><div>Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function <em>f</em> from its spectrogram, i.e., the magnitudes of its short-time Fourier transform <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi></math></span> with window function <em>g</em>. While it is known that for appropriate windows, any function <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> can be reconstructed from the full spectrogram <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>f</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo></math></span>, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>∈</mo><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> is determined up to a global phase from<span><span><span><math><mrow><mo>(</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mi>f</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>)</mo></mrow></math></span></span></span> whenever <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> satisfies the density condition <span><math><mo>|</mo><mi>det</mi><mo></mo><mi>A</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≥</mo><mn>4</mn></math></span>. For real","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110733"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659892","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 : 2024-11-06DOI: 10.1016/j.jfa.2024.110735
Yemon Choi , Mahya Ghandehari , Hung Le Pham
The AMNM property for commutative Banach algebras is a form of Ulam stability for multiplicative linear functionals. We show that on any semilattice of infinite breadth, one may construct a weight for which the resulting weighted convolution algebra fails to have the AMNM property. Our work is the culmination of a trilogy started in [4] and continued in [5]. In particular, we obtain a refinement of the main result of [5], by establishing a dichotomy for union-closed set systems that has a Ramsey-theoretic flavour.
{"title":"Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth","authors":"Yemon Choi , Mahya Ghandehari , Hung Le Pham","doi":"10.1016/j.jfa.2024.110735","DOIUrl":"10.1016/j.jfa.2024.110735","url":null,"abstract":"<div><div>The AMNM property for commutative Banach algebras is a form of Ulam stability for multiplicative linear functionals. We show that on any semilattice of infinite breadth, one may construct a weight for which the resulting weighted convolution algebra fails to have the AMNM property. Our work is the culmination of a trilogy started in <span><span>[4]</span></span> and continued in <span><span>[5]</span></span>. In particular, we obtain a refinement of the main result of <span><span>[5]</span></span>, by establishing a dichotomy for union-closed set systems that has a Ramsey-theoretic flavour.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110735"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.jfa.2024.110736
Jiayu Li , Xiangrong Zhu
Suppose that u is a map from to a compact smooth Riemannian manifold N with bounded energy. We show that there exists a constant which depends only on N and such that if the tension field τ belongs to Zygmund class , then the Hopf differential of u belongs to the Zygmund class and the norm depends only on and . As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence with bounded energy and bounded in .
假设 u 是一个从 D8 到紧凑光滑黎曼流形 N 的有界能的映射。我们将证明存在一个常数 λ>；0，使得如果张力场τ属于齐格蒙类 LlnλL(D8)，那么u 的霍普夫微分属于齐格蒙类 Lln3L(D1)，且规范‖h‖Lln3L(D1)只取决于 N、E(u,D8) 和‖τ‖LlnλL(D8)。作为直接推论，我们得到了在 LlnλL(D8) 中具有有界能量 E(un) 和有界 τ(un) 的炸裂序列 un 的能量同一性和无颈性。
{"title":"Approximated harmonic maps with tension fields in Zygmund class","authors":"Jiayu Li , Xiangrong Zhu","doi":"10.1016/j.jfa.2024.110736","DOIUrl":"10.1016/j.jfa.2024.110736","url":null,"abstract":"<div><div>Suppose that <em>u</em> is a map from <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span> to a compact smooth Riemannian manifold <em>N</em> with bounded energy. We show that there exists a constant <span><math><mi>λ</mi><mo>></mo><mn>0</mn></math></span> which depends only on <em>N</em> and <span><math><mi>E</mi><mo>(</mo><mi>u</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> such that if the tension field <em>τ</em> belongs to Zygmund class <span><math><mi>L</mi><msup><mrow><mi>ln</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo></mo><mi>L</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span>, then the Hopf differential of <em>u</em> belongs to the Zygmund class <span><math><mi>L</mi><msup><mrow><mi>ln</mi></mrow><mrow><mn>3</mn></mrow></msup><mo></mo><mi>L</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> and the norm <span><math><msub><mrow><mo>‖</mo><mi>h</mi><mo>‖</mo></mrow><mrow><mi>L</mi><msup><mrow><mi>ln</mi></mrow><mrow><mn>3</mn></mrow></msup><mo></mo><mi>L</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msub></math></span> depends only on <span><math><mi>N</mi><mo>,</mo><mi>E</mi><mo>(</mo><mi>u</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mo>‖</mo><mi>τ</mi><mo>‖</mo></mrow><mrow><mi>L</mi><msup><mrow><mi>ln</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo></mo><mi>L</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></mrow></msub></math></span>. As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with bounded energy <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> and bounded <span><math><mi>τ</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> in <span><math><mi>L</mi><msup><mrow><mi>ln</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo></mo><mi>L</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110736"},"PeriodicalIF":1.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659887","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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