权力扩张系统{𝑓(𝑧^{𝑘})}_{𝑘∈ℕ}在Dirichlet-type空间中

Pub Date : 2023-06-07 DOI:10.1090/spmj/1762

H. Dan, K. Guo

{"title":"权力扩张系统{𝑓(𝑧^{𝑘})}_{𝑘∈ℕ}在Dirichlet-type空间中","authors":"H. Dan, K. Guo","doi":"10.1090/spmj/1762","DOIUrl":null,"url":null,"abstract":"<p>Power dilation systems <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace f left-parenthesis z Superscript k Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>z</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">N</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{f(z^k)\\}_{k\\in \\mathbb {N}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in Dirichlet-type spaces <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D Subscript t Baseline left-parenthesis t element-of double-struck upper R right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mtext> </mml:mtext>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {D}_t\\ (t\\in \\mathbb {R})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are treated. When <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t not-equals 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>t</mml:mi>\n <mml:mo>≠</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">t\\neq 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, it is proved that a system of functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace f left-parenthesis z Superscript k Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>z</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">N</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{f(z^k)\\}_{k\\in \\mathbb {N}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is orthogonal in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D Subscript t\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {D}_t</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> only if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f equals c z Superscript upper N\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>c</mml:mi>\n <mml:msup>\n <mml:mi>z</mml:mi>\n <mml:mi>N</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f=cz^N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for some constant <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c\">\n <mml:semantics>\n <mml:mi>c</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">c</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and some positive integer <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\n <mml:semantics>\n <mml:mi>N</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Power dilation systems {𝑓(𝑧^{𝑘})}_{𝑘∈ℕ} in Dirichlet-type spaces\",\"authors\":\"H. Dan, K. Guo\",\"doi\":\"10.1090/spmj/1762\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Power dilation systems <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace f left-parenthesis z Superscript k Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>z</mml:mi>\\n <mml:mi>k</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>∈</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">N</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{f(z^k)\\\\}_{k\\\\in \\\\mathbb {N}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in Dirichlet-type spaces <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper D Subscript t Baseline left-parenthesis t element-of double-struck upper R right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">D</mml:mi>\\n </mml:mrow>\\n <mml:mi>t</mml:mi>\\n </mml:msub>\\n <mml:mtext> </mml:mtext>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {D}_t\\\\ (t\\\\in \\\\mathbb {R})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> are treated. When <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"t not-equals 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>t</mml:mi>\\n <mml:mo>≠</mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">t\\\\neq 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, it is proved that a system of functions <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace f left-parenthesis z Superscript k Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>z</mml:mi>\\n <mml:mi>k</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>∈</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">N</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{f(z^k)\\\\}_{k\\\\in \\\\mathbb {N}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is orthogonal in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper D Subscript t\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">D</mml:mi>\\n </mml:mrow>\\n <mml:mi>t</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {D}_t</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> only if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"f equals c z Superscript upper N\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>f</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mi>c</mml:mi>\\n <mml:msup>\\n <mml:mi>z</mml:mi>\\n <mml:mi>N</mml:mi>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">f=cz^N</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for some constant <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"c\\\">\\n <mml:semantics>\\n <mml:mi>c</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">c</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and some positive integer <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N\\\">\\n <mml:semantics>\\n <mml:mi>N</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">N</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-06-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1762\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1762","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

对dirichlet型空间中幂展开式系统{f(z k)} k∈N \{f(z k)\}_{k\in \mathbb {N}}中的dt (t∈R)\ mathcal {D}_t\ (t\in \mathbb {R})进行处理。当t≠0 t\neq 0时，证明了一个函数系统{f(z k)} k∈N \{f(z k)\}_{k\in \mathbb {N}}在dt \mathbb {D}_t中是正交的，只有当f=cz N f=cz^N对于某个常数c c和某个正整数N N。给出了狄利克雷型空间的幂膨胀系统所形成的无条件基和框架的完备刻画。最后，将这些结果应用于dirichlet型空间上矩问题的算符理论情形。

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Power dilation systems {𝑓(𝑧^{𝑘})}_{𝑘∈ℕ} in Dirichlet-type spaces

Power dilation systems { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} in Dirichlet-type spaces D t ( t ∈ R ) \mathcal {D}_t\ (t\in \mathbb {R}) are treated. When t ≠ 0 t\neq 0 , it is proved that a system of functions { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} is orthogonal in D t \mathcal {D}_t only if f = c z N f=cz^N for some constant c c and some positive integer N N . Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.

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