Spectrality of infinite convolutions with three-element digit sets

Bibliographic Details
Authors and Corporations: Fu, Yan-Song
Title: Spectrality of infinite convolutions with three-element digit sets
In: Monatshefte für Mathematik, 183, 2017, 3, p. 465-485
published:
Springer Vienna
Physical Description:465-485
ISSN/ISBN: 0026-9255
1436-5081
EISSN:1436-5081
Summary:Let <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq2.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$0&lt; \rho &lt;1$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mn>0</mn><mo>&lt;</mo><mi mathvariant="italic">ρ</mi><mo>&lt;</mo><mn>1</mn></mrow></math> </EquationSource></InlineEquation> and let <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq3.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\{a_n, b_n\}_{n=1}^\infty $$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><msubsup><mrow><mo stretchy="false">{</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math> </EquationSource></InlineEquation> be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq4.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\mu _{\rho , \{0, a_n, b_n\}}$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><msub><mi mathvariant="italic">μ</mi><mrow><mi mathvariant="italic">ρ</mi><mo>,</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow></msub></math> </EquationSource></InlineEquation> generated by the following infinite convolution product <Equation ID="Equ30"><MediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_Equ30.gif" Color="BlackWhite"/></MediaObject><EquationSource Format="TEX">$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$</EquationSource><EquationSource Format="MATHML"> <math display="block" xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mtable columnspacing="0.5ex"><mtr><mtd columnalign="right"><mrow><msub><mi mathvariant="italic">μ</mi><mrow><mi mathvariant="italic">ρ</mi><mo>,</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow></msub><mo>=</mo><msub><mi mathvariant="italic">δ</mi><mrow><mi mathvariant="italic">ρ</mi><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">}</mo></mrow></msub><mrow/><mo>∗</mo><msub><mi mathvariant="italic">δ</mi><mrow><msup><mi mathvariant="italic">ρ</mi><mn>2</mn></msup><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow></mrow></msub><mrow/><mo>∗</mo><msub><mi mathvariant="italic">δ</mi><mrow><msup><mi mathvariant="italic">ρ</mi><mn>3</mn></msup><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>b</mi><mn>3</mn></msub><mo stretchy="false">}</mo></mrow></mrow></msub><mrow/><mo>∗</mo><mo>⋯</mo></mrow></mtd></mtr></mtable></mrow></math> </EquationSource></Equation>in the weak convergence, where <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq5.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><msub><mi mathvariant="italic">δ</mi><mi>E</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mo>#</mo><mi>E</mi></mrow></mfrac><msub><mo>∑</mo><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></msub><msub><mi mathvariant="italic">δ</mi><mi>e</mi></msub></mrow></math> </EquationSource></InlineEquation> and <InlineEquation ID="IEq6"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq6.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\hbox {gcd}(a_n, b_n)=1$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mtext>gcd</mtext><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></math> </EquationSource></InlineEquation> for all <InlineEquation ID="IEq7"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq7.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$n \in {{\mathbb {N}}}$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></math> </EquationSource></InlineEquation>. In this paper, we show that <InlineEquation ID="IEq8"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq8.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$L^2(\mu _{\rho , \{0, a_n, b_n\}})$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="italic">μ</mi><mrow><mi mathvariant="italic">ρ</mi><mo>,</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></math> </EquationSource></InlineEquation> admits an exponential orthonormal basis if and only if <InlineEquation ID="IEq9"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq9.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\rho ^{-1} \in 3{{\mathbb {N}}}$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><msup><mi mathvariant="italic">ρ</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>∈</mo><mn>3</mn><mi mathvariant="double-struck">N</mi></mrow></math> </EquationSource></InlineEquation> and  <InlineEquation ID="IEq10"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq10.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mrow><mo stretchy="false">{</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><mo>≡</mo><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><mspace width="4pt"/><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">mod</mi><mspace width="4pt"/><mn>3</mn><mo stretchy="false">)</mo></mrow></mrow></math> </EquationSource></InlineEquation> for all <InlineEquation ID="IEq11"><InlineMediaObject><ImageObject Type="Linedraw" Rendition="HTML" Format="GIF" FileRef="605_2017_1026_Article_IEq11.gif" Color="BlackWhite"/></InlineMediaObject><EquationSource Format="TEX">$$n \in {{\mathbb {N}}}$$</EquationSource><EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></math> </EquationSource></InlineEquation>.
Type of Resource:E-Article
Source:Springer Journals
Language: English