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• W3201916090 abstract "Abstract In this work, we study the spectral property of a class of self-affine measures μ M , D on <?CDATA ${mathbb{R}}^{2}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> generated by an expanding real matrix <?CDATA $M=mathrm{diag}left({rho }_{1}^{-1},{rho }_{2}^{-1}right)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>diag</mml:mi> <mml:mfenced close=) open=(> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:mfenced> </mml:math> and a non-collinear integer digit set <?CDATA $D=left{{(0,0)}^{t},{({alpha }_{1},{alpha }_{2})}^{t},{({beta }_{1},{beta }_{2})}^{t}right}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy=false>{</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>}</mml:mo> </mml:mrow> </mml:math> with <?CDATA ${alpha }_{i}-2{beta }_{i}notin 3mathbb{Z}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∉</mml:mo> <mml:mn>3</mml:mn> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:math> , i = 1, 2. We give the sufficient and necessary conditions so that μ M , D becomes a spectral measure, i.e., there exists a countable subset <?CDATA ${Lambda}subset {mathbb{R}}^{2}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi mathvariant=normal>Λ</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> such that { e 2 πi ⟨ λ , x ⟩ : λ ∈ Λ} forms an orthonormal basis for L 2 ( μ M , D ). This extends the results of Dai, Fu and Yan (2021 Appl. Comput. Harmon. Anal. 52 63–81) and Deng and Lau (2015 J. Funct. Anal. 269 1310–1326)." @default.
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• W3201916090 title "Spectrality of a class of self-affine measures on R2 *" @default.
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