Spectrality of the planar Sierpinski family

2015 
Abstract Let μ be a Borel probability measure with compact support in R 2 . μ is called a spectral measure if there exists a countable set Λ ⊂ R 2 such that E Λ = { e − 2 π i 〈 λ , x 〉 : λ ∈ Λ } is an orthonormal basis for L 2 ( μ ) . In this note we prove that the integral Sierpinski measure μ A , D is a spectral measure if and only if ( A , D ) is admissible. This completely settles the spectrality of integral Sierpinski measures in R 2 .
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