Fourier bases on self-similar measures*

作者：Yin, Feng-Li; He, Xing-Gang^*; Zhang, Min-Min^*

来源：Chaos, Solitons and Fractals, 2023, 173: 113723.

DOI：10.1016/j.chaos.2023.113723

• Fourier bases
• Hadamard triple
• Self-similar measures
• Sierpinski measures
• Spectral measures

Let l ??????l > 1 be a real number and let ??????be a 2 x 2 orthogonal and involutory matrix. Let R = {0, ??????, ??????} & SUB; Z2 such that det(??????, ??????) # 0 and 3 does not divide det(??????, ??????). In this paper, we prove that the Hilbert space ??????2(??????????????????,R) generated by the general self-similar Sierpinski measure ??????????????????,R has an orthonormal basis of the form ????????????= {??????2 ????????????(??????,??????) & COLRATIO; ??????& ISIN; ??????} with ?????? & SUB; R2, i.e., ??????????????????,R is a spectral measure, if and only if there exist three integers ??????, ??????, ??????such that I/1 3 ?????? 3 ??????) ??????2 divide (??????2 + ??????2) and ????????????= ?????? ?????? in the standard expression. @@@ This provides new and different 3 ???????????? -3 ?????? ?????? characterization of spectrality of self-similar measures in the plane and gives a counterexample to Laba-Wang's conjecture in two dimension.