×

On spectra and spectral eigenmatrix problems of the planar Sierpinski measures. (English) Zbl 1487.42060

Summary: A Sierpinski-type measure \(\mu\) is a self-similar measure which satisfies that \[ \mu(E)=\frac{1}{3}\mu(3qE)+\frac{1}{3}\mu(3qE-e_1)+\frac{1}{3}\mu(3qE-e_2), \] for any Borel set \(E\subset\mathbb{R}^2\), where \(q\ge1\) is an integer, and \(\{e_1,e_2\}\) is a basis of \(\mathbb{R}^2\). Another form of expression for Sierpinski-type measures \(\mu\) is the infinite convolution \[ \mu=\delta_{(3q)^{-1}\{0,e_1,e_2\}}*\delta_{(3q)^{-1}\{0,e_1,e_2\}}*\delta_{(3q)^{-1}\{0,e_1,e_2\}}*\cdots, \] where \(\delta_E=(1/\#E)\sum_{e\in E}\delta_e\) is the linear combination of standard Dirac measures with equality weight and the convergence is in a weak sense. Let \[ E_{\Lambda}=\{e^{-2\pi i\langle\lambda, x\rangle}: \lambda\in\Lambda\} \] be a family of exponentials in \(\mathbb{R}^2\) for any set \(\Lambda\subset\mathbb{R}^2\). This paper gives a characterization for \(E_{\Lambda}\) to be a maximal orthogonal family in \(L^2(\mu)\). As its applications, some sufficient conditions are obtained for a maximal orthogonal family \(E_{\Lambda}\) to be or not to be an orthogonal basis of \(L^2(\mu)\). Moreover, several simple criteria are given for a matrix \(R\in M_2(\mathbb{R})\) so that there exists an orthogonal basis \(E_{\Lambda}\) satisfying that the family \(E_{R\Lambda}\) is also an orthogonal basis of \(L^2(\mu)\).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
28A80 Fractals
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A65 Completeness of sets of functions in one variable harmonic analysis
42A85 Convolution, factorization for one variable harmonic analysis
60A10 Probabilistic measure theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. X. AN and X. G. HE, A class of spectral Moran measures, J. Funct. Anal. 266 (2014), no. 1, 343-354. http://dx.doi.org/10.1016/j.jfa.2013.08.031. MR3121733 · Zbl 1303.28009
[2] L. X. AN, X. G. HE, and K. S. LAU, Spectrality of a class of infinite convolutions, Adv. Math. 283 (2015), 362-376. http://dx.doi.org/10.1016/j.aim.2015.07.021. MR3383806 · Zbl 1323.28007
[3] L. X. AN, X. G. HE, and L. TAO, Spectrality of the planar Sierpinski family, J. Math. Anal. Appl. 432 (2015), no. 2, 725-732. http://dx.doi.org/10.1016/j.jmaa.2015.06. 064. MR3378387 · Zbl 1364.42030
[4] X. R. DAI, When does a Bernoulli convolution admit a spectrum?, Adv. Math. 231 (2012), no. 3-4, 1681-1693. http://dx.doi.org/10.1016/j.aim.2012.06.026. MR2964620 · Zbl 1303.28009
[5] , Spectra of Cantor measures, Math. Ann. 366 (2016), no. 3-4, 1621-1647. http://dx. doi.org/10.1007/s00208-016-1374-5. MR3563247 · Zbl 1323.28007
[6] X. R. DAI, X. G. HE, and C. K. LAI, Spectral property of Cantor measures with consecu-tive digits, Adv. Math. 242 (2013), 187-208. http://dx.doi.org/10.1016/j.aim.2013.04. 016. MR3055992 · Zbl 1364.42030
[7] X. R. DAI, X. G. HE, and K. S. LAU, On spectral N-Bernoulli measures, Adv. Math. 259 (2014), 511-531. http://dx.doi.org/10.1016/j.aim.2014.03.026. MR3197665 · Zbl 1266.42012
[8] X. H. DONG and K. S. LAU, Cauchy transforms of self-similar measures: The Laurent coefficients, J. Funct. Anal. 202 (2003), no. 1, 67-97. http://dx.doi.org/10.1016/ S0022-1236(02)00069-1 . MR1994765 · Zbl 1352.42009
[9] , An integral related to the Cauchy transform on the Sierpinski gasket, Experi-ment. Math. 13 (2004), no. 4, 415-419. http://dx.doi.org/10.1080/10586458.2004. 10504549. MR2118265 · Zbl 1277.28009
[10] X. H. DONG, K. S. LAU, and J. C. LIU, Cantor boundary behavior of analytic func-tions, Adv. Math. 232 (2013), 543-570. http://dx.doi.org/10.1016/j.aim.2012.09. 021. MR2989993 · Zbl 1032.28005
[11] X. H. DONG, K. S. LAU, and H. H. WU, Cauchy transforms of self-similar measures: Starlikeness and univalence, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4817-4842. http://dx.doi.org/ 10.1090/tran/6819. MR3632551 · Zbl 1059.28006
[12] D. E. DUTKAY, D. HAN, and Q. SUN, On the spectra of a Cantor measure, Adv. Math. 221 (2009), no. 1, 251-276. http://dx.doi.org/10.1016/j.aim.2008.12.007. MR2509326 · Zbl 1272.30009
[13] , Divergence of the mock and scrambled Fourier series on fractal measures, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2191-2208. · Zbl 1272.30009
[14] D. E. DUTKAY and J. HAUSSERMANN, Number theory problems from the harmonic analysis of a fractal, J. Number Theory 159 (2016), 7-26. http://dx.doi.org/10.1016/j.jnt.2015.07. 009. MR3412709 · Zbl 1456.28005
[15] D. E. DUTKAY, J. HAUSSERMANN, and C. K. LAI, Hadamard triples generate self-affine spectral measures, Trans. Amer. Math. Soc. 371 (2019), no. 2, 1439-1481. http://dx.doi.org/10. 1090/tran/7325. MR3885185 · Zbl 1406.42008
[16] D. E. DUTKAY and P. E. T. JORGENSEN, Analysis of orthogonality and of orbits in affine iter-ated function systems, Math. Z. 256 (2007), no. 4, 801-823. http://dx.doi.org/10.1007/ s00209-007-0104-9. MR2308892 · Zbl 1129.28006
[17] , Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (2007), no. 1, 110-137. http://dx.doi.org/10.1016/j.jfa.2007.03.002. MR2319756 · Zbl 1128.42013
[18] , Fourier series on fractals: A parallel with wavelet theory, Radon Transforms, Geometry, and Wavelets, Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 75-101. http://dx.doi.org/10.1090/conm/464/09077. MR2440130
[19] , Fourier duality for fractal measures with affine scales, Math. Comp. 81 (2012), no. 280, 2253-2273. http://dx.doi.org/10.1090/S0025-5718-2012-02580-4 . MR2945155 · Zbl 1129.28006
[20] D. E. DUTKAY and C. K. LAI, Uniformity of measures with Fourier frames, Adv. Math. 252 (2014), 684-707. http://dx.doi.org/10.1016/j.aim.2013.11.012. MR3144246 · Zbl 1128.42013
[21] K. FALCONER, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990. MR1102677 · Zbl 1256.42048
[22] B. FUGLEDE, Commuting self-adjoint partial differential operators and a group theo-retic problem, J. Functional Analysis 16 (1974), 101-121. http://dx.doi.org/10.1016/ 0022-1236(74)90072-x. MR0470754 · Zbl 1277.47033
[23] R. GREENFELD and N. LEV, Fuglede’s spectral set conjecture for convex polytopes, Anal. PDE 10 (2017), no. 6, 1497-1538. http://dx.doi.org/10.2140/apde.2017.10.1497. MR3678495 · Zbl 1369.28008
[24] X. G. HE, C. K. LAI, and K. S. LAU, Exponential spectra in L 2 (µ), Appl. Comput. Har-mon. Anal. 34 (2013), no. 3, 327-338. http://dx.doi.org/10.1016/j.acha.2012.05. 003. MR3027906 · Zbl 1264.42010
[25] X. G. HE, M. W. TANG, and Z. Y. WU, Spectral structure and spectral eigenvalue problems of a class of self-similar spectral measures, J. Funct. Anal. 277 (2019), no. 10, 3688-3722. http://dx. doi.org/10.1016/j.jfa.2019.05.019. MR4001085 · Zbl 1377.42014
[26] P. E. T. JORGENSEN, K. A. KORNELSON, and K. L. SHUMAN, Families of spectral sets for Bernoulli convolutions, J. Fourier Anal. Appl. 17 (2011), no. 3, 431-456. http://dx.doi.org/ 10.1007/s00041-010-9158-x. MR2803943 · Zbl 1264.42010
[27] P. E. T. JORGENSEN and S. PEDERSEN, Dense analytic subspaces in fractal L 2 -spaces, J. Anal. Math. 75 (1998), 185-228. http://dx.doi.org/10.1007/BF02788699. MR1655831 · Zbl 0959.28008
[28] J. KIGAMI, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge Univer-sity Press, Cambridge, 2001. http://dx.doi.org/10.1017/CBO9780511470943. MR1840042 · Zbl 1440.42027
[29] I. ŁABA and Y. WANG, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409-420. http://dx.doi.org/10.1006/jfan.2001.3941. MR1929508 · Zbl 1235.28008
[30] H. J. LANDAU, Necessary density conditions for sampling and interpolation of certain entire func-tions, Acta Math. 117 (1967), 37-52. http://dx.doi.org/10.1007/BF02395039. MR222554 · Zbl 0959.28008
[31] J. P. LUND, R. S. STRICHARTZ, and J. P. VINSON, Cauchy transforms of self-similar measures, Experiment. Math. 7 (1998), no. 3, 177-190. http://dx.doi.org/10.1080/10586458.1998. 10504368. MR1676691 · Zbl 0959.28006
[32] P. MATTILA, Fourier Analysis and Hausdorff Dimension, Cambridge Studies in Advanced Math-ematics, vol. 150, Cambridge University Press, Cambridge, 2015. http://dx.doi.org/10. 1017/CBO9781316227619. MR3617376 · Zbl 1016.28009
[33] R. S. STRICHARTZ, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000), 209-238. http://dx.doi.org/10.1007/BF02788990. MR1785282 · Zbl 0959.28006
[34] , Convergence of mock Fourier series, J. Anal. Math. 99 (2006), 333-353. http://dx. doi.org/10.1007/BF02789451. MR2279556 · Zbl 0959.28006
[35] , Differential Equations on Fractals: A tutorial, Princeton University Press, Princeton, NJ, 2006. MR2246975 · Zbl 1332.28001
[36] , Piecewise linear wavelets on Sierpinski gasket type fractals, J. Fourier Anal. Appl. 3 (1997), no. 4, 387-416. http://dx.doi.org/10.1007/BF02649103. MR1468371 · Zbl 0976.42020
[37] T. TAO, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251-258. http://dx.doi.org/10.4310/MRL.2004.v11.n2.a8. MR2067470 · Zbl 1134.42308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.