分形与框架的相关问题研究
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摘要
分形与框架的相关问题是分形几何中有趣而重要的课题.本论文研究了分形级数,刻画了Fourier框架与Tilings的一种关系,并且利用迭代函数系统构造了一类广义连续框架.
本论文总共有三部分.
第一部分我们描述了一种自然的方法利用d-维欧氏空间中的绝对收敛级数来构造分形集,并对这类集合的结构特点作出了一些描述.然后,对一维直线上的情况,我们给出两个条件分别得到了区间和齐次Moran集.最后,对高维的情况,我们给出了一个充分条件来计算具有重叠结构的集合的Hausdorff维数,并且给出了一些例子.
第二部分我们首先给出了一个可分Hilbert空间的广义连续框架的概念.在得出广义连续框架的一些基本结果之后,我们利用迭代函数系统、概率函数以及窗口函数构造了一类广义连续框架.
论文的最后一部分介绍了谱集猜想.在研究谱集猜想的过程中,Jorgensen和Pedersen提出了一个猜想.受此驱动,Lagarias、Reeds和Wang得到了谱与Tilings的一个关系刻画,而且结合Keller标准就能证明Jorgensen和Pedersen猜想.Iosevich、Pedersen和Kolountzakis利用另外的技巧也证明了Jorgensen和Pedersen猜想.在本文中,我们考虑Fourier框架并得到了Fourier框架与Tilings的一个关系,由此刻画了谱与Tilings的一个更一般的关系.
The related problems of fractals and frames are both important and interesting in fractal geometry. In this thesis, we study the fractal series, obtain some characterizations of Fourier frames and tilings, and construct a class of generalized continuous frames by using iterated function systems.
This thesis contains three parts.
In the first part of this thesis, we describe a natural way to associate fractals to a certain class of absolutely convergent series in Rd, and portray the structures of the sets. In the case for d= 1, we give two sufficient conditions for the set associated to series being an interval or a homogeneous Moran set, respectively. Finally, for d≥1, we give a sufficient condition to calculate the Hausdorff dimension of some fractals with overlapping, and present some examples.
In the second part of this thesis, we give the definition of a generalized continuous frame for a separable Hilbert space. For such a generalization, after obtaining some basic results about these frames, we construct a class of generalized continuous frames by using iterated function systems, probability functions and window functions.
In the last part of this thesis, we introduce the spectral set conjecture. In their investi-gation and study of the conjecture, Jorgensen and Pedersen give a conjecture. Motivated by the conjecture, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove the conjecture of Jorgensen and Pedersen by Keller's criterion. Different techniques to prove these facts have also been developed by Iosevich, Pedersen and Kolountzakis. In this part, we present an elementary approach to give some characterizations of Fourier frames and tilings, and obtain a more general form of spectra and tilings.
本论文总共有三部分.
第一部分我们描述了一种自然的方法利用d-维欧氏空间中的绝对收敛级数来构造分形集,并对这类集合的结构特点作出了一些描述.然后,对一维直线上的情况,我们给出两个条件分别得到了区间和齐次Moran集.最后,对高维的情况,我们给出了一个充分条件来计算具有重叠结构的集合的Hausdorff维数,并且给出了一些例子.
第二部分我们首先给出了一个可分Hilbert空间的广义连续框架的概念.在得出广义连续框架的一些基本结果之后,我们利用迭代函数系统、概率函数以及窗口函数构造了一类广义连续框架.
论文的最后一部分介绍了谱集猜想.在研究谱集猜想的过程中,Jorgensen和Pedersen提出了一个猜想.受此驱动,Lagarias、Reeds和Wang得到了谱与Tilings的一个关系刻画,而且结合Keller标准就能证明Jorgensen和Pedersen猜想.Iosevich、Pedersen和Kolountzakis利用另外的技巧也证明了Jorgensen和Pedersen猜想.在本文中,我们考虑Fourier框架并得到了Fourier框架与Tilings的一个关系,由此刻画了谱与Tilings的一个更一般的关系.
The related problems of fractals and frames are both important and interesting in fractal geometry. In this thesis, we study the fractal series, obtain some characterizations of Fourier frames and tilings, and construct a class of generalized continuous frames by using iterated function systems.
This thesis contains three parts.
In the first part of this thesis, we describe a natural way to associate fractals to a certain class of absolutely convergent series in Rd, and portray the structures of the sets. In the case for d= 1, we give two sufficient conditions for the set associated to series being an interval or a homogeneous Moran set, respectively. Finally, for d≥1, we give a sufficient condition to calculate the Hausdorff dimension of some fractals with overlapping, and present some examples.
In the second part of this thesis, we give the definition of a generalized continuous frame for a separable Hilbert space. For such a generalization, after obtaining some basic results about these frames, we construct a class of generalized continuous frames by using iterated function systems, probability functions and window functions.
In the last part of this thesis, we introduce the spectral set conjecture. In their investi-gation and study of the conjecture, Jorgensen and Pedersen give a conjecture. Motivated by the conjecture, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove the conjecture of Jorgensen and Pedersen by Keller's criterion. Different techniques to prove these facts have also been developed by Iosevich, Pedersen and Kolountzakis. In this part, we present an elementary approach to give some characterizations of Fourier frames and tilings, and obtain a more general form of spectra and tilings.
引文
[1]Ali S T, Antoine J P, Gazeau J P. Coherent States, Wavelets and Their Generalizations. New York: Springer-Verlag,2000.
[2]Ali S T, Antoine J P, Gazeau J P. Continuous frames in Hilbert spaces. Annals of Physics,1993, 222:1-37.
[3]Askari-Hemmat A, Dehghan M A, Radjabalipour M. Generalized frames and their redundancy. Proc. Amer. Math. Soc.,2001,129(4):1143-1147.
[4]Christensen O. An Introduction to Frames and Riesz Bases. Boston:Birkhauser,2002.
[5]Casazza P G. Modern tools for Weyl-Heisenberg (Gabor) frame theory. Adv. Imag. Elect. Phs., 2001,115:1-27.
[6]Chui Ch K. An Introduction to Wavelets. New York:Academic Press,1992.
[7]Daubechies I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory,1990,36:961-1005.
[8]Daubechies I. The Lectures on Wavelets. SIAM. Vol.61,1992.
[9]Daubechies I, Grossman A, Meyer Y. Painless nonorthogonal expansions. J. Mate. Phys.,1986,27: 1271-1283.
[10]Duffin R J, Schaeffer A C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc.,1952, 72:147-158.
[11]Daubechies I. The wavelet transformation, time-frequency localization and signal analysis. IEEE Trans. Infor. Theory,1990,36:961-1005.
[12]Edgar G A. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics, New York:Springer-Verlag,1990.
[13]Edgar G A. Integral, Probability, and Fractal Measures. New York:Springer-Verlag,1998.
[14]Falconer K J. The Geometry of Fractal Sets. Cambridge:Cambridge University Press,1985.
[15]Falconer K J. Fractal Geometry:Mathematical Foundations and Applications. John Wiley & Sons, 1990.
[16]Falconer K J. Techniques in Fractal Geometry. Chichester:John Wiley and Sons, Ltd.,1997.
[17]Federer H. Geometry Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, New York:Springer-Verlag,1969.
[18]Foranasier M, Rauhut H. Continuous frames, function spaces, and the discretization problem. J. Four. Anal. Appl.,2005,113:245-286.
[19]Feng D J, Wen Z Y, Wu J. Some dimensional results for homogeneous Moran sets. Science of China (Series A),1997,40(5):475-482.
[20]Feng D J, Rao H, Wu J. The net measure properties for symmetric Cantor set and their applications. Prog. Natur. Sci.,1997,7(3):172.
[21]Fuglede B. Commuting self-adjoint partial differential operators and a group theoritic problem. J. Funct. Anal.,1974,16:101-121.
[22]Farkas B, Matolcsi M, Mora P. On Fuglede's conjecture and the existence of universal spectra. J. Fourier Anal. Appl.,2006,12(5):483-494.
[23]Farkas B, Revesz S Gy. Tiles with no spectra in dimension 4. Math. Scand.,2006,98(1):44-52.
[24]Grochenig K. Describing functions:atomic decomposition versus frames. Monatshefte fur Math., 1991,112(1):1-41.
[25]Gabardo J P, Han D. Frames associated with measurable space. Adv. Comp. Math.,2003,18(3): 127-147.
[26]Hutchinson J E. Fractals and self-similarity. Indiana Univ. Math. J.,1981,30:713-747.
[27]Hua S. On the dimension of generalized self-similar sets. Acta Math. Appl. Sinica,1994,17:551.
[28]Hua S, Li W X. Packing dimension of generalized Moran sets. Prog. Natur. Sci. (English Ed.), 1996,6(2):148-152.
[29]Heil C E, Walnut D F. Continuous and discrete wavelet transforms. SIAM Rev.,1989,31:628-666.
[30]Hua S, Rao H, Wen Z Y. On the structures and dimensions of Moran sets. Sci. in China, Ser A., 2000,43:836.
[31]Heil C E. History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl., 2007,13:113-166.
[32]Iosevich A, Katz N, Tao T. Convex bodies with a point of curvature do not have Fourier bases. Amer. J. Math.,2001,123(1):115-120.
[33]Iosevich A, Katz N, Tao T. The Fuglede spectral conjecture holds for convex planar domains. Math. Res. Lett.,2003,10(5-6):559-569.
[34]Iosevich A, Pedersen S. Spectral and tiling properties of the unit cube. Internat. Math. Res. Notices, 1998,16:819-828.
[35]Jorgensen P E T, Pedersen S. Spectral pairs in Cartsian coordinates. J. Fourier Anal. Appl.,1999, 5:285-302.
[36]Kigami J. Analysis on Fractals. Cambridge:Cambridge University Press,2001.
[37]Kaiser G. A Friendly Guide to Wavelets. Boston:Birkhauser,1994.
[38]Kolountzakis M N, Matolcsi M. Complex Hadamard matrices and the spectral set conjecture. Collect. Math. Vol. Extra,2006:281-291.
[39]Kolountzakis M N, Matolcsi M. Tiles with no spectra. Forum Math.,2006,18:519-528.
[40]Kolountzakis M N, Laba I. Tiling and spectral properties of near-cubic domains. Studia Math., 2004,160(3):287-299.
[41]Kolountzakis M N. Non-symmetric convex domains have no basis of exponentials. Illinois J. Math., 2000,44:542-550.
[42]Kolountzakis M N. The study of translational tiling with Fourier analysis, in Fourier Analysis and Convexity. Appl. Numer. Harmon. Anal., Boston:Birkhauser,2004:131-187.
[43]Kolountzakis M N. Packing, tiling, orthogonality and completeness. Bull. London Math. Soc., 2000,32:589-599.
[44]Laba I. Fuglede's conjecture for a union of two intervals. Proc. Amer. Math. Soc.,2001,129: 2965-2972.
[45]Laba I. The spectral set conjecture and multiplicative properties of roots of polynomials. J. London Math. Soc.,2002,65(3):661-671.
[46]Lagarias J C, Reeds J A, Wang Y. Orthonormal bases of exponentials for the n-cube. Duke Math. J.,2000,103:25-37.
[47]Liu Q H, Wen Z Y. On dimension of multitype Moran sets. Math. Proc. Camb. Phil. Soc.,2005, 139:541-553.
[48]Li J L. On a characterization of spectra and tilings. J. Math. Anal. Appl.,2004,289:244-247.
[49]Li J L. On characterizations of spectra and tilings. Journal of Functional Analysis,2004,213: 31-44.
[50]Li J L. Spectral self-affine measures in Rn. Proc. Edinb. Math. Soc.,2007,50:197-217.
[51]Li J L. μM,D-orthogonality and compatible pair. J. Funct. Anal.,2007,244:628-638.
[52]Li J L. Orthogonal exponentials on the generalized plane Sierpinski gasket. J. Appr. Theory,2008, 153:161-169.
[53]Li J L. Non-spectral problem for a class of planar self-affine measures. J. Funct. Anal.,2008,255: 3125-3148.
[54]Li J L. Non-spectrality of planar self-affine measures with three-elements digit set. J. Funct. Anal., 2009,257:537-552.
[55]Matolcsi M. Fuglede's conjecture fails in dimension 4. Proc. Amer. Math. Soc.,2005,133:3021-3026.
[56]Moran P A P. Additive functions of intervals and Husdorff measure. Proc. Camb. Philo. Soc.,1946, 42:15-23.
[57]Mattila P. Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol.44, Cambridge:Cambridge University Press,1995, Fractals and rectifiability.
[58]Moran M. Fractal series. Mathematika,1989,36:334-348.
[59]Moran M. Dimension function for fractal sets associated to series. Proceedings of the American Mathematical Society,1994,120:749-754.
[60]Matolcsi M. Fuglede's conjecture fails in dimension 4. Proc. Amer. Math. Soc.,2005,133:3021-3026.
[61]Murphy G J. C*-algebras and Operator Theory. San Diego:Academic Press,1990.
[62]Rahimi A, Najati A, Dehghan Y N. Continuous frames in Hilbert spaces. Methods of Functional Analysis and Topology,2006,12(2):170-182.
[63]Rogers C A. Hausdorff Measures. Cambridge Mathematical Library, Cambridge:Cambridge University Press,1998, Reprint of the 1970 original, With a foreword by K. J. Falconer.
[64]Rao H, Wen Z Y. Some studies of a class of self-similar fractals with overlap structure. Adv. appl. Math.,1998,20:50-72.
[65]Rudin W. Real and Complex Analysis. Third Edition. McGraw-Hill Companies, Inc.,1987.
[66]Rudin W. Functional Analysis. Second Edition. McGraw-Hill Companies, Inc.,1991.
[67]Thirulogasanthar K, Bahsoun W. Frames built on fractal sets. Journal of Geometry and physics, 2004,50:79-98.
[68]Thirulogasanthar K, Bahsoun W. Continuous frames on Julia sets. Journal of Geometry and physics, 2004,51:183-194.
[69]Tricot C. Two definitions of fractal dimension. Math. proc. Camb. philo. Soc,1982,92:54-74.
[70]Taylor S J. The measure theory of random fractals. Math. proc. Camb. philo. Soc.,1986,100: 333-406.
[71]Tao T. Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Lett.,2004,11:251-258.
[72]Wen Z Y. Moran sets and Moran classes. Chinese Science Bulletin,2001,46(22):1849-1856.
[73]Young R. An Introduction to Nonharmonic Fourier Series. New York:Springer-Verlag,1977.
[74]Yosida K. Functional Analysis. New York:Springer-Verlag,1980.
[75]刘培德.泛函分析基础.武汉：武汉大学出版社,2001.
[76]李登峰,薛明志.Banach空间上的基和框架.北京：科学出版社,2007.
[77]文志英,文志雄,苏维宜等.分形几何理论与应用.上海：上海科技教育出版社,2000.
[78]文志英.分形几何的数学基础.杭州：浙江科学技术出版社,1998.
[2]Ali S T, Antoine J P, Gazeau J P. Continuous frames in Hilbert spaces. Annals of Physics,1993, 222:1-37.
[3]Askari-Hemmat A, Dehghan M A, Radjabalipour M. Generalized frames and their redundancy. Proc. Amer. Math. Soc.,2001,129(4):1143-1147.
[4]Christensen O. An Introduction to Frames and Riesz Bases. Boston:Birkhauser,2002.
[5]Casazza P G. Modern tools for Weyl-Heisenberg (Gabor) frame theory. Adv. Imag. Elect. Phs., 2001,115:1-27.
[6]Chui Ch K. An Introduction to Wavelets. New York:Academic Press,1992.
[7]Daubechies I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory,1990,36:961-1005.
[8]Daubechies I. The Lectures on Wavelets. SIAM. Vol.61,1992.
[9]Daubechies I, Grossman A, Meyer Y. Painless nonorthogonal expansions. J. Mate. Phys.,1986,27: 1271-1283.
[10]Duffin R J, Schaeffer A C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc.,1952, 72:147-158.
[11]Daubechies I. The wavelet transformation, time-frequency localization and signal analysis. IEEE Trans. Infor. Theory,1990,36:961-1005.
[12]Edgar G A. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics, New York:Springer-Verlag,1990.
[13]Edgar G A. Integral, Probability, and Fractal Measures. New York:Springer-Verlag,1998.
[14]Falconer K J. The Geometry of Fractal Sets. Cambridge:Cambridge University Press,1985.
[15]Falconer K J. Fractal Geometry:Mathematical Foundations and Applications. John Wiley & Sons, 1990.
[16]Falconer K J. Techniques in Fractal Geometry. Chichester:John Wiley and Sons, Ltd.,1997.
[17]Federer H. Geometry Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, New York:Springer-Verlag,1969.
[18]Foranasier M, Rauhut H. Continuous frames, function spaces, and the discretization problem. J. Four. Anal. Appl.,2005,113:245-286.
[19]Feng D J, Wen Z Y, Wu J. Some dimensional results for homogeneous Moran sets. Science of China (Series A),1997,40(5):475-482.
[20]Feng D J, Rao H, Wu J. The net measure properties for symmetric Cantor set and their applications. Prog. Natur. Sci.,1997,7(3):172.
[21]Fuglede B. Commuting self-adjoint partial differential operators and a group theoritic problem. J. Funct. Anal.,1974,16:101-121.
[22]Farkas B, Matolcsi M, Mora P. On Fuglede's conjecture and the existence of universal spectra. J. Fourier Anal. Appl.,2006,12(5):483-494.
[23]Farkas B, Revesz S Gy. Tiles with no spectra in dimension 4. Math. Scand.,2006,98(1):44-52.
[24]Grochenig K. Describing functions:atomic decomposition versus frames. Monatshefte fur Math., 1991,112(1):1-41.
[25]Gabardo J P, Han D. Frames associated with measurable space. Adv. Comp. Math.,2003,18(3): 127-147.
[26]Hutchinson J E. Fractals and self-similarity. Indiana Univ. Math. J.,1981,30:713-747.
[27]Hua S. On the dimension of generalized self-similar sets. Acta Math. Appl. Sinica,1994,17:551.
[28]Hua S, Li W X. Packing dimension of generalized Moran sets. Prog. Natur. Sci. (English Ed.), 1996,6(2):148-152.
[29]Heil C E, Walnut D F. Continuous and discrete wavelet transforms. SIAM Rev.,1989,31:628-666.
[30]Hua S, Rao H, Wen Z Y. On the structures and dimensions of Moran sets. Sci. in China, Ser A., 2000,43:836.
[31]Heil C E. History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl., 2007,13:113-166.
[32]Iosevich A, Katz N, Tao T. Convex bodies with a point of curvature do not have Fourier bases. Amer. J. Math.,2001,123(1):115-120.
[33]Iosevich A, Katz N, Tao T. The Fuglede spectral conjecture holds for convex planar domains. Math. Res. Lett.,2003,10(5-6):559-569.
[34]Iosevich A, Pedersen S. Spectral and tiling properties of the unit cube. Internat. Math. Res. Notices, 1998,16:819-828.
[35]Jorgensen P E T, Pedersen S. Spectral pairs in Cartsian coordinates. J. Fourier Anal. Appl.,1999, 5:285-302.
[36]Kigami J. Analysis on Fractals. Cambridge:Cambridge University Press,2001.
[37]Kaiser G. A Friendly Guide to Wavelets. Boston:Birkhauser,1994.
[38]Kolountzakis M N, Matolcsi M. Complex Hadamard matrices and the spectral set conjecture. Collect. Math. Vol. Extra,2006:281-291.
[39]Kolountzakis M N, Matolcsi M. Tiles with no spectra. Forum Math.,2006,18:519-528.
[40]Kolountzakis M N, Laba I. Tiling and spectral properties of near-cubic domains. Studia Math., 2004,160(3):287-299.
[41]Kolountzakis M N. Non-symmetric convex domains have no basis of exponentials. Illinois J. Math., 2000,44:542-550.
[42]Kolountzakis M N. The study of translational tiling with Fourier analysis, in Fourier Analysis and Convexity. Appl. Numer. Harmon. Anal., Boston:Birkhauser,2004:131-187.
[43]Kolountzakis M N. Packing, tiling, orthogonality and completeness. Bull. London Math. Soc., 2000,32:589-599.
[44]Laba I. Fuglede's conjecture for a union of two intervals. Proc. Amer. Math. Soc.,2001,129: 2965-2972.
[45]Laba I. The spectral set conjecture and multiplicative properties of roots of polynomials. J. London Math. Soc.,2002,65(3):661-671.
[46]Lagarias J C, Reeds J A, Wang Y. Orthonormal bases of exponentials for the n-cube. Duke Math. J.,2000,103:25-37.
[47]Liu Q H, Wen Z Y. On dimension of multitype Moran sets. Math. Proc. Camb. Phil. Soc.,2005, 139:541-553.
[48]Li J L. On a characterization of spectra and tilings. J. Math. Anal. Appl.,2004,289:244-247.
[49]Li J L. On characterizations of spectra and tilings. Journal of Functional Analysis,2004,213: 31-44.
[50]Li J L. Spectral self-affine measures in Rn. Proc. Edinb. Math. Soc.,2007,50:197-217.
[51]Li J L. μM,D-orthogonality and compatible pair. J. Funct. Anal.,2007,244:628-638.
[52]Li J L. Orthogonal exponentials on the generalized plane Sierpinski gasket. J. Appr. Theory,2008, 153:161-169.
[53]Li J L. Non-spectral problem for a class of planar self-affine measures. J. Funct. Anal.,2008,255: 3125-3148.
[54]Li J L. Non-spectrality of planar self-affine measures with three-elements digit set. J. Funct. Anal., 2009,257:537-552.
[55]Matolcsi M. Fuglede's conjecture fails in dimension 4. Proc. Amer. Math. Soc.,2005,133:3021-3026.
[56]Moran P A P. Additive functions of intervals and Husdorff measure. Proc. Camb. Philo. Soc.,1946, 42:15-23.
[57]Mattila P. Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol.44, Cambridge:Cambridge University Press,1995, Fractals and rectifiability.
[58]Moran M. Fractal series. Mathematika,1989,36:334-348.
[59]Moran M. Dimension function for fractal sets associated to series. Proceedings of the American Mathematical Society,1994,120:749-754.
[60]Matolcsi M. Fuglede's conjecture fails in dimension 4. Proc. Amer. Math. Soc.,2005,133:3021-3026.
[61]Murphy G J. C*-algebras and Operator Theory. San Diego:Academic Press,1990.
[62]Rahimi A, Najati A, Dehghan Y N. Continuous frames in Hilbert spaces. Methods of Functional Analysis and Topology,2006,12(2):170-182.
[63]Rogers C A. Hausdorff Measures. Cambridge Mathematical Library, Cambridge:Cambridge University Press,1998, Reprint of the 1970 original, With a foreword by K. J. Falconer.
[64]Rao H, Wen Z Y. Some studies of a class of self-similar fractals with overlap structure. Adv. appl. Math.,1998,20:50-72.
[65]Rudin W. Real and Complex Analysis. Third Edition. McGraw-Hill Companies, Inc.,1987.
[66]Rudin W. Functional Analysis. Second Edition. McGraw-Hill Companies, Inc.,1991.
[67]Thirulogasanthar K, Bahsoun W. Frames built on fractal sets. Journal of Geometry and physics, 2004,50:79-98.
[68]Thirulogasanthar K, Bahsoun W. Continuous frames on Julia sets. Journal of Geometry and physics, 2004,51:183-194.
[69]Tricot C. Two definitions of fractal dimension. Math. proc. Camb. philo. Soc,1982,92:54-74.
[70]Taylor S J. The measure theory of random fractals. Math. proc. Camb. philo. Soc.,1986,100: 333-406.
[71]Tao T. Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Lett.,2004,11:251-258.
[72]Wen Z Y. Moran sets and Moran classes. Chinese Science Bulletin,2001,46(22):1849-1856.
[73]Young R. An Introduction to Nonharmonic Fourier Series. New York:Springer-Verlag,1977.
[74]Yosida K. Functional Analysis. New York:Springer-Verlag,1980.
[75]刘培德.泛函分析基础.武汉：武汉大学出版社,2001.
[76]李登峰,薛明志.Banach空间上的基和框架.北京：科学出版社,2007.
[77]文志英,文志雄,苏维宜等.分形几何理论与应用.上海：上海科技教育出版社,2000.
[78]文志英.分形几何的数学基础.杭州：浙江科学技术出版社,1998.