半直线伸缩调制框架集

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半直线伸缩调制框架集李云章, 王雅慧北京工业大学应用数理学院北京 100124 The Dilation-and-Modulation Frame Sets on the Half Real Line Yun Zhang LI, Ya Hui WANGCollege of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China

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摘要本文研究右半直线平方可积函数空间L²（R₊）中的一类伸缩调制系.实际问题中时间变量不可取负值，L²（R₊）可模拟因果信号空间.但因R₊按加法不能作成一个群，它不容许小波与Gabor系.我们研究L²（R₊）中由特征函数生成的伸缩调制系（MD-系）框架，引入了R₊中MD-框架集的概念，利用"伸缩等价"与"基数函数"方法刻画了L²（R₊）中MD-Bessel集与完备集；得到了关于MD-Riesz基集的两个充分条件，并证明了通过对MD-Riesz基集进行有限可测分解可得到MD-框架集.

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收稿日期: 2019-01-05

MR (2010):

O174.2

基金资助:国家自然科学基金资助项目（11971043）

通讯作者:李云章E-mail: yzlee@bjut.edu.cn

作者简介: 王雅慧,E-mail:wangyahui@emails.bjut.edu.cn

引用本文:

李云章, 王雅慧. 半直线伸缩调制框架集[J]. 数学学报, 2020, 63(1): 45-60. Yun Zhang LI, Ya Hui WANG. The Dilation-and-Modulation Frame Sets on the Half Real Line. Acta Mathematica Sinica, Chinese Series, 2020, 63(1): 45-60.

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[1] Arambaši? L., Baki? D., Raji? R., Dimension functions, scaling sequences, and wavelet sets, Studia Math., 2010, 198:1-32.
[2] Auscher P., Solution of two problems on wavelets, J. Geom. Anal., 1995, 5:181-236.
[3] Baggett L. W. H., Medina A., Merrill K. D., Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn, J. Fourier Anal. Appl., 1999, 5:563-573.
[4] Benedetto J. J., Benedetto R. L., The Construction of Wavelet Sets, Wavelets and Multiscale Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2011:17-56.
[5] Benedetto J. J., Sumetkijakan S., Tight frames and geometric properties of wavelet sets, Adv. Comput. Math., 2006, 24:35-56.
[6] Benedetto J. J., King E. J., Smooth functions associated with wavelet sets on R^d, d ≥ 1, and frame bound gaps, Acta Appl. Math., 2009, 107:121-142.
[7] Bownik M., Ross K., The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl., 2015, 21:849-884.
[8] Cabrelli C., Paternostro V., Shift-invariant spaces on LCA groups, J. Funct. Anal., 2010, 258:2034-2059.
[9] Casazza P., Kalton N., Roots of complex polynomials and Weyl-Heisenberg frames, Proc. Amer. Math. Soc., 2002, 130:2313-2318.
[10] Christensen O., An Introduction to Frames and Riesz Bases, Springer, Birkhäuser, 2016.
[11] Christensen O., Goh S. S., Fourier-like frames on locally compact abelian groups, J. Approx. Theory, 2015, 192:82-101.
[12] Dai X., Diao Y., Gu Q., et al., Frame wavelet sets in R^d, J. Comput. Appl. Math., 2003, 155:69-82.
[13] Dai X., Diao Y., Gu Q., Frame wavelets with frame set support in the frequency domain, Illinois J. Math., 2004, 48:539-558.
[14] Dai X., Diao Y., Gu Q., et al., The existence of subspace wavelet sets, J. Comput. Appl. Math., 2003, 155:83-90.
[15] Dai X., Larson D. R., Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc., 1998, 134:68.
[16] Dai X., Larson D. R., Speegle D. M., Wavelet sets in Rⁿ, J. Fourier Anal. Appl., 1997, 3:451-456.
[17] Dai X. R., Sun Q., The abc-problem for Gabor systems, Mem. Amer. Math. Soc., 2016, 244:99.
[18] Fang X., Wang X., Construction of minimally supported frequency wavelets, J. Fourier Anal. Appl., 1995, 2:315-327.
[19] Feichtinger H. G., Gröchenig K. H., Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal., 1989, 86:307-340.
[20] Feichtinger H. G., Strohmer T., Gabor Analysis and Algorithms, Theory and Applications, Birkhäuser, Boston, 1998.
[21] Feichtinger H. G., Strohmer T., Advances in Gabor Analysis, Birkhäuser, Boston, 2003.
[22] Fu X., Gabardo J. P., Construction of wavelet sets using integral self-affine multi-tiles, J. Fourier Anal. Appl., 2014, 20:234-257.
[23] Gabardo J. P., Yu X. J., Construction of wavelet sets with certain self-similarity properties, J. Geom. Anal., 2004, 14:629-651.
[24] Gröchenig K., Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
[25] Gu Q., Han D., When a characteristic function generates a Gabor frame, Appl. Comput. Harmon. Anal., 2008, 24:290-309.
[26] Han B., Framelets and Wavelets, Algorithms, Analysis, and Applications, Cham, Springer, 2017.
[27] Han D., Larson, D.R., Frames, bases and group representations, Mem. Amer. Math. Soc., 2000, 147:94.
[28] Heil C., A Basis Theory Primer, Expanded edition, Birkhäuser, New York, 2011.
[29] Heil C., History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl., 2007, 13:113-166.
[30] Hernández E., Weiss G., A First Course on Wavelets, Boca Raton, CRC Press, 1996.
[31] Hernández E., Wang X., Weiss G., Smoothing minimally supported frequency (MSF) wavelets:Part I, J. Fourier Anal. Appl., 1996, 2:329-340.
[32] Hernández E., Wang X., Weiss G., Smoothing minimally supported frequency wavelets:Part II, J. Fourier Anal. Appl., 1997, 3:23-41.
[33] Ionascu E. J., Larson D. R., Pearcy C. M., On wavelet sets, J. Fourier Anal. Appl., 1998, 4:711-721.
[34] Ionascu E. J., A new construction of wavelet sets, Real Anal. Exchange, 2002, 28:593-609.
[35] Ionascu E. J., Wang Y., Simultaneous translational and multiplicative tiling and wavelet sets in R², Indiana Univ. Math. J., 2006, 55:1935-1949.
[36] Jakobsen M. S., Lemvig J., Reproducing formulas for generalized translation invariant systems on locally compact abelian groups, Trans. Amer. Math. Soc., 2016, 368:8447-8480.
[37] Janssen A. J. E. M., Representations of Gabor frame Operators, Twentieth Century Harmonic Analysis-A Celebration (Il Ciocco, 2000), NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer Acad. Publ., Dordrecht, 2001, 73-101.
[38] Janssen A. J. E. M., Zak transforms with few zeros and the tie, Advances in Gabor Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2003:31-70.
[39] Kutyniok G., Labate D., The theory of reproducing systems on locally compact abelian groups, Colloq. Math., 2006, 106:197-220.
[40] Li Y. Z., Wang Y. H., The density theorem of a class of dilation-and-modulation systems on the half real line, arXiv:1712.02606.
[41] Li Y. Z., Zhang W., Dilation-and-modulation systems on the half real line, J. Inequal. Appl., 2016, 186:11.
[42] Li Y. Z., Zhang W., Multi-window dilation-and-modulation frames on the half real line, Sci. China Math., 2017, doi:10.1007/s11425-018-9468-8.
[43] Seip K., Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc., 1993, 117:213-220.
[44] Volkmer H., Frames of wavelets in Hardy space, Analysis, 1995, 15:405-421.
[45] Wang Y. H., Li Y. Z., A class of vector-valued dilation-and-modulation frames on the half real line, Math. Meth. Appl. Sci., 2018, 41:3900-3912.
[46] Young R. M., An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
[47] Yu X. J., Gabardo J. P., Nonuniform wavelets and wavelet sets related to one-dimensional spectral pairs, J. Approx. Theory, 2007, 145:133-139.

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