Acta mathematica scientia, Series B >
MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES
Received date: 2012-07-11
Revised date: 2013-10-09
Online published: 2014-05-20
Supported by
This work was supported by Macao FDCT (098/2012/A3), Research Grant of the University of Macau (UL017/08-Y4/MAT/QT01/FST), National Natural Science Funds for Young Scholars (10901166), and Sun Yat-sen University Operating Costs of Basic Research Projects to Cultivate Young Teachers (11lgpy99).
Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries con-sisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary
dimensions.
QIAN Tao , WANG Jin-Xun , YANG Yan . MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES[J]. Acta mathematica scientia, Series B, 2014 , 34(3) : 660 -672 . DOI: 10.1016/S0252-9602(14)60038-2
[1] Mallat S G, Zhang Z. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 1993, 41(12): 3397–3415
[2] Qian T, Wang Y B. Adaptive Fourier series–a variation of greedy algorithm. Advances in Computational Mathematics, 2011, 34: 279–293
[3] Qian T. Intrinsic mono-component decomposition of functions: An advance of Fourier theory. Math Meth Appl Sci, 2010, 33: 880–891
[4] DeVore R A, Temlyakov V N. Some remarks on greedy algorithms. Advances in Computational Mathematics, 1996, 5: 173–187
[5] Qian T, Tan L H, Wang Y B. Adaptive decomposition by weighted inner functions: a generalization of Fourier series. J Fourier Anal Appl, 2011, 17: 175–190
[6] Qian T, Wegert E. Optimal approximation by Blaschke forms. Complex Variables and Elliptic Equations, 2013, 58: 123–133
[7] Qian T, Spr¨oßig W, Wang J X. Adaptive Fourier decomposition of functions in quaternionic Hardy spaces. Math Meth Appl Sci, 2012, 35: 43–64
[8] Brackx F, Delanghe R, Sommen F. Clifford Analysis. Boston: Pitman Advanced Publishing Program, 1982
[9] Cerejeiras P, Ferreira M, K¨ahler U, Sommen F. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Commun Pure Appl Anal, 2007, 6: 619–641
[10] Sakaguchi F. A larger class of wavepacket eigenfunction systems which contains Cauchy wavelets and coherent states//Proceedings of 8th IEEE Signal Processing Workshop on Statistical Signal and Array Processing. 1996: 436–439
[11] Mi W, Qian T, Wan F. A fast adaptive model reduction method based on Takenaka–Malmquist systems. Systems & Control Letters, 2012, 61: 223–230
[12] Yang Y, Qian T, Sommen F. Phase derivative of monogenic signals in higher dimensional spaces. Complex Anal Oper Theory, 2012, 6: 987–1010
[13] Gilbert J E, Murray A M. Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge: Cambridge
University Press, 1991
[14] Mitrea M. Clifford Wavelets, Singular Integrals, and Hardy Spaces. Berlin: Springer-Verlag, 1994
[15] Qian T, Yang Y. Hilbert transforms on the sphere with the Clifford algebra setting. J Fourier Anal Appl, 2009, 15: 753–774
[16] Brackx F, De Knock B, De Schepper H, Eelbode D. On the interplay between the Hilbert transform and conjugate harmonic functions. Math Meth Appl Sci, 2006, 29: 1435–1450
[17] Brackx F, Van Acker N. A conjugate Poisson kernel in Euclidean space. Simon Stevin, 1993, 67: 3–14
[18] Ahlfors L V. M¨obius Transformations in Several Dimensions. Minneapolis: University of Minnesota, 1981
[19] Qian T, Ryan J. Conformal transformations and Hardy spaces arising in Clifford analysis. J Operator Theory, 1996, 35: 349–372
/
| 〈 |
|
〉 |