Approximation order of two-direction multiscaling functions.(English)Zbl 1436.42035

A matrix $$A$$ satisfies condition E if it has a simple eigenvalue of 1, and all other eigenvalues are smaller than 1 in absolute value. Condition E for two-direction multiscaling functions $$\phi$$ is necessary for the stability of the multiresolution approximation produced by $$\phi$$ or for the existence of $$\phi$$ in the matrix refinement equation of $$\phi$$. Condition E for $$\Phi(x) := [\phi(x);\phi(-x)]^{T}$$ is used as an equivalent condition for condition E for $$\phi$$. In this article the author obtains a simple and efficient criterion for condition E for $$\phi$$, criteria for the basic regularity conditions for $$\phi$$ as well as a criterion for approximation order of $$\phi$$.
The author also gives examples for illustrating the general theory.

MSC:

 42C15 General harmonic expansions, frames 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:

References:

 [1] S. Du and D. Yuan, The description of two-directional biorthogonal finitely supported wavelet packets with poly-scale dilation, pp. 1171-1176 in Advanced Measurement and Test X, Key Engineering Materials 439, Trans Tech Publications, Zürich (2010). [2] B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001), 18-53. @articleHan2001, MRKEY = MR1826084, AUTHOR = Han, Bin, TITLE = Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, JOURNAL = J. Approx. Theory, FJOURNAL = Journal of Approximation Theory, VOLUME = 110, YEAR = 2001, NUMBER = 1, PAGES = 18-53, ISSN = 0021-9045, MRCLASS = 41A25 (42C40 65T60), MRNUMBER = 1826084, MRREVIEWER = Tomas Sauer, DOI = 10.1006/jath.2000.3545, URL =, · Zbl 0986.42020 [3] B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003), 44-88. @articleHan2003, MRKEY = MR2010780, AUTHOR = Han, Bin, TITLE = Vector cascade algorithms and refinable function vectors in Sobolev spaces, JOURNAL = J. Approx. Theory, FJOURNAL = Journal of Approximation Theory, VOLUME = 124, YEAR = 2003, NUMBER = 1, PAGES = 44-88, ISSN = 0021-9045, MRCLASS = 42C15 (39B12 42C40), MRNUMBER = 2010780, MRREVIEWER = Gerald B. Folland, DOI = 10.1016/S0021-9045(03)00120-5, URL =, · Zbl 1028.42019 [4] B. Han, Framelets and wavelets: algorithms, analysis, and applications, Applied and Numerical Harmonic Analysis, Springer (2017). @bookHan2017, MRKEY = MR3752124, AUTHOR = Han, Bin, TITLE = Framelets and wavelets, SERIES = Applied and Numerical Harmonic Analysis, NOTE = Algorithms, analysis, and applications, PUBLISHER = Birkhäuser/Springer, Cham, YEAR = 2017, PAGES = xxxiii + 724, ISBN = 978-3-319-68529-8; 978-3-319-68530-4, MRCLASS = 42-02 (41A30 42C15 42C40 65T60), MRNUMBER = 3752124, MRREVIEWER = R. A. Zalik, DOI = 10.1007/978-3-319-68530-4, URL =, · Zbl 1387.42001 [5] F. Keinert, Wavelets and multiwavelets, Studies in Advanced Mathematics, Chapman & Hall (2004). @bookKeinert2004, MRKEY = MR2035222, AUTHOR = Keinert, Fritz, TITLE = Wavelets and multiwavelets, SERIES = Studies in Advanced Mathematics, PUBLISHER = Chapman & Hall/CRC, Boca Raton, FL, YEAR = 2004, PAGES = xii+275, ISBN = 1-58488-304-9, MRCLASS = 42-02 (42C40 65T60 94A12), MRNUMBER = 2035222, MRREVIEWER = Morten Nielsen, [6] F. Keinert and S.-G. Kwon, Point values and normalization of two-direction multiwavelets and their derivatives, Kyungpook Math. J. 55 (2015), 1053-1067. @articleKeinertKwon2015, MRKEY = MR3597348, AUTHOR = Keinert, Fritz and Kwon, Soon-Geol, TITLE = Point values and normalization of two-direction multiwavelets and their derivatives, JOURNAL = Kyungpook Math. J., FJOURNAL = Kyungpook Mathematical Journal, VOLUME = 55, YEAR = 2015, NUMBER = 4, PAGES = 1053-1067, ISSN = 1225-6951, MRCLASS = 42C40 (30G35), MRNUMBER = 3597348, MRREVIEWER = Mawardi Bahri, DOI = 10.5666/KMJ.2015.55.4.1053, URL =, · Zbl 1348.42043 [7] S.-G. Kwon, Characterization of orthonormal high-order balanced multiwavelets in terms of moments, Bull. Korean Math. Soc. 46 (2009), 183-198. @articleKwon2009, MRKEY = MR2488513, AUTHOR = Kwon, Soon-Geol, TITLE = Characterization of orthonormal high-order balanced multiwavelets in terms of moments, JOURNAL = Bull. Korean Math. Soc., FJOURNAL = Bulletin of the Korean Mathematical Society, VOLUME = 46, YEAR = 2009, NUMBER = 1, PAGES = 183-198, ISSN = 1015-8634, MRCLASS = 42C40 (42C15 94A12), MRNUMBER = 2488513, MRREVIEWER = Gerlind Plonka, DOI = 10.4134/BKMS.2009.46.1.183, URL =, [8] S.-G. Kwon, Two-direction multiwavelet moments, Appl. Math. Comput. 219 (2012), 3530-3540. @articleKwon2012, MRKEY = MR2996795, AUTHOR = Kwon, Soon-Geol, TITLE = Two-direction multiwavelet moments, JOURNAL = Appl. Math. Comput., FJOURNAL = Applied Mathematics and Computation, VOLUME = 219, YEAR = 2012, NUMBER = 8, PAGES = 3530-3540, ISSN = 0096-3003, MRCLASS = 42C40, MRNUMBER = 2996795, DOI = 10.1016/j.amc.2012.09.034, URL =, · Zbl 1311.42092 [9] S.-G. Kwon, Orthogonal two-direction wavelets of order 2 from orthogonal symmetric/antisymmetric multiwavelets, J. Appl. Math. Inform. 35 (2017), 181-189. @articleKwon2017, MRKEY = MR3603230, AUTHOR = Kwon, Soon-Geol, TITLE = Orthogonal two-direction wavelets of order 2 from orthogonal symmetric/antisymmetric multiwavelets, JOURNAL = J. Appl. Math. Inform., FJOURNAL = Journal of Applied Mathematics & Informatics, VOLUME = 35, YEAR = 2017, NUMBER = 1-2, PAGES = 181-189, ISSN = 1598-5857, MRCLASS = 42C40, MRNUMBER = 3603230, DOI = 10.14317/jami.2017.181, URL =, · Zbl 1388.42081 [10] J. Morawiec, On $$L\sp 1$$-solutions of a two-direction refinement equation, J. Math. Anal. Appl. 354 (2009), 648-656. @articleMorawiec2009, MRKEY = MR2515246, AUTHOR = Morawiec, Janusz, TITLE = On $$L^1$$-solutions of a two-direction refinement equation, JOURNAL = J. Math. Anal. Appl., FJOURNAL = Journal of Mathematical Analysis and Applications, VOLUME = 354, YEAR = 2009, NUMBER = 2, PAGES = 648-656, ISSN = 0022-247X, MRCLASS = 39B12 (42C40), MRNUMBER = 2515246, MRREVIEWER = Paşc Găvruţa, DOI = 10.1016/j.jmaa.2009.01.041, URL =, · Zbl 1165.39001 [11] G. Plonka and V. Strela, From wavelets to multiwavelets, pp. 375-399 in Mathematical methods for curves and surfaces, II (Lillehammer, 1997), Innov. Appl. Math., Vanderbilt Univ. Press (1998). @incollectionPlonka1998, MRKEY = MR1640571, AUTHOR = Plonka, Gerlind and Strela, Vasily, TITLE = From wavelets to multiwavelets, BOOKTITLE = Mathematical methods for curves and surfaces, II (Lillehammer, 1997), SERIES = Innov. Appl. Math., PAGES = 375-399, PUBLISHER = Vanderbilt Univ. Press, Nashville, TN, YEAR = 1998, MRCLASS = 42C15 (65T20), MRNUMBER = 1640571, MRREVIEWER = Eugenio Hernández, · Zbl 0905.65138 [12] G. Wang, X. Zhou and B. Wang, The construction of orthogonal two-direction multiwavelet from orthogonal two-direction wavelet, preprint. [13] C. Xie and S. Yang, Orthogonal two-direction multiscaling functions, Front. Math. China 1 (2006), 604-611. @articleXieYang2006, MRKEY = MR2257198, AUTHOR = Xie, Changzhen and Yang, Shouzhi, TITLE = Orthogonal two-direction multiscaling functions, JOURNAL = Front. Math. China, FJOURNAL = Frontiers of Mathematics in China, VOLUME = 1, YEAR = 2006, NUMBER = 4, PAGES = 604-611, ISSN = 1673-3452, MRCLASS = 42C15 (42C40 94A12), MRNUMBER = 2257198, DOI = 10.1007/s11464-006-0031-9, URL =, · Zbl 1222.42032 [14] S. Yang and Y. Li, Two-direction refinable functions and two-direction wavelets with dilation factor $$m$$, Appl. Math. Comput. 188 (2007), 1908-1920. @article MR2335044, AUTHOR = Shouzhi, Yang and Youfa, Li, TITLE = Two-direction refinable functions and two-direction wavelets with dilation factor $$m$$, JOURNAL = Appl. Math. Comput., FJOURNAL = Applied Mathematics and Computation, VOLUME = 188, YEAR = 2007, NUMBER = 2, PAGES = 1908-1920, ISSN = 0096-3003, MRCLASS = 42C40, MRNUMBER = 2335044, DOI = 10.1016/j.amc.2006.11.078, URL =, · Zbl 1132.65120 [15] S. Yang and Y. Li, Two-direction refinable functions and two-direction wavelets with high approximation order and regularity, Sci. China Ser. A 50 (2007), 1687-1704. @articleYangLi2007SciChina, MRKEY = MR2390482, AUTHOR = Yang, Shou-zhi and Li, You-fa, TITLE = Two-direction refinable functions and two-direction wavelets with high approximation order and regularity, JOURNAL = Sci. China Ser. A, FJOURNAL = Science in China. Series A. Mathematics, VOLUME = 50, YEAR = 2007, NUMBER = 12, PAGES = 1687-1704, ISSN = 1006-9283, MRCLASS = 42C40 (41A30 42C15 94A12), MRNUMBER = 2390482, MRREVIEWER = Bin Han, DOI = 10.1007/s11425-007-0091-7, URL =, · Zbl 1137.42012 [16] S. Yang and C. Xie, A class of orthogonal two-direction refinable functions and two-direction wavelets, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 883-894. @articleYangXie2008, MRKEY = MR2589303, AUTHOR = Yang, Shouzhi and Xie, Changzhen, TITLE = A class of orthogonal two-direction refinable functions and two-direction wavelets, JOURNAL = Int. J. Wavelets Multiresolut. Inf. Process., FJOURNAL = International Journal of Wavelets, Multiresolution and Information Processing, VOLUME = 6, YEAR = 2008, NUMBER = 6, PAGES = 883-894, ISSN = 0219-6913, MRCLASS = 42C15 (94A12), MRNUMBER = 2589303, DOI = 10.1142/S0219691308002653, URL =, · Zbl 1283.42051
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.