Multivariate matrix refinable functions with arbitrary matrix dilation
Author:
Qingtang Jiang
Journal:
Trans. Amer. Math. Soc. 351 (1999), 24072438
MSC (1991):
Primary 39B62, 42B05, 41A15; Secondary 42C15
Published electronically:
February 15, 1999
MathSciNet review:
1650101
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function with arbitrary matrix dilation are provided in terms of the eigenvalue and eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of is equivalent to the order of the vanishing moment conditions of the matrix refinement mask . The restricted transition operator associated with the matrix refinement mask is represented by a finite matrix , with and being the Kronecker product of matrices and . The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.
 1.
C. Cabrelli, C. Heil and U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998), 552. CMP 99:01
 2.
Charles
K. Chui and Jianao
Lian, A study of orthonormal multiwavelets, Appl. Numer.
Math. 20 (1996), no. 3, 273–298. Selected
keynote papers presented at 14th IMACS World Congress (Atlanta, GA, 1994).
MR
1402703 (98g:42051), http://dx.doi.org/10.1016/01689274(95)001115
 3.
Albert
Cohen, Ingrid
Daubechies, and Gerlind
Plonka, Regularity of refinable function vectors, J. Fourier
Anal. Appl. 3 (1997), no. 3, 295–324. MR 1448340
(98e:42031), http://dx.doi.org/10.1007/BF02649113
 4.
M.
J. Collins, Representations and characters of finite groups,
Cambridge Studies in Advanced Mathematics, vol. 22, Cambridge
University Press, Cambridge, 1990. MR 1050762
(91f:20001)
 5.
Ingrid
Daubechies, Ten lectures on wavelets, CBMSNSF Regional
Conference Series in Applied Mathematics, vol. 61, Society for
Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
(93e:42045)
 6.
Carl
de Boor, Ronald
A. DeVore, and Amos
Ron, The structure of finitely generated shiftinvariant spaces in
𝐿₂(𝑅^{𝑑}), J. Funct. Anal.
119 (1994), no. 1, 37–78. MR 1255273
(95g:46050), http://dx.doi.org/10.1006/jfan.1994.1003
 7.
C. de Boor, R. DeVore and A. Ron, Approximation orders of FSI spaces in , I, II, Constr. Approx. 14 (1998), 411427, 631652. CMP 98:14; CMP 99:01
 8.
Jeffrey
S. Geronimo, Douglas
P. Hardin, and Peter
R. Massopust, Fractal functions and wavelet expansions based on
several scaling functions, J. Approx. Theory 78
(1994), no. 3, 373–401. MR 1292968
(95h:42033), http://dx.doi.org/10.1006/jath.1994.1085
 9.
T. N. Goodman, Pairs of refinable bivariate splines, Advanced Topics in Multivariate Approximation (F. Fontanelle, K. Jetter and L. L. Schumaker, eds.), World Sci. Publ. Co., 1996.
 10.
T.
N. T. Goodman, S.
L. Lee, and W.
S. Tang, Wavelets in wandering
subspaces, Trans. Amer. Math. Soc.
338 (1993), no. 2,
639–654. MR 1117215
(93j:42017), http://dx.doi.org/10.1090/S00029947199311172150
 11.
Christopher
Heil, Gilbert
Strang, and Vasily
Strela, Approximation by translates of refinable functions,
Numer. Math. 73 (1996), no. 1, 75–94. MR 1379281
(97c:65033), http://dx.doi.org/10.1007/s002110050185
 12.
Roger
A. Horn and Charles
R. Johnson, Topics in matrix analysis, Cambridge University
Press, Cambridge, 1991. MR 1091716
(92e:15003)
 13.
C.
K. Chui and L.
L. Schumaker (eds.), Approximation theory VIII. Vol. 2, Series
in Approximations and Decompositions, vol. 6, World Scientific
Publishing Co., Inc., River Edge, NJ, 1995. Wavelets and multilevel
approximation. MR 1471770
(98d:41002)
 14.
R. Jia, Shiftinvariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259288.
 15.
R. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. (to appear).
 16.
Rong
Qing Jia and Charles
A. Micchelli, Using the refinement equations for the construction
of prewavelets. II. Powers of two, Curves and surfaces
(ChamonixMontBlanc, 1990) Academic Press, Boston, MA, 1991,
pp. 209–246. MR 1123739
(93e:65024)
 17.
R. Jia, S. Riemenschneider and D. Zhou, Approximation by multiple refinable functions, Canadian J. Math. 49 (1997), 944962. CMP 98:08
 18.
Rong
Qing Jia and Zuowei
Shen, Multiresolution and wavelets, Proc. Edinburgh Math. Soc.
(2) 37 (1994), no. 2, 271–300. MR 1280683
(95h:42035), http://dx.doi.org/10.1017/S0013091500006076
 19.
Q. Jiang, On the regularity of matrix refinable functions, SIAM J. Math. Anal. 29 (1998), 11571176. CMP 98:11
 20.
Q. Jiang and S. L. Lee, Matrix continuous refinement equations, preprint, 1996.
 21.
Q. Jiang and Z. Shen, On existence and weak stability of matrix refinable functions, Constr. Approx., to appear.
 22.
W.
Lawton, S.
L. Lee, and Zuowei
Shen, Stability and orthonormality of multivariate refinable
functions, SIAM J. Math. Anal. 28 (1997), no. 4,
999–1014. MR 1453317
(98d:41027), http://dx.doi.org/10.1137/S003614109528815X
 23.
Ruilin
Long, Wen
Chen, and Shenglan
Yuan, Wavelets generated by vector multiresolution analysis,
Appl. Comput. Harmon. Anal. 4 (1997), no. 3,
317–350. MR 1454406
(98m:42052), http://dx.doi.org/10.1006/acha.1997.0216
 24.
C. Micchelli and T. Sauer, Regularity of multiwavelets, Advances in Comp. Math., 7 (1997), 455456. CMP 98:01
 25.
G.
Plonka, Approximation order provided by refinable function
vectors, Constr. Approx. 13 (1997), no. 2,
221–244. MR 1437211
(98c:41023), http://dx.doi.org/10.1007/s003659900039
 26.
Z. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), 235250. CMP 98:11
 27.
G. Strang and G. Fix, A Fourier analysis of finiteelement variational method, Constructive Aspects of Functional Analysis, (G. Geymonat ed.), C.I.M.E, 1973, pp. 793840.
 28.
H.
Levy and F.
Lessman, Finite difference equations, Dover Publications,
Inc., New York, 1992. Reprint of the 1961 edition. MR 1217083
(94e:39002)
 1.
 C. Cabrelli, C. Heil and U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998), 552. CMP 99:01
 2.
 C. K. Chui and J. Lian, A study on orthonormal multiwavelets, Appl. Numer. Math., 20 (1996), 273298. MR 98g:42051
 3.
 A. Cohen, I. Daubechies and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. and Appl., 3 (1997), 295324. MR 98e:42031
 4.
 M. Collins, Representations and characters of finite groups, Cambridge Univ. Press, Cambridge, 1990. MR 91f:20001
 5.
 I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR 93e:42045
 6.
 C. de Boor, R. DeVore and A. Ron, The structure of finitely generated shiftinvariant spaces in , J. Funct. Anal., 119 (1994), 3778. MR 95g:46050
 7.
 C. de Boor, R. DeVore and A. Ron, Approximation orders of FSI spaces in , I, II, Constr. Approx. 14 (1998), 411427, 631652. CMP 98:14; CMP 99:01
 8.
 J. Geronimo, D. Hardin and P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), 373401. MR 95h:42033
 9.
 T. N. Goodman, Pairs of refinable bivariate splines, Advanced Topics in Multivariate Approximation (F. Fontanelle, K. Jetter and L. L. Schumaker, eds.), World Sci. Publ. Co., 1996.
 10.
 T. N. Goodman, S. L. Lee and W. S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc., 338 (1993), 639654. MR 93j:42017
 11.
 C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), 7594. MR 97c:65033
 12.
 R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991. MR 92e:15003
 13.
 R. Jia, Refinable shiftinvariant spaces:from splines to wavelets, Approximation Theory VIII, vol. 2 (C. K. Chui and L. L. Schumaker, eds.), 1995, pp. 179208. MR 98d:41002
 14.
 R. Jia, Shiftinvariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259288.
 15.
 R. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. (to appear).
 16.
 R. Jia and C. Micchelli, Using the refinement equation for the construction of prewavelets II: Powers of two, Curves and Surfaces (P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 209246. MR 93e:65024
 17.
 R. Jia, S. Riemenschneider and D. Zhou, Approximation by multiple refinable functions, Canadian J. Math. 49 (1997), 944962. CMP 98:08
 18.
 R. Jia and Z. Shen, Multiresolution and wavelets, Proc. Edinburgh Math. Soc., 37 (1994), 271300. MR 95h:42035
 19.
 Q. Jiang, On the regularity of matrix refinable functions, SIAM J. Math. Anal. 29 (1998), 11571176. CMP 98:11
 20.
 Q. Jiang and S. L. Lee, Matrix continuous refinement equations, preprint, 1996.
 21.
 Q. Jiang and Z. Shen, On existence and weak stability of matrix refinable functions, Constr. Approx., to appear.
 22.
 W. Lawton, S. L. Lee and Z. Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal., 28 (1997), 9991014. MR 98d:41027
 23.
 R. Long, W. Chen and S. Yuan, Wavelets generated by vector multiresolution analysis, Appl. Comput. Harmon. Anal., 4 (1997), 317350, MR 98m:42052
 24.
 C. Micchelli and T. Sauer, Regularity of multiwavelets, Advances in Comp. Math., 7 (1997), 455456. CMP 98:01
 25.
 G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx., 13 (1997), 221244. MR 98c:41023
 26.
 Z. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), 235250. CMP 98:11
 27.
 G. Strang and G. Fix, A Fourier analysis of finiteelement variational method, Constructive Aspects of Functional Analysis, (G. Geymonat ed.), C.I.M.E, 1973, pp. 793840.
 28.
 L. Villemoes, Energy moments in time and frequency for twoscale difference equation solutions and wavelets, SIAM J. Math. Anal., 23 (1992), 15191543. MR 94e:39002
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
39B62,
42B05,
41A15,
42C15
Retrieve articles in all journals
with MSC (1991):
39B62,
42B05,
41A15,
42C15
Additional Information
Qingtang Jiang
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 and Department of Mathematics, Peking University, Beijing 100871, China
Email:
qjiang@haar.math.nus.edu.sg
DOI:
http://dx.doi.org/10.1090/S0002994799024496
PII:
S 00029947(99)024496
Keywords:
Matrix refinable function,
transition operator,
stability,
orthonormality,
approximation order,
regularity
Received by editor(s):
September 26, 1996
Published electronically:
February 15, 1999
Additional Notes:
The author was supported by an NSTB postdoctoral research fellowship at the National University of Singapore.
Article copyright:
© Copyright 1999
American Mathematical Society
