## Vector subdivision schemes and multiple wavelets

HTML articles powered by AMS MathViewer

- by Rong-Qing Jia, S. D. Riemenschneider and Ding-Xuan Zhou PDF
- Math. Comp.
**67**(1998), 1533-1563 Request permission

## Abstract:

We consider solutions of a system of refinement equations written in the form \begin{equation*}\phi = \sum _{\alpha \in \mathbb {Z}} a(\alpha )\phi (2\cdot -\alpha ),\end{equation*} where the vector of functions $\phi =(\phi ^{1},\ldots ,\phi ^{r})^{T}$ is in $(L_{p}(\mathbb {R}))^{r}$ and $a$ is a finitely supported sequence of $r\times r$ matrices called the refinement mask. Associated with the mask $a$ is a linear operator $Q_{a}$ defined on $(L_{p}(\mathbb {R}))^{r}$ by $Q_{a} f := \sum _{\alpha \in \mathbb {Z}} a(\alpha )f(2\cdot -\alpha )$. This paper is concerned with the convergence of the subdivision scheme associated with $a$, i.e., the convergence of the sequence $(Q_{a}^{n}f)_{n=1,2,\ldots }$ in the $L_{p}$-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask $a$ in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the $L_{2}$-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.## References

- Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli,
*Stationary subdivision*, Mem. Amer. Math. Soc.**93**(1991), no. 453, vi+186. MR**1079033**, DOI 10.1090/memo/0453 - C. K. Chui and J. A. Lian,
*A study of orthonormal multi-wavelets,*J. Applied Numerical Math.**20**(1996), 273–298. - A. Cohen, I. Daubechies, and G. Plonka,
*Regularity of refinable function vectors*, J. Fourier Anal. Appl. 3 (1997), 295-324. - A. Cohen, N. Dyn, and D. Levin,
*Stability and inter-dependence of matrix subdivision schemes*, in Advanced Topics in Multivariate Approximation, F. Fontanella, K. Jetter and P.-J. Laurent (eds.), 1996, pp. 33-45. - W. Dahmen and C. A. Micchelli,
*Biorthogonal wavelet expansions*, Constr. Approx.**13**(1997), 293–328. - Ingrid Daubechies and Jeffrey C. Lagarias,
*Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals*, SIAM J. Math. Anal.**23**(1992), no. 4, 1031–1079. MR**1166574**, DOI 10.1137/0523059 - George C. Donovan, Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust,
*Construction of orthogonal wavelets using fractal interpolation functions*, SIAM J. Math. Anal.**27**(1996), no. 4, 1158–1192. MR**1393432**, DOI 10.1137/S0036141093256526 - Nira Dyn, John A. Gregory, and David Levin,
*Analysis of uniform binary subdivision schemes for curve design*, Constr. Approx.**7**(1991), no. 2, 127–147. MR**1101059**, DOI 10.1007/BF01888150 - T. N. T. Goodman, R. Q. Jia, and C. A. Micchelli,
*On the spectral radius of a bi-infinite periodic and slanted matrix*, Southeast Asian Bull. Math., to appear. - T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward,
*Spectral radius formulas for subdivision operators*, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR**1244611** - B. Han and R. Q. Jia,
*Multivariate refinement equations and subdivision schemes*, SIAM J. Math. Anal., to appear. - Christopher Heil and David Colella,
*Matrix refinement equations: existence and uniqueness*, J. Fourier Anal. Appl.**2**(1996), no. 4, 363–377. MR**1395770** - Christopher Heil, Gilbert Strang, and Vasily Strela,
*Approximation by translates of refinable functions*, Numer. Math.**73**(1996), no. 1, 75–94. MR**1379281**, DOI 10.1007/s002110050185 - Loïc Hervé,
*Multi-resolution analysis of multiplicity $d$: applications to dyadic interpolation*, Appl. Comput. Harmon. Anal.**1**(1994), no. 4, 299–315. MR**1310654**, DOI 10.1006/acha.1994.1017 - T. A. Hogan,
*Stability and linear independence of the shifts of finitely many refinable functions*, J. Fourier Anal. Appl.**3**(1997), 757–774. - Rong Qing Jia,
*Subdivision schemes in $L_p$ spaces*, Adv. Comput. Math.**3**(1995), no. 4, 309–341. MR**1339166**, DOI 10.1007/BF03028366 - Rong-Qing Jia,
*Shift-invariant spaces on the real line*, Proc. Amer. Math. Soc.**125**(1997), no. 3, 785–793. MR**1350950**, DOI 10.1090/S0002-9939-97-03586-7 - Rong Qing Jia and Charles A. Micchelli,
*On linear independence for integer translates of a finite number of functions*, Proc. Edinburgh Math. Soc. (2)**36**(1993), no. 1, 69–85. MR**1200188**, DOI 10.1017/S0013091500005903 - R. Q. Jia, S. Riemenschneider, and D. X. Zhou,
*Approximation by multiple refinable functions*, Canadian J. Math.**49**(1997), 944-962. - Rong Qing Jia and Zuowei Shen,
*Multiresolution and wavelets*, Proc. Edinburgh Math. Soc. (2)**37**(1994), no. 2, 271–300. MR**1280683**, DOI 10.1017/S0013091500006076 - Rong Qing Jia and Jianzhong Wang,
*Stability and linear independence associated with wavelet decompositions*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 1115–1124. MR**1120507**, DOI 10.1090/S0002-9939-1993-1120507-8 - W. Lawton, S. L. Lee, and Zuowei Shen,
*An algorithm for matrix extension and wavelet construction*, Math. Comp.**65**(1996), no. 214, 723–737. MR**1333319**, DOI 10.1090/S0025-5718-96-00714-4 - W. Lawton, S. L. Lee, and Z. W. Shen,
*Stability and orthonormality of multivariate refinable functions*, SIAM J. Math. Anal.**28**(1997), 999–1014. - W. Lawton, S. L. Lee, and Z. W. Shen,
*Convergence of multidimensional cascade algorithm*, Numer. Math.**78**(1998), 427–438. - R. L. Long, W. Chen, and S. L. Yuan,
*Wavelets generated by vector multiresolution analysis*, Appl. Comput. Harmon. Anal.**4**(1997), no. 3, 293–316. - R. L. Long and Q. Mo,
*$L^{2}$-convergence of vector cascade algorithm*, manuscript. - Charles A. Micchelli and Hartmut Prautzsch,
*Uniform refinement of curves*, Linear Algebra Appl.**114/115**(1989), 841–870. MR**986909**, DOI 10.1016/0024-3795(89)90495-3 - G. Plonka,
*Approximation order provided by refinable function vectors*, Constr. Approx.**13**(1997), 221–244. - Gian-Carlo Rota and Gilbert Strang,
*A note on the joint spectral radius*, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math.**22**(1960), 379–381. MR**0147922**, DOI 10.1016/S1385-7258(60)50046-1 - Z. W. Shen,
*Refinable function vectors*, SIAM J. Math. Anal.**29**(1998), 235–250. - Lars F. Villemoes,
*Wavelet analysis of refinement equations*, SIAM J. Math. Anal.**25**(1994), no. 5, 1433–1460. MR**1289147**, DOI 10.1137/S0036141092228179 - J. Z. Wang,
*$\;$Stability and linear independence associated with scaling vectors*, SIAM J. Math. Anal., to appear - Ding-Xuan Zhou,
*Stability of refinable functions, multiresolution analysis, and Haar bases*, SIAM J. Math. Anal.**27**(1996), no. 3, 891–904. MR**1382838**, DOI 10.1137/0527047 - D. X. Zhou,
*Existence of multiple refinable distributions*, Michigan Math. J.**44**(1997), 317–329.

## Additional Information

**Rong-Qing Jia**- Email: jia@xihu.math.ualberta.ca
**S. D. Riemenschneider**- Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
- Email: sherm@approx.math.ualberta.ca
**Ding-Xuan Zhou**- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
- Email: mazhou@math.cityu.edu.hk
- Received by editor(s): December 12, 1996
- Additional Notes: Research supported in part by NSERC Canada under Grants # OGP 121336 and A7687.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1533-1563 - MSC (1991): Primary 39B12, 41A25, 42C15, 65F15
- DOI: https://doi.org/10.1090/S0025-5718-98-00985-5
- MathSciNet review: 1484900