## Abstract

The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets.

The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary *L*_{p} spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]M. Buhmann and C.A. Micchelli, Using two slanted matrices for subdivision, Proc. London Math. Soc. 69 (1994) 428–448.MATHMathSciNetGoogle Scholar
- [2]A.S. Cavaretta, W. Dahmen and C.A. Micchelli,
*Stationary Subdivision*, Memoirs of the AMS 453 (1991).Google Scholar - [3]A.S. Cavaretta, W. Dahmen, C.A. Micchelli and P. Smith, A factorization theorem for banded matrices, Linear Algebra Appl. 39 (1981) 229–245.MATHMathSciNetCrossRefGoogle Scholar
- [4]C.K. Chui and J. Lian, A study of orthonormal multi-wavelets, Preprint (1995).Google Scholar
- [5]A. Cohen, I. Daubechies and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl., to appear.Google Scholar
- [6]W. Dahmen and C.A. Micchelli, On stationary subdivision and the construction of compactly supported orthonormal wavelets, in:
*International Series of Numerical Mathematics*94, eds. W. Haussmann and K. Jetter (Birkhäuser, Basel, 1990) pp. 69–81.Google Scholar - [7]W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., to appear.Google Scholar
- [8]I. Daubechies,
*Ten Lectures on Wavelets*, CBMS-NSF 61 (SIAM, Philadelphia, PA, 1992).MATHGoogle Scholar - [9]G. Donovan, J.S. Geronimo and D.P. Hardin, Intertwining multiresolution analysis and the construction of piecewise polynomial wavelets, Preprint (1995).Google Scholar
- [10]G. Donovan, J.S. Geronimo, D.P. Hardin and P.R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal., to appear.Google Scholar
- [11]G.B. Follard,
*Real Analysis*(Wiley, New York, 1984).Google Scholar - [12]M. Gasca and C.A. Micchelli, eds.,
*Total Positivity and its Applications*(Kluwer, 1996).Google Scholar - [13]J.S. Geronimo, D.P. Hardin and P.R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1995) 373–401.MathSciNetCrossRefGoogle Scholar
- [14]T.N.T. Goodman, R.Q. Jia and C.A. Micchelli, The
*ℓ*_{2}(ℤ) spectral radius of an*N*-periodic two slanted bi-infinite matrix is an eigenvalue, Preprint (1996).Google Scholar - [15]T.N.T. Goodman and S.L. Lee, Wavelets of multiplicity
*r*, Trans. Amer. Math. Soc. 342 (1994) 307–324.MATHMathSciNetCrossRefGoogle Scholar - [16]T.N.T. Goodman, S.L. Lee and W.S. Tang, Wavelets in wandering spaces, Trans. Amer. Math. Soc. (1993) 639–654.Google Scholar
- [17]C. Heil and D. Colella, Matrix refinement equations: Existence and uniqueness, Preprint (1994).Google Scholar
- [18]C. Heil, P.H. Heller, G. Strang, V. Strela and P. Topiwala, The application of multiwavelet filter banks to signal and image processing, Preprint (1995).Google Scholar
- [19]C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996) 75–94.MATHMathSciNetCrossRefGoogle Scholar
- [20]L. Hervé, Multiresolution analysis of multiplicity
*d*: applications to dyadic interpolation, Appl. Comput. Harmon. Anal. 1 (1994) 299–315.MATHMathSciNetCrossRefGoogle Scholar - [21]R.Q. Jia, Total positivity of the discrete spline collocation matrix, J. Approx. Theory 39 (1983) 11–23.MATHMathSciNetCrossRefGoogle Scholar
- [22]R.Q. Jia, Subdivisions schemes in
*L*_{p}spaces, Adv. Comput. Math. 3 (1995) 309–341.MATHMathSciNetCrossRefGoogle Scholar - [23]R.Q. Jia and C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets II: powers of two, in:
*Curves and Surfaces*, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, Boston, 1991) pp. 209–258.Google Scholar - [24]R.Q. Jia and C.A. Micchelli, On the linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. 36 (1992) 69–85.MathSciNetCrossRefGoogle Scholar
- [25]R.Q. Jia, S. Riemenschneider and D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, Preprint (1997).Google Scholar
- [26]T.Y. Lam,
*Serre's Conjecture*, Lecture Notes in Mathematics 635 (Springer, New York, 1978).MATHGoogle Scholar - [27]W. Lawton, S.L. Lee and Z. Shen, Stability and orthonormality of multiwavelets, Preprint (1995).Google Scholar
- [28]W. Lawton, S.L. Lee and Z. Shen, Algorithm for matrix extension and wavelet construction, Math. Comp. 65 (1996) 723–737.MATHMathSciNetCrossRefGoogle Scholar
- [29]W. Lawton, S.L. Lee and Z. Shen, Convergence of multidimensional cascade algorithm, Numer. Math., to appear.Google Scholar
- [30]R. Long, W. Chen and S. Yuan, Wavelets generated by vector multiresolution analysis, Preprint (1995).Google Scholar
- [31]J.L. Merrien, A family of Hermite interpolants by bisection algorithms, Numer. Algorithms 2 (1992) 187–200.MATHMathSciNetCrossRefGoogle Scholar
- [32]C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets VI: Shift invariant subspaces, in:
*Approximation Theory, Spline Functions and Applications*, ed. S.P. Singh (Kluwer Academic Publishers, 1992) pp. 213–222.Google Scholar - [33]C.A. Micchelli,
*Mathematical Aspects of Geometric Modeling*, CBMS-NSF 65 (SIAM, Philadelphia, PA, 1995).MATHGoogle Scholar - [34]C.A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approx. Theory 86 (1996) 41–71.MATHMathSciNetCrossRefGoogle Scholar
- [35]C.A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989) 841–870.MathSciNetCrossRefGoogle Scholar
- [36]C.A. Micchelli and T. Sauer, On vector subdivision, Preprint (1997).Google Scholar
- [37]C.A. Micchelli and T. Sauer, Sobolev norm convergence of stationary subdivision schemes, in:
*Surface Fitting and Multiresolution Methods*, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, 1997) pp. 245–261.Google Scholar - [38]C.A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmon. Anal. 4 (1994) 391–401.MATHMathSciNetCrossRefGoogle Scholar
- [39]G. Plonka, Spline wavelets with higher defect, in:
*Curves and Surfaces II*, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, Boston, 1994) pp. 387–398.Google Scholar - [40]G. Plonka, Approximation properties of multi-scaling functions: A Fourier approach, Rostock. Math. Kolloq. 49 (1995) 115–126.MATHMathSciNetGoogle Scholar
- [41]G. Plonka, Two-scale symbol and autocorrelation symbol for B-splines with multiple knots, Adv. Comput. Math. 3 (1995) 297–306.MathSciNetCrossRefGoogle Scholar
- [42]G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx., to appear.Google Scholar
- [43]G. Plonka and V. Strela, Construction of multi-scaling functions with approximation and symmetry, SIAM J. Math. Anal., to appear.Google Scholar
- [44]W. Rudin,
*Functional Analysis*(McGraw-Hill, New York, 1973).MATHGoogle Scholar - [45]L.L. Schumaker,
*Spline Functions: Basic Theory*(Wiley, New York, 1981).MATHGoogle Scholar - [46]Z. Shen, Refinable function vectors, Preprint (1995).Google Scholar
- [47]G. Strang and V. Strela, Finite element multiwavelets, Preprint (1995).Google Scholar
- [48]G. Strang and V. Strela, Short wavelets and matrix dilation equations, IEEE Trans. Signal Processing 43 (1995).Google Scholar