A cardinal spline approach to wavelets

Authors:
Charles K. Chui and Jian-zhong Wang

Journal:
Proc. Amer. Math. Soc. **113** (1991), 785-793

MSC:
Primary 41A15; Secondary 42C05

MathSciNet review:
1077784

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: While it is well known that the th order -spline with integer knots generates a multiresolution analysis, , with the th order of approximation, we prove that , where denotes the th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces . Note that for , when the -spline is the characteristic function of the unit interval , our basic wavelet is simply the well-known Haar wavelet. In proving that , we give the exact formulation of , in terms of integer translates of and . This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.

**[1]**Guy Battle,*A block spin construction of ondelettes. I. Lemarié functions*, Comm. Math. Phys.**110**(1987), no. 4, 601–615. MR**895218****[2]**Charles K. Chui,*Multivariate splines*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR**1033490****[3]**Ingrid Daubechies,*Orthonormal bases of compactly supported wavelets*, Comm. Pure Appl. Math.**41**(1988), no. 7, 909–996. MR**951745**, 10.1002/cpa.3160410705**[4]**Pierre Gilles Lemarié,*Ondelettes à localisation exponentielle*, J. Math. Pures Appl. (9)**67**(1988), no. 3, 227–236 (French, with English summary). MR**964171****[5]**Stephane G. Mallat,*Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅)*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 69–87. MR**1008470**, 10.1090/S0002-9947-1989-1008470-5**[6]**-,*Multifrequency channel decompositions of images and wavelet models*, IEEE Trans. Acoust. Speech Signal Process.**37**(1989), 2091-2110.**[7]**Y. Meyer,*Ondelettes et functions splines*, Seminaire Equations aux Derivees Partielles, École Polytechnique, Paris (Dec. 1986).**[8]**I. J. Schoenberg,*Cardinal spline interpolation*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. MR**0420078**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
41A15,
42C05

Retrieve articles in all journals with MSC: 41A15, 42C05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1077784-X

Keywords:
Wavelets,
cardinal splines,
Euler-Frobenius polynomials,
wavelet decompositions,
algorithms

Article copyright:
© Copyright 1991
American Mathematical Society