## Characterization of scaling functions in a multiresolution analysis

HTML articles powered by AMS MathViewer

- by P. Cifuentes, K. S. Kazarian and A. San Antolín
- Proc. Amer. Math. Soc.
**133**(2005), 1013-1023 - DOI: https://doi.org/10.1090/S0002-9939-04-07786-X
- Published electronically: November 19, 2004
- PDF | Request permission

## Abstract:

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map $A: \mathbb {R}^n\rightarrow \mathbb {R}^n$ such that $A(\mathbb {Z}^n) \subset \mathbb {Z}^n$ and all (complex) eigenvalues of $A$ have absolute value greater than $1.$ In the general case the conditions depend on the map $A.$ We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when $A$ is a diagonal matrix with all numbers in the diagonal equal to $2.$ There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.## References

- Andrew M. Bruckner,
*Differentiation of real functions*, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR**507448**, DOI 10.1007/BFb0069821 - Carl de Boor, Ronald A. DeVore, and Amos Ron,
*On the construction of multivariate (pre)wavelets*, Constr. Approx.**9**(1993), no. 2-3, 123–166. MR**1215767**, DOI 10.1007/BF01198001 - Charles K. Chui,
*An introduction to wavelets*, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR**1150048** - Ole Christensen,
*An introduction to frames and Riesz bases*, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR**1946982**, DOI 10.1007/978-0-8176-8224-8 - Xingde Dai, David R. Larson, and Darrin M. Speegle,
*Wavelet sets in $\mathbf R^n$*, J. Fourier Anal. Appl.**3**(1997), no. 4, 451–456. MR**1468374**, DOI 10.1007/BF02649106 - Ingrid Daubechies,
*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107**, DOI 10.1137/1.9781611970104 - K. Gröchenig and W. R. Madych,
*Multiresolution analysis, Haar bases, and self-similar tilings of $\textbf {R}^n$*, IEEE Trans. Inform. Theory**38**(1992), no. 2, 556–568. MR**1162214**, DOI 10.1109/18.119723 - Eugenio Hernández and Guido Weiss,
*A first course on wavelets*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR**1408902**, DOI 10.1201/9781420049985 - Wally R. Madych,
*Some elementary properties of multiresolution analyses of $L^2(\textbf {R}^n)$*, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 259–294. MR**1161255** - Stephane G. Mallat,
*Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 69–87. MR**1008470**, DOI 10.1090/S0002-9947-1989-1008470-5 - Yves Meyer,
*Ondelettes et opérateurs. I*, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. MR**1085487** - I. P. Natanson;
*Theory of functions of a real variable,*London, vol. II, 1960. - Robert S. Strichartz,
*Construction of orthonormal wavelets*, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 23–50. MR**1247513** - P. Wojtaszczyk,
*A mathematical introduction to wavelets*, London Mathematical Society Student Texts, vol. 37, Cambridge University Press, Cambridge, 1997. MR**1436437**, DOI 10.1017/CBO9780511623790

## Bibliographic Information

**P. Cifuentes**- Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: patricio.cifuentes@uam.es
**K. S. Kazarian**- Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: kazaros.kazarian@uam.es
**A. San Antolín**- Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: angel.sanantolin@uam.es
- Received by editor(s): March 6, 2003
- Received by editor(s) in revised form: June 19, 2003
- Published electronically: November 19, 2004
- Additional Notes: The first two authors were partially supported by BFM2001-0189
- Communicated by: David R. Larson
- © Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**133**(2005), 1013-1023 - MSC (2000): Primary 42C15; Secondary 42C30
- DOI: https://doi.org/10.1090/S0002-9939-04-07786-X
- MathSciNet review: 2117202