# zbMATH — the first resource for mathematics

A characterization of dimension functions of wavelets. (English) Zbl 0979.42018
A finite set $$\Psi = \{ \psi^1, \ldots, \psi^L \} \subset L^2(R^d)$$ is called an orthonormal multiwavelet, if the collection $\{ \psi^l_{j,k} (x) = |\det A|^{j/2} \psi^l (A^j x-k) : j\in Z,\;k \in Z^d,\;l=1, \ldots, L \}$ forms an orthonormal basis for $$L^2(R^d)$$. Here, $$A$$ is an expansive matrix which preserves $$Z^d$$. The dimension function of such a multiwavelet is defined as $D_\Psi (\xi) = \sum_{l=1}^L \sum_{j=1}^\infty \sum_{k\in Z^d} |\widehat \psi^l (B^j(\xi+k))|^2,$ where $$B=A^T$$.
It is well known that dimension functions are measurable, nonnegative, $$Z^d$$-periodic and integer-valued. In this paper the authors provide a full characterization of all dimension functions, by adding to the above properties the following conditions: $\int_{(-1/2,1/2]^d} D(\xi) d\xi = L/(q-1), \quad \text{where }q = |\det A|,\tag{1}$
$\liminf_{n\to \infty} D(B^{-n}\xi) \geq 1,\tag{2}$
$\sum_{k \in Z^d} \chi_\Delta(\xi+k) \geq D(\xi), \text{ a.e. for }\Delta = \{ \xi \in R^d: D(B^{-j}\xi) \geq 1;\;j\in N\cup\{0\} \},\tag{3}$
$\sum_{d\in {\mathcal D}} D(\xi+B^{-1}d) = D(B\xi)+L, \quad \text{a.e.} .\tag{4}$ Here, $${\mathcal D}= \{d_1, \dots, d_q \}$$ denotes the set of representatives of $$Z^d/BZ^d$$, with $$d_1=0$$.
The authors further study the properties of the dimension function using these characterizing conditions, and provide several examples of wavelets for given dimension functions. In particular, they show that for any dimension function $$D$$, there exists an MSF wavelet whose dimension function is $$D$$.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:
##### References:
  Auscher, P., Solution of two problems on wavelets, J. geom. anal., 5, 181-236, (1995) · Zbl 0843.42015  L. W. Baggett, An abstract approach to the wavelet dimension function using group representations, J. Fourier Anal. Appl, to appear. · Zbl 0964.42023  Baggett, L.W.; Medina, H.A.; Merrill, K.D., Generalized multiresolution analyses, and a construction procedure for all wavelet sets in $$R$$^{n}, J. Fourier anal. appl., 5, 563-573, (1999) · Zbl 0972.42021  Baggett, L.W.; Merrill, K.D., Abstract harmonic analysis and wavelets in $$R$$^{n}, The functional and harmonic analysis of wavelets and frames, san antonio, TX, 1999, (1999), American Math. Society Providence, p. 17-27 · Zbl 0957.42021  B. Behera, and, S. Madan, On a class of band-limited wavelets not associated with an MRA, preprint. · Zbl 1096.42024  Bownik, M., A characterization of affine dual frames in L2($$R$$n), Appl. comput. harmon. anal, 8, 203-221, (2000) · Zbl 0961.42018  Calogero, A., Wavelets on general lattices, associated with general expanding maps on $$R$$^{n}, (1998), Università di Milano · Zbl 0914.42026  Dai, X.; Larson, D.R., Wandering vectors for unitary systems and orthogonal wavelets, Mem. amer. math. soc., 134, (1998) · Zbl 0990.42022  Dai, X.; Larson, D.R.; Speegle, D.M., Wavelet sets in $$R$$^{n}, J. Fourier anal. appl., 3, 451-456, (1997) · Zbl 0881.42023  Frazier, M.; Garrigós, G.; Wang, K.; Weiss, G., A characterization of functions that generate wavelet and related expansion, Proceedings of the conference dedicated to Professor miguel de guzmán (el escorial, 1996), (1997), p. 883-906 · Zbl 0896.42022  Gripenberg, G., A necessary and sufficient condition for the existence of a father wavelet, Studia math., 114, 207-226, (1995) · Zbl 0838.42012  Gu, Q.; Han, D., On multiresolution analysis (MRA) wavelets in $$R$$^{N}, J. Fourier anal. appl., 6, 437-447, (2000) · Zbl 0964.42021  Hernández, E.; Weiss, G., A first course on wavelets, Studies in advanced mathematics, (1996), CRC Press Boca Raton  Ionascu, E.J.; Pearcy, C.M., On subwavelet sets, Proc. amer. math. soc., 126, 3549-3552, (1998) · Zbl 0918.42028  Lemarié-Rieusset, P.-G., Existence de “fonction-père” pour LES ondelettes à support compact, C.R. acad. sci. Paris. ser. I math., 314, 17-19, (1992) · Zbl 0752.42017  Lemarié-Rieusset, P.-G., Sur lrexistence des analyses multi-résolutions en théorie des ondelettes, Rev. mat. iberoamer., 8, 457-474, (1992) · Zbl 0779.42019  Madych, W.R., Orthogonal wavelet bases for L2($$R$$n), Fourier analysis, orono, ME, 1992, (1994), Dekker New York, p. 243-302 · Zbl 0807.42023  Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2($$R$$), Trans. amer. math. soc., 315, 69-87, (1989) · Zbl 0686.42018  Papadakis, M.; Šikić, H.; Weiss, G., The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts, J. Fourier anal. appl., 5, 495-521, (1999) · Zbl 0935.42023  Speegle, D.M., The s-elementary wavelets are path-connected, Proc. amer. math. soc., 127, 223-233, (1999) · Zbl 0907.46015  Walters, P., An introduction to ergodic theory, (1982), Springer-Verlag New York · Zbl 0475.28009  Wang, X., The study of wavelets from the properties of their Fourier transforms, (1995), Washington University St. Louis  Weber, E., Applications of the wavelet multiplicity function, The functional and harmonic analysis of wavelets and frames, san antonio, TX, 1999, (1999), American Math. Society Providence, p. 297-306 · Zbl 0960.42014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.