# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. (English) Zbl 1021.42020
For any $d\times d$ dilation matrix $M$ it is proved how to construct compactly supported tight wavelet frames and orthonormal wavelet bases having exponential decay; the bases have the form $\psi_{j,k}= |\text{det } M|^{j/2}\psi(M^j \cdot -k), j\in \bbfZ, k\in \bbfZ^d$ for some functions $\psi \in L^2(\bbfR^d)$; they are derived from refinable functions $\phi$, in the sense that they have the form $\psi = |\text{det } M|\sum_{k\in \bbfZ^d}b_k \phi (M \cdot -k)$ for some sequence $\{b_k\}_{k\in \bbfZ^d}$. One of the main results is as follows. Given any positive integer $r$, there exists a collection $\Psi$ of at most $(3/2)^d|\text{det } M|$ functions in $C^r(\bbfR^d)$, derived from a refinable function with compact support, such that $\Psi$ has vanishing moments of order $r$ and generates a tight wavelet frame for $L^2(\bbfR^d)$.

##### MSC:
 42C40 Wavelets and other special systems
Full Text:
##### References:
 [1] Battle, G.: A block spin construction of ondelettes, ilemarié functions. Comm. math. Phys. 110, 601-615 (1987) [2] Bownik, M.: The construction of r-regular wavelets for arbitrary dilations. J. Fourier anal. Appl. 7, 489-506 (2001) · Zbl 1003.42021 [3] C.K. Chui, W. He, J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal., to appear. [4] Chui, C. K.; Shi, X. L.: Bessel sequences and affine frames. Appl. comput. Harmon. anal. 1, 29-49 (1993) · Zbl 0788.42011 [5] Cohen, A.; Daubechies, I.: A stability criterion for biorthogonal wavelet bases and their related subband coding scheme. Duke math. J. 68, 313-335 (1992) · Zbl 0784.42022 [6] Cohen, A.; Gröchenig, K.; Villemoes, L.: Regularity of multivariate refinable functions. Constr. approx. 15, 241-255 (1999) · Zbl 0937.42017 [7] Dai, X.; Diao, Y.; Gu, Q.: Frame wavelet sets in R. Proc. amer. Math. soc. 129, 2045-2055 (2001) · Zbl 0973.42029 [8] Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. pure appl. Math. 41, 906-996 (1988) · Zbl 0644.42026 [9] Daubechies, I.: Ten lectures on wavelets. CBMS-NSF regional conference series in applied mathematics 61 (1992) [10] I. Daubechies, B. Han, A. Ron, Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear. · Zbl 1035.42031 [11] I. Daubechies, B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx., to appear. · Zbl 1055.42025 [12] Daubechies, I.; Han, B.: The canonical dual frame of a wavelet frame. Appl. comput. Harmon. anal. 12, 269-285 (2002) · Zbl 1013.42023 [13] Gröchenig, K.; Ron, A.: Tight compactly supported wavelet frames of arbitrarily high smoothness. Proc. amer. Math. soc. 126, 1101-1107 (1998) · Zbl 0911.42014 [14] Han, B.: On dual wavelet tight frames. Appl. comput. Harmon. anal. 4, 380-413 (1997) · Zbl 0880.42017 [15] Han, B.; Jia, R. Q.: Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. anal. 29, 1177-1199 (1998) · Zbl 0915.65143 [16] Han, B.: Symmetry property and construction of wavelets with a general dilation matrix. Linear algebra appl. 353, 207-225 (2002) · Zbl 0999.42020 [17] Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix anal. Appl. 24, 693-714 (2003) · Zbl 1032.42036 [18] Hernández, E.; Weiss, G.: A first course on wavelets. (1996) · Zbl 0885.42018 [19] R.Q. Jia, Refinable functions and wavelets associated with arbitrary dilation matrices, plenary talk in the International Conference of Computational Harmonic Analysis, June 4--8, Hong Kong. [20] Jia, R. Q.: Approximation properties of multivariate wavelets. Math. comput. 67, 647-665 (1998) · Zbl 0889.41013 [21] Jia, R. Q.: Characterization of smoothness of multivariate refinable functions in Sobolev spaces. Trans. amer. Math. soc. 351, 4089-4112 (1999) · Zbl 1052.42029 [22] Jia, R. Q.; Micchelli, C. A.: Using the refinement equation for the construction of pre-wavelets II: Power of two. Curves and surfaces, 209-246 (1991) · Zbl 0777.41013 [23] Q.T. Jiang, Parameterization of masks for tight affine frames with two symmetric/antisymmetric generators (2001), preprint. [24] Lawton, W. M.: Tight frames of compactly supported affine wavelets. J. math. Phys. 31, 1898-1901 (1990) · Zbl 0708.46020 [25] Lemarié, P. -G.: Ondelettes à localisation exponentielles. J. math. Pures appl. 67, 227-236 (1988) [26] Narasimhan, R.: Several complex variables. (1971) · Zbl 0223.32001 [27] Ron, A.; Shen, Z. W.: Affine systems in $L2(Rd)$the analysis of the analysis operator. J. funct. Anal. 148, No. 2, 408-447 (1997) · Zbl 0891.42018 [28] Ron, A.; Shen, Z. W.: Compactly supported tight affine spline frames in $L2(Rd)$. Math. comput. 67, 191-207 (1998) · Zbl 0892.42018 [29] Strichartz, R.: Wavelets and self-affine tilings. Constr. approx. 9, 327-346 (1993) · Zbl 0813.42021 [30] Villemoes, L.: Sobolev regularity of wavelets and stability of iterated filter banks. Progress in wavelet analysis and applications, 243-251 (1993) · Zbl 0878.42024