# zbMATH — the first resource for mathematics

Wavelets on general lattices, associated with general expanding maps of $$\mathbf R^n$$. (English) Zbl 0914.42026
Summary: In the context of a general lattice $$\Gamma$$ in $${\mathbb{R}}^n$$ and a strictly expanding map $$M$$ which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity $$d\geq 1,$$ and all the scaling functions. Moreover, we give several examples: in particular, we construct a single MRA and $$C^\infty({\mathbb{R}}^n)$$ wavelet, which is nonseparable and with compactly supported Fourier transform.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:
##### References:
 [1] Bagget W.B., Medina H.A., Merrill K.D.: Generalized multiresolution analyses, and a construction procedure for all wavelet sets in $${\mathbf R}^n$$, preprint. [2] Calogero A.: A characterization of wavelets on general lattices, Journal of Geometric Analysis, to appear. · Zbl 1057.42025 [3] Calogero A.: A characterization of scaling functions of multiresolution analyses on general lattices, preprint (1998). [4] Calogero A.: Wavelets on general lattices, associated with general expanding maps of $${\mathbf R}^n.$$ Ph. D. Thesis, Universitá di Milano (1998). [5] Calogero A., Garrigós G.: A characterization of wavelet families arising from biorthogonal MRA’s of multiplicity $$d$$, preprint (1998). · Zbl 0994.42020 [6] A. Cohen, Ondelettes et traitement numérique du signal, RMA: Research Notes in Applied Mathematics, vol. 25, Masson, Paris, 1992 (French). · Zbl 0826.42024 [7] Albert Cohen and Ingrid Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), no. 1, 51 – 137. · Zbl 0792.42021 [8] Xingde Dai, David R. Larson, and Darrin M. Speegle, Wavelet sets in \?$$^{n}$$, J. Fourier Anal. Appl. 3 (1997), no. 4, 451 – 456. · Zbl 0881.42023 [9] Dai X., Larson D., Speegle D.: Wavelet sets in $${\mathbf R}^n$$, II. Contemporary Mathematics 216, 15-40 (1998). CMP 98:11 [10] L. De Michele and P. M. Soardi, On multiresolution analysis of multiplicity \?, Monatsh. Math. 124 (1997), no. 3, 255 – 272. · Zbl 0908.42021 [11] Frazier M., Garrigós G., Wang K., Weiss G.: A characteritazion of functions that generate wavelet and related expansion, Journal of Fourier Analysis and Applications, Vol. 3 (special issue), (1997). CMP 98:09 · Zbl 0896.42022 [12] Garrigós G.: The characterization of wavelets and related functions and connectivity of $$\alpha$$-localized wavelets on $${\mathbf R}$$, Ph.D. Thesis, Washington University in St. Louis (1998). [13] Gustaf Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995), no. 3, 207 – 226. · Zbl 0838.42012 [14] Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), no. 3, 373 – 401. · Zbl 0806.41016 [15] K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of \?$$^{n}$$, IEEE Trans. Inform. Theory 38 (1992), no. 2, 556 – 568. · Zbl 0742.42012 [16] Loïc Hervé, Multi-resolution analysis of multiplicity \?: applications to dyadic interpolation, Appl. Comput. Harmon. Anal. 1 (1994), no. 4, 299 – 315. · Zbl 0814.42017 [17] 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. · Zbl 0885.42018 [18] Eugenio Hernández, Xihua Wang, and Guido Weiss, Smoothing minimally supported frequency wavelets. I, J. Fourier Anal. Appl. 2 (1996), no. 4, 329 – 340. · Zbl 0944.42021 [19] Eugenio Hernández, Xihua Wang, and Guido Weiss, Smoothing minimally supported frequency wavelets. II, J. Fourier Anal. Appl. 3 (1997), no. 1, 23 – 41. · Zbl 0944.42022 [20] Hernández E., Wang X., Weiss G.: Characterization of wavelets, scaling function and wavelets associated with multiresolution analysis. Washington University in St. Louis, Preprint (1995). [21] Kahane J. P., Lemarié-Rieusset P.G.: Fourier series and wavelets. Gordon and Breach (1995). · Zbl 0966.42001 [22] P. G. Lemarié , Les ondelettes en 1989, Lecture Notes in Mathematics, vol. 1438, Springer-Verlag, Berlin, 1990 (French). Abstracts from the Seminar on Harmonic Analysis held at the Université de Paris-Sud, Orsay, January – March 1989. [23] Lemarié-Rieusset P.G.: Ondelettes à localisation exponentielle, J. Math. Pure et Appl. 67 (1988), 227-236. [24] Wally R. Madych, Some elementary properties of multiresolution analyses of \?²(\?$$^{n}$$), Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 259 – 294. · Zbl 0760.41030 [25] Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. · Zbl 0694.41037 [26] Soardi P.M., Weiland D.: Single wavelets in $$n$$-dimensions, The Journal of Fourier Analysis and Applications 4, 299-315 (1998). CMP 99:03 · Zbl 0911.42013 [27] Gilbert Strang and Truong Nguyen, Wavelets and filter banks, Wellesley-Cambridge Press, Wellesley, MA, 1996. · Zbl 1254.94002 [28] Robert S. Strichartz, Construction of orthonormal wavelets, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 23 – 50. · Zbl 0891.42017 [29] Robert S. Strichartz, Wavelets and self-affine tilings, Constr. Approx. 9 (1993), no. 2-3, 327 – 346. · Zbl 0813.42021 [30] Wang X.: The study of wavelets from the properties of their Fourier trasforms, Ph.D. Thesis, Washington University in St. Louis (1995).
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.