Abstract
We give a simple orthonormal wavelet system associated to the quincunx matrix. That system has four generators and those generators are sum of two constant functions on triangles. Our construction is based on the structure of a multiresolution analysis. Moreover, we prove that such a system is complete in \(L^p({\mathbb {R}}^2)\), \(1< p < \infty \). Finally, we give a filter bank algorithm associated to the proposed wavelet system.
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Ayache, A.: Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases. Appl. Comput. Harmon. Anal. 10(1), 99–111 (2001)
Ayache, A.: Construction of non-separable dyadic compactly supported orthonormal wavelet bases for \(L^2(\mathbb{R}^2)\) of arbitrarily high regularity. Rev. Mat. Iberoam. 15(1), 37–58 (1999)
Belogay, E., Wang, Y.: Arbitrarily smooth orthogonal nonseparable wavelets in \(\mathbb{R}^2\). SIAM J. Math. Anal. 30(3), 678–697 (1999)
Bownik, M.: The construction of \(r\)-regular wavelets for arbitrary dilations. J. Fourier Anal. Appl. 7(5), 489–506 (2001)
Cohen, A., Daubechies, I.: Non-separable bidimensional wavelet bases. Rev. Mat. Iberoam. 9(1), 51–137 (1993)
Gröchenig, K., Madych, W.R.: Multiresolution analysis, Haar bases and self-similar tillings of \(\mathbb{R}^n\). IEEE Trans. Inf. Theory 38(2), 556–568 (1992)
Han, B.: Symmetry property and construction of wavelets with a general dilation matrix. Linear Algebra Appl. 353, 207–225 (2002)
Jia, R.Q.: Multiresolution of \(Lp\) spaces. J. Math. Anal. Appl. 184(3), 620–639 (1994)
Karoui, A.: A general construction of nonseparable multivariate orthonormal wavelet bases. Cent. Eur. J. Math. 6(4), 504–525 (2008)
Krishtal, I.A., Robinson, B.D., Weiss, G.L., Wilson, E.N.: Some simple Haar-type wavelets in higher dimensions. J. Geom. Anal. 17(1), 87–96 (2007)
Kovacevic, J., Vetterli, M.: Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for \(\mathbb{R}^n\). IEEE Trans. Inf. Theory 38(2), 533–555 (1992)
Krommweh, J.: An orthonormal basis of directional Haar wavelets on triangles. Results Math. 53(3–4), 323–331 (2009)
Krommweh, J., Plonka, G.: Directional Haar wavelet frames on triangles. Appl. Comput. Harmon. Anal. 27(2), 215–234 (2009)
Lagarias, J.C., Wang, Y.: Haar type orthonormal wavelet bases in \(\mathbb{R}^2\). J. Fourier Anal. Appl. 2(1), 1–14 (1995)
Lai, M.-J.: Construction of multivariate compactly supported orthonormal wavelets. Adv. Comput. Math 25(1–3), 41–56 (2006)
Lan, L., Zhengxing, C., Yongdong, H.: Construction of a class of trivariate nonseparable compactly supported wavelets with special dilation matrix. Bull. Iran. Math. Soc. 38(1), 39–54 (2012)
Maass, P.: Families of orthogonal two-dimensional wavelets. SIAM J. Math. Anal. 27(5), 1454–1481 (1996)
Peng, S.L.: Construction of two-dimensional compactly supported orthogonal wavelets filters with linear phase. Acta Math. Sin. (Engl. Ser.) 18(4), 719–726 (2002)
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The author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain)
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San Antolín, A. A Simple Quincunx Wavelet with Support on Triangles. Results Math 73, 112 (2018). https://doi.org/10.1007/s00025-018-0869-7
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DOI: https://doi.org/10.1007/s00025-018-0869-7