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Continuous wavelet transforms on the space $$L^{2}(\mathbf R,\mathbb H; dx)$$. (English) Zbl 1040.42031
Summary: Let $$\mathbf P$$ be the affine group of the real line $$\mathbf R$$, and let $$\mathbb H$$ be the set of all quaternions. Thus, $$L^2 (\mathbf R, \mathbb H; dx)$$ denotes the space of all square integrable $$\mathbb H$$-valued functions. From the viewpoint of square integrable group representations, we study the theory of continuous wavelet transforms on $$L^2(\mathbf R,\mathbb H; dx)$$ associated with the group $$\mathbf P$$, and give the Calderón reproducing formula.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 43A80 Analysis on other specific Lie groups
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##### References:
 [1] Feichtinger, H.G.; Grochenig, K.H., Banach spaces related to integrable group representations and their atomic decompositions I, J. funct. anal., 86, 307-340, (1989) · Zbl 0691.46011 [2] Grossmann, A.; Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. math. anal., 15, 723-736, (1984) · Zbl 0578.42007 [3] Grossman, A.; Morlet, J.; Paul, T., Wavelet transforms associated to square integrable representations II, Ann Henri. poincare, 45, 293-309, (1986) · Zbl 0601.22001 [4] Antonie, J.P.; Vandergheynst, P., Wavelets on the n-sphere and related manifolds, J. math. phys., 39, 3987-4008, (1998) · Zbl 0929.42029 [5] He, J.X.; Liu, H.P., Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane, J. math. anal. appl., 208, 58-70, (1997) · Zbl 0885.22015 [6] Liu, H.P.; Peng, L.Z., Admissible wavelets associated with the Heisenberg group, Pacific J. math, 180, 101-123, (1997) · Zbl 0885.22012 [7] Johnson, N.W.; Weiss, A.I., Quaternionic modular groups, Linear algebra and its appl., 295, 159-189, (1999) · Zbl 0960.20031 [8] Pfaff, F.R., A commutative multiplication of number triplets, Amer. math. monthly., 107, 156-162, (2000) · Zbl 1076.30522 [9] Sudbery, A., Quaternionic analysis, (), 199-225 · Zbl 0399.30038 [10] Duval, P., Homographies, quaternions, and rotations, () [11] Deavours, C.A., The quaternion calculus, Amer. math. monthly, 80, 995-1008, (1973) · Zbl 0282.30040 [12] Qian, T., Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. ann., 310, 601-630, (1998) · Zbl 0921.42012 [13] He, J.X., Wavelet transforms associated to square integrable group representations on L2 (\bfc, H; dz), Applicable analysis, 81, 495-512, (2002) · Zbl 1018.42025 [14] Xia, X.G.; Suter, B.W., Vector-valued wavelets and vector filter banks, IEEE trans. signal process, 44, 508-518, (1996) [15] Walden, A.T.; Serroukh, A., Wavelet analysis of matrix-valued time-series, (), 157-179 · Zbl 1010.62087 [16] Jiang, Q.T.; Peng, L.Z., Wavelet transform and Toeplitz-Hankel type operators, Math. scand., 70, 247-264, (1992) · Zbl 0763.42019
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