×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
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.