Bohnke, G. Wavelet lattices associated with Lorentz groups. (Treillis d’ondelettes associés aux groupes de Lorentz.) (French) Zbl 0743.43005 Ann. Inst. Henri Poincaré, Phys. Théor. 54, No. 3, 245-259 (1991). By using the action of a Lorentz group on a cone in \(\mathbb{R}^{n+1}\), a tight frame of wavelets for functions on the cone is constructed. The cone is \[ C:=\{x=(x_ 0,x_ 1,\dots,x_ n)\in\mathbb{R}^{n+1}:\;x_ 0>0,\;x^ 2_ 0>x_ 1^ 2+x_ 2^ 2+\dots +x_ n^ 2\}. \] The group \(G\) that acts on \(C\) is a semi-direct product of \(\{t\in\mathbb{R}: t>0\}\) (under multiplication for the dilation action) with \(SO_ 0(1,n)\) and \(\mathbb{R}^{n+1}\). The idea is to construct one basic function with compact support in \(C\) and a set of countably many \(G\)-translates of it which form a tight frame (a certain kind of spanning set for a Hilbert space of functions on \(C\)). For example when \(n=1\) the Hilbert space on \(C\) is \(L^ 2(C,(x^ 2_ 0-x_ 1^ 2)^{-1}dx_ 0dx_ 1)\). After some details on the group action the paper presents constructions of wavelets for both \(n=1\) and the general case \(n>1\). Reviewer: C.F.Dunkl (Charlottesville) Cited in 7 Documents MSC: 43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc. 22E15 General properties and structure of real Lie groups Keywords:Lorentz group; frame of wavelets; semi-direct product; dilation action; group action × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] S.T. Ali , J.P. Antoine et J.P. Gazeau , Square Integrability of Group Representations on Homogeneous Spaces. I. Reproducing Triples and Frames. II. Generalized Square Integrability and Equivalent Families of Coherent States , U.C.L.-I.P.T. 89 18, preprints, décembre 1989 . [2] J. Bertrand et P. Bertrand , A Relativistic Wigner Function Affiliated with the Weyl-Poincaré Group , preprint. · Zbl 0850.22006 [3] I. Daubechies , Orthonormal Bases of Compactly Supported Wavelets , Comm. Pure Appl. Math. , vol. XLI , 1988 , p. 909 - 996 . MR 951745 | Zbl 0644.42026 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705 [4] I. Daubechies , The Wavelet Transform, Time-Frequency Localization and Signal Analysis , preprint. MR 1066587 · Zbl 0738.94004 [5] H. Feichtinger et K. Gröchenig , A Unified Approach to Atomic Decompositions Trough Integrable Group Representations , preprint. · Zbl 0658.22007 [6] A. Grossmann , J. Morlet et T. Paul , Transforms Associated to Square Integrable Group Representations. I. General results , J. Math. Phys. , vol. 26 , ( 10 ), 1985 , p. 2473 - 2479 . MR 803788 | Zbl 0571.22021 · Zbl 0571.22021 · doi:10.1063/1.526761 [7] A. Grossmann , J. Morlet et T. Paul , Transforms Associated to Square Integrable Group Representations. II. Examples , Ann. Inst. Henri-Poincaré , vol. 45 , n^\circ 3 , 1986 , p. 293 - 309 . Numdam | MR 868528 | Zbl 0601.22001 · Zbl 0601.22001 [8] P.G. Lemarie et Y. Meyer , Ondelettes et bases hilbertiennes , Revista Matemática Iberoamericana , vol. 2 , n^\circ 12 , 1986 , p. 1 - 18 . MR 864650 | Zbl 0657.42028 · Zbl 0657.42028 [9] M.S. Raghunathan , Discrete Subgroups of Lie Groups , Ergebnisse der Mathematik , vol. 68 , Springer , 1972 . MR 507234 | Zbl 0254.22005 · Zbl 0254.22005 [10] P.J. Sally Jr , Analytic Continuation of the Irreducible Unitary Representations of the Universal Covering Group of SL (2, R) , Memoirs of the A.M.S. , n^\circ 69 , 1967 . Zbl 0157.20702 · Zbl 0157.20702 [11] A. Unterberger et J. Unterberger , La série discrète de SL (2, R) et les opérateurs pseudo-différentiels sur une demi-droite , Ann. Sci. Ec. Norm. Sup. , vol. 17 , 1984 , p. 83 - 116 . Numdam | MR 744069 | Zbl 0549.35119 · Zbl 0549.35119 [12] A. Unterberger et J. Unterberger , A Quantization of the Cartan Domain BDI (q=2) and Operators on the Light Cone , J. Funct. Anal. , vol. 72 , n^\circ 2 , 1987 , p. 279 - 319 . MR 886815 | Zbl 0632.58033 · Zbl 0632.58033 · doi:10.1016/0022-1236(87)90090-5 [13] A. Unterberger , Analyse harmonique et analyse pseudo-différentielle du cône de lumière , Astérisque , n^\circ 156 , S.M.F., 1987 . MR 947371 | Zbl 0643.35118 · Zbl 0643.35118 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.