Abstract
Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk \(\mathbb{D}_{1}\)), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from N samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from N samples is still possible and the accuracy of the approximation, which tends to be exact in the limit N→∞.
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Ali, S.T., Antoine, J.-P., Gazeau, J.P.: Coherent States, Wavelets and Their Generalizations. Springer, Berlin (2000)
Antoine, J.-P., Hohouto, A.L.: Discrete frames of Poincaré coherent states in 1+3 dimensions. J. Fourier Anal. Appl. 9, 141–173 (2003)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses. Springer, Berlin (2003)
Calixto, M., Guerrero, J.: Wavelet transform on the circle and the real line: a unified group-theoretical treatment. Appl. Comput. Harmon. Anal. 21, 204–229 (2006)
Calixto, M., Guerrero, J., Sánchez-Monreal, J.C.: Sampling theorem and discrete Fourier transform on the Riemann sphere. J. Fourier Anal. Appl. 14, 538–567 (2008)
Calixto, M., Guerrero, J., Sánchez-Monreal, J.C.: Sampling theorems and discrete Fourier transforms on curved phase spaces of constant curvature: a unified treatment. In progress
Chirikjian, G.S., Kyatkin, A.: Engineering Applications of Noncommutative Harmonic Analysis. CRC Press, Boca Raton (2001)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
Daoud, M., Jellal, A.: Quantum Hall droplets on disk and effective Weiss-Zumino-Witten action for edge states. Int. J. Geom. Methods Mod. Phys. 4, 1187–1204 (2007)
Davis, P.J.: Circulant Matrices. Chelsea, New York (1994)
Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: Analogues of sampling theorems for some homogeneous spaces. Hiroshima Math. J. 36, 125–140 (2006)
Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: On sampling formulas on symmetric spaces. J. Fourier Anal. Appl. 12, 1–15 (2006)
Feichtinger, H., Pesenson, I.: A reconstruction method for band-limited signals on the hyperbolic plane. Sampl. Theory Signal Image Process. 4, 107–119 (2005)
Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer, Berlin (2005)
Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations I. General results. J. Math. Phys. 26, 2473–2479 (1985)
Guerrero, J., Aldaya, V.: Invariant measures on polarized submanifolds in group quantization. J. Math. Phys. 41, 6747–6765 (2000)
Holschneider, M.: Wavelets: an Analysis Tool. Oxford University Press, London (1998)
Klauder, J.R., Skagerstam, Bo-Sture: Coherent States: Applications in Physics and Mathematical Physics. Singapore, World Scientific (1985)
Kyatkin, A., Chirikjian, G.S.: Computation of robot configuration and workspaces via the Fourier transform on the discrete motion-group. Int. J. Robot. Res. 18 601–615 (1999)
Kyatkin, A., Chirikjian, G.S.: Algorithms for fast convolutions on motion groups. Appl. Comput. Harmon. Anal. 9, 220–241 (2000)
Maslen, D.: Efficient computation of Fourier transforms on compact groups. J. Fourier Anal. Appl. 4, 19–52 (1998)
Maslen, D.: Sampling of functions and sections for compact Groups. Mod. Signal Process. 46, 247–280 (2003)
Perelomov, A.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)
Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352, 4257–4269 (2000)
Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley-Wiener functions on manifolds. J. Geom. Anal. 14, 101–121 (2004)
Pesenson, I.: Deconvolution of band limited functions on non-compact symmetric spaces. Houst. J. Math. 32, 183–204 (2006)
Pesenson, I.: Paley-Wiener approximations and multiscale approximations in Sobolev and Besov spaces on manifolds. J. Geom. Anal. 19, 390–419 (2009)
Stenzel, M.B.: A reconstruction theorem for Riemannian symmetric spaces of noncompact type. J. Fourier Anal. Appl. 15, 839–856 (2009)
Wigner, E.P., Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959)
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Communicated by T. Strohmer.
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Calixto, M., Guerrero, J. & Sánchez-Monreal, J.C. Sampling Theorem and Discrete Fourier Transform on the Hyperboloid. J Fourier Anal Appl 17, 240–264 (2011). https://doi.org/10.1007/s00041-010-9142-5
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DOI: https://doi.org/10.1007/s00041-010-9142-5