Abstract
The objective of this paper is to construct Gabor frame on a positive half-line. A necessary condition and two sufficient conditions for Gabor frame on a positive half-line are given in the time domain.
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P. G. Casazza and O. Christensen, “Weyl-Heisenberg frames for subspaces of L 2(ℝ)”, Proc. Amer. Math. Soc., 129, 145–154, 2001.
O. Christensen, “Frames, Riesz bases, and discrete Gabor/wavelet expansions”, Bull. Amer. Math. Soc. (New Series), 38(3) 273–291, 2001.
O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003).
C. K. Chui and X. L. Shi, “Inequalities of Littlewood-Paley type for frames and wavelets”, SIAM J. Math. Anal., 24, 263–277, 1993.
I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conferences in Applied Mathematics, SIAM, Philadelphia, 1992.
I. Daubechies, A. Grossmann and Y. Meyer, “Painless non-orthogonal expansions”, J. Math. Phys. 27(5) 1271–1283, 1986.
R. J. Duffin and A. C. Shaeffer, “A class of nonharmonic Fourier series”, Trans. Amer. Math. Soc., 72, 341–366, 1952.
Yu. A. Farkov, “On wavelets related to Walsh series”, J. Approx. Theory, 161(1) 259–279, 2009.
H. G. Feichtinger and T. Strohmer, Gabor Analysis and Algorithms: Theory and Applications (Birkhäuser, Boston, 1998).
H. G. Feichtinger and T. Strohmer, Advances in Gabor Analysis (Birkhäuser, Boston, 2003).
D. Gabor, “Theory of communications”, J. Inst. Elect. Engn., 93, 429–457, 1946.
B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications (Kluwer, Dordrecht, 1991).
K. Gröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001).
D. F. Li, G. Wu and X. Zhang, “Two sufficient conditions in frequency domain for Gabor frames”, Appl. Math. Lett., 24, 506–511, 2011.
F. Schipp, W. R. Wade and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol and New York, 1990).
F. A. Shah and L. Debnath, “Dyadic wavelet frames on a half-line using the Walsh-Fourier transform”, Integ. Transf. Spec. Funct., 22(7) 477–486, 2011.
X. L. Shi and F. Chen, “Necessary conditions for Gabor frames”, Science in China: Series A, 50(2) 276–284, 2007.
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Original Russian Text © F. A. Shah, 2012, published in Izvestiya NAN Armenii. Matematika, 2012, No. 5, pp. 65–76.
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Shah, F.A. Gabor frames on a half-line. J. Contemp. Mathemat. Anal. 47, 251–260 (2012). https://doi.org/10.3103/S1068362312050056
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DOI: https://doi.org/10.3103/S1068362312050056