Abstract
Given a real sequence {λn}n∈ℤ. Suppose that\(\left\{ {e^{i\lambda _n x} } \right\}_{n \in \mathbb{Z}}\) is a frame for L2[−π, π] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {μn}n∈ℤ with ¦μn −λn¦ ≤δ <L,\(\left\{ {e^{i\mu _n x} } \right\}_{n \in \mathbb{Z}}\) is also a frame for L2[−π, π]. Balan [1] obtained\(L_R = \tfrac{1}{4} - \tfrac{1}{\pi }\)arcsin\(\left( {\tfrac{1}{{\sqrt 2 }}\left( {1 - \sqrt {\tfrac{A}{B}} } \right)} \right)\). This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem\(\left( {L_R = \tfrac{1}{4}ifA = B} \right)\) and is better than\(L_{DS} = \tfrac{1}{\pi }\ln \left( {1 + \sqrt {\tfrac{A}{B}} } \right)\) (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.
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References
Balan, R. (1997). Stability theorems for Fourier frames and wavelet Riesz bases,J. Four. Anal. Appl.,3, 499–504.
Christensen, O. (1997). Perturbation of frames and applications to Gabor frames, inGabor Analysis and Algorithms: Theory and Applications, Feichtinger, H.G. and Strohmer, T., Eds., Birkhäuser, Ch. 5, 193–209.
Duffin, R.J. and Schaeffer, A.C. (1952). A class of nonharmonic Fourier series,Trans. Amer. Math. Soc.,72, 341–366.
Kadec, M.I. (1964). The exact value of the Paley-Wiener constant,Sov. Math. doklady,5, 559–561.
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Communicated by Ingrid Daubechies
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Su, W., Zhou, X. A sharper stability bound of Fourier frames. The Journal of Fourier Analysis and Applications 5, 67–71 (1999). https://doi.org/10.1007/BF01274189
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DOI: https://doi.org/10.1007/BF01274189