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Stability of frames generated by nonlinear Fourier atoms. (English) Zbl 1086.42018

For a complex number \(a\), let \(\theta_a(t)= t+2\arctan\left(\frac{| a| \sin(t -Arg \;a)}{1-| a| \cos(t- Arg \;a)}\right).\) Extending Kadec’ 1/4-theorem, it is proved that the exponentials \(\{e^{i\lambda_n \theta_a(t)}\}_{n\in Z}\) form a Riesz basis for \(L^2(0,2\pi)\) (and a weighted version hereof) if \(\sup| \lambda_n-n| <1/4.\) Multiplying the functions with translates of an appropriate window function leads to a frame for \(L^2(\mathbf R)\).

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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