Localization of the window functions of dual and tight Gabor frames generated by the Gaussian function. (English. Russian original) Zbl 07891402
Sb. Math. 215, No. 3, 364-382 (2024); translation from Mat. Sb. 215, No. 3, 80-99 (2024).
Authors’ abstract: Gabor frames generated by the Gaussian function are considered. The localization of the window functions of dual frames is estimated in terms of the uncertainty constants, its dependence on the relation between the parameters of the time-frequency window, and the degree of overcompleteness. It is shown that localization worsens rapidly with the increasing disproportion in the parameters of the window. On the other hand, the higher the system of functions forming the frame is overdetermined, the better the window function of the dual frame is localized. For a tight frame, the localization of the window function with the same set of parameters is much better than that for the dual frame. This problem is closely related to the problem of interpolation because we have uniform Gaussian function shifts. The nodal interpolation function and the window function of the dual frame are constructed from the same coefficients. These coefficients also play an important role in deriving formulae for the uncertainty constants. This is why their properties related to sign alternation and the monotonicity of decrease of the absolute value are considered in the paper.
Reviewer: Paşc Găvruţă (Timişoara)
MSC:
| 42C15 | General harmonic expansions, frames |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65T60 | Numerical methods for wavelets |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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