Limit Properties of Systems of Integer Translates and Functions Generating Tight Gabor Frames

Киселев, Евгений Александрович; Minin, L. A.; Novikov, Igor

doi:10.1134/s0001434619070071

Math Notes

2019

DOI: 10.1134/s0001434619070071

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Limit Properties of Systems of Integer Translates and Functions Generating Tight Gabor Frames

Евгений Александрович Киселев

L. A. Minin

Igor Novikov

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2024

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Cited by 2 publications

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Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Kiselev,

Minin,

Novikov

et al. 2024

Математический сборник

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Рассматриваются фреймы Габора, порожденные функцией Гаусса. С помощью констант неопределенности оценивается локализация функций двойственных фреймов в зависимости от соотношения параметров частотно-временного окна и степени переполненности. Общий вывод таков: при увеличении диспропорции окна локализация быстро ухудшается. С другой стороны, чем более переопределена исходная система функций, тем лучше локализованы функции двойственного фрейма. Для жесткого фрейма локализация при одном и том же наборе параметров существенно лучше, чем для двойственного фрейма. Рассматриваемая задача тесно связана с задачей интерполяции по равномерным сдвигам функции Гаусса. Построение узловой функции при интерполяции и функции окна двойственного фрейма осуществляется с помощью одних и тех же коэффициентов. Эти коэффициенты играют важную роль и при выводе формул для констант неопределенности. Поэтому в работе изучаются их свойства, связанные со знакочередуемостью и монотонностью убывания по модулю. Библиография: 38 названий.

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Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Kiselev,

Minin,

Novikov

et al. 2024

Математический сборник

View full text Add to dashboard Cite

show abstract

Localization of the window functions of dual and tight Gabor frames generated by the Gaussian function

Kiselev,

Minin,

Novikov

et al. 2024

Sb. Math.

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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, it 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 by we have uniform shifts of the Gaussian function. Both the nodal interpolation function and the window function of the dual frame are constructed from the same coefficients. These coefficients play an important role also in the derivation of 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. Bibliography: 38 titles.

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Limit Properties of Systems of Integer Translates and Functions Generating Tight Gabor Frames

Cited by 2 publications

References 9 publications

Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Localization of the window functions of dual and tight Gabor frames generated by the Gaussian function

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