Functions of variable bandwidth via time-frequency analysis tools

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:382

Article

Aceska, R.¹  Feichtinger, H. G.² 

[1] Univ Ss Cyril & Methodius, Dept Math & Comp Sci, Fac Mech Engn, MK-1001 Skopje, North Macedonia

[2] Univ Vienna, Fac Math, NuHAG, A-1090 Vienna, Austria

关键词: Gabor frames;    Modulation spaces;    Short-time Fourier transform;    Time-frequency analysis;    Variable bandwidth;   

DOI  :  10.1016/j.jmaa.2011.04.044

来源: Elsevier

【 摘 要 】

This paper is motivated by the apparent lack of a precise mathematical description of the very natural idea of variable bandwidth for a function, defined on the real line. Different existing concepts suffer from serious shortcomings. We will present a new and mathematically well justified function space model, which is based on the theory of coorbit spaces, in a time-frequency context. Making use of the flexibility of this approach we define a family of Hilbert spaces which can be viewed as generalised modulation spaces. More precisely, we define a family of Banach spaces of functions with variable bandwidth (VB-functions) by imposing a weighted mixed-norm condition on the short-time Fourier transform of their elements, punishing the contributions outside a strip of variable width described by the band-width function b. Similar bandwidth functions define the same space with equivalent norms. Any good Gabor frame (e.g. with Schwartz windows) can be used to characterize the membership of a function in such a space. Moreover, various classes of functions, e.g. those obtained by a time-variant filter, are shown to belong to such a space. In addition, error estimates are given when approximating certain subclasses. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free