Frames Adapted to a Phase-Space Cover

Abstract

Abstract. We construct frames adapted to a given cover of the time-frequency or time-scale plane. The main feature is that we allow for quite general and possibly irregular covers. The frame members are obtained by maximizing their concentration in the respective regions of phase-space. We present applications in time-frequency, wavelet and Gabor analysis.

Multi-window weaving frames

“…We will now have a closer look at the time-frequency localization operator H ηγ : L 2 (R) → L 2 (R). Assuming that the family of symbols {η γ | γ ∈ Γ} is non-negative and well-spread we can use the results in [4] which state that, under these conditions, the localization operator H ηγ is positive and trace class and, hence, can be diagonalized. Therefore, we have…”

“…In a series of papers, cf. [4] for references, it has been shown, that, if a family of localization operators H ηγ ,φ is well-spread and H ηγ ,φ is invertible, one has the norm equivalence f 2 2 ≈ γ∈Γ H ηγ ,φ f 2 2 . Thereby, however, the window φ defining the localization operator remains the same for all γ.…”

“…Proof : We start by recalling from [4], that for (2) to hold, the family of operators need to fulfill two conditions. First, they must be well-spread and second, the sum of operators must be invertible, i.e., we require, for some A > 0, that…”

“…Multi-window Gabor frames. In [4] it was shown that the norm equivalence of the sum of localized functions as stated in Prop. 3.3 and Prop.3.5 is maintained if the full spectral representation of the operators is replaced by truncated versions.…”

Sharp rates of convergence for accumulated spectrograms

“…If Ω ⊆ R 2d is compact, then H Ω is a compact and positive operator on L 2 (R d ) [9,10,16,39]. Hence H Ω can be diagonalized as…”