Wavelet multipliers on 𝐿^{𝑝}(ℝⁿ)

Abstract

Abstract. We give results on the boundedness and compactness of wavelet multipliers on L p (R n ), 1 ≤ p ≤ ∞.

Frames Adapted to a Phase-Space Cover

“…In the context of time-scale analysis, one can define operators in analogy to time-frequency localization operators using the wavelet transform in (17) instead of the STFT. These are called time-scale localization operators or wavelet multipliers [18,53,40]. With an adequate choice of E, S E and P , time-scale localization operators are unitarily equivalent to phase-space multipliers.…”

“…Remark 5.11. When η γ is the characteristic function of a set Ω γ , the condition in (40) holds whenever the sets satisfy: B r (γ) ⊆ Ω γ ⊆ B R (γ), with R > r > 0 and Γ a relatively separated set.…”

“…Hence, the non-zero eigenvalues of H Br form an infinite non-increasing sequence λ R k > 0, k ≥ 1. From(40) it follows that H ηγ ≥ cH Br+γ (cf. Proposition 2.10).…”

Characterization of coorbit spaces with phase-space covers

“…The operator M m is clearly bounded by Proposition 3 and the solidity of E. When the space S is taken to be the range of the abstract wavelet transform associated with an unitary representation of G, these operators are called localization operators or wavelet multipliers (see for example [36,50,39]). (More precisely, the operators M m are unitary equivalent to localization operators, see Section 9.1 for further details).…”

“…Of course, the rigorous interpretation of M m (f ) is problematic since, in general, T M m (f ) = mT (f ). When T is the abstract wavelet transform (representation-coefficients function) associated with an unitary action of a group, these operators are know as localization operators or wavelet multipliers [36,50,39]. In the case of time-frequency analysis these operators are known as time-frequency localization operators or multipliers of the short-time Fourier transform [8,5,6,3].…”