Decay and smoothness for eigenfunctions of localization operators

An excursion to multiplications and convolutions on modulation spaces

“…The general multiplication and convolution properties in Section 2 also overlap with results by Bastianoni, Cordero and Nicola in [1], by Bastianoni and Teofanov in [2], and by Guo, Chen, Fan and Zhao in [33].…”

“…There are several approaches to multiplication and convolution in the case when the involved Lebesgue exponents belong to [1, ∞] (see [9,17,20,33,44,49]). Here we consider the case when these exponents belong (0, ∞) (see also [1,2,26,42,43,51]). In addition, and in order to keep the survey style of our exposition, we focus on the bilinear case, and refer to [55] for extension of these results to multi-linear products as well as allowing the Lebesgue exponents to belong to the full interval (0, ∞].…”

“…Proposition 2.4. Let s > 1, p, q ∈ (0, ∞], ω ∈ P E (R 2d ) be such that (2.12) holds for some r > 0, φ, ψ ∈ Σ s (R d ) with values in [0,1] be such that (2.14), (2.15) and (2.16) hold true, and let f ∈ Σ ′ s (R d ). Then the following is true:…”

“…To show Proposition 0.1 in the quasi-Banach setting, apart from the usual use of of Hölder's and Young's inequalities, additional arguments are needed. In our situation we discretize the situations in similar ways as in [1] by using Gabor analysis for modulation spaces, and then apply some further arguments, valid in non-convex analysis. This approach is slightly different compared to what is used in [33] which follows the discretization technique introduced in [56], and which has some traces of Gabor analysis.…”

Daubechies’ Time–Frequency Localization Operator on Cantor Type Sets I

“…Since its conception, this operator has not only been studied as an operator between L 2spaces, but also more broadly as an operator between modulation spaces, and therein questions regarding boundedness and properties of eigenfunctions and egenvalues remain relevant (see [10], [11], [12]). If we stick to the L 2 (R)-context, and assume the weight to be real-valued and integrable, the operator P F,φ : L 2 (R) → L 2 (R) becomes self-adjoint, compact.…”