Abstract:We characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.
“…and W(ω, ℓ p,q ), respectively, at each occurrence. (For r = ∞ , see [29] when p, q ∈ [1, ∞], [26,51] when p, q ∈ (0, ∞], and for r ∈ (0, ∞], see [54]. )…”
We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.
“…and W(ω, ℓ p,q ), respectively, at each occurrence. (For r = ∞ , see [29] when p, q ∈ [1, ∞], [26,51] when p, q ∈ (0, ∞], and for r ∈ (0, ∞], see [54]. )…”
We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.
“…and W(ω, ℓ p,q ), respectively, at each occurrence. (For r = ∞ , see [14] when p, q ∈ [1, ∞], [11,29] when p, q ∈ (0, ∞], and for r ∈ (0, ∞], see [32]. )…”
Section: Mixed Norm Space Of Lebesgue Typesmentioning
We prove the boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.
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