Step multipliers, Fourier step multipliers and multiplications on quasi-Banach modulation spaces

Toft, Joachim

doi:10.48550/arxiv.2011.06941

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2021

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“…Theorem 25 [46,Theorem 3.2] Let there be given p j , q j ∈ [1, ∞], j = 0, 1, 2, such that R(p) ≤ 1 and R(q) ≤ 0, and let w 1 , w 2 ∈ P E (R 2d ). If w 0 ∈ P E (R 2d ) satisfies (51), then the map…”

Section: Extension Of the Productsmentioning

confidence: 99%

Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation

Dias¹,

Prata²,

Teofanov³

2021

Preprint

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We show that the short-time Fourier transform of the pointwise product of two functions f and h can be written as a suitable product of the short-time Fourier transforms of f and h. The same result is then shown to be valid for the Wigner wave-packet transform. We study the main properties of the new products. We then use these products to derive integro-differential equations on the time-frequency space equivalent to, and generalizing, the cubic nonlinear Schrödinger equation. We also obtain the Weyl-Wigner-Moyal equation satisfied by the Wigner-Ville function associated with the solution of the nonlinear Schrödinger equation. The new equation resembles the Boltzmann equation.

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Section: Extension Of the Productsmentioning

confidence: 99%

Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation

Dias¹,

Prata²,

Teofanov³

2021

Preprint

View full text Add to dashboard Cite

show abstract

Step multipliers, Fourier step multipliers and multiplications on quasi-Banach modulation spaces

Abstract: 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.

Cited by 1 publication

References 20 publications

Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation

Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation

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