Sentence examples for modulation spaces have from inspiring English sources

Recently Hardy type modulation spaces have been proposed in [13].

The modulation spaces have been well known as the 'right' spaces in time-frequency analysis.

The Fourier multipliers for the modulation spaces have been developed in many papers [14, 17, 24 26] where the so-called unimodular Fourier multipliers were studied and applied into PDEs.

During the last ten years, modulation spaces have not only become useful function spaces for time-frequency analysis, they have also been employed to study boundedness properties of pseudo-differential operators, Fourier multipliers, Fourier integral operators and well-posedness of solutions to PDE's.

The Weyl product on Banach modulation spaces has been studied in e. g.

[13, 16] studied the Hardy type modulation spaces and have shown a similar estimate, namely ∥ e i t | D | α f ∥ μ p, q s − γ ≲ 〈 t 〉 n | 1 2 − 1 p | ∥ f ∥ μ p, q s, where γ = n ( α − 2 ) | 1 2 − 1 p |.

Inspired by this idea, modulation and Wiener amalgam spaces have been introduced and used to measure the time-frequency concentration of a function or a tempered distribution (see [3 7]).

We also have the definition of discrete form for modulation spaces, which is very useful in studying unimodular Fourier multipliers.

Motivated by the work of Cheng-Zhang [4] on the modulation spaces, one naturally expects that the strongly singular integral operators (T_{n, beta, gamma}) have the boundedness property on the α-modulation spaces for all (0leqalphaleq1).

We use modulation spaces as appropriate symbols classes.

The results extend known results for Banach modulation spaces.