\(L^p(\mathbb R)\) boundedness and compactness of localization operators associated with the Stockwell transform. (English) Zbl 1204.42019
The Stockwell transform of a function \(f\) on \(\mathbb R=(-\infty,\infty)\) is a hybrid of the Gabor transform and the wavelet transform:
\[
S_\varphi f (b,\xi)=(2\pi)^{-1/2} |\xi| \int_{-\infty}^{\infty}e^{-ix\xi} f(x) \overline{\varphi (\xi (x-b))}\, dx,
\]
\(b \in \mathbb R\), \(\xi \in \mathbb R\). The authors study the boundedness and compactness of localization operators associated to \(S_\varphi\) on \(L^p(\mathbb R)\). The results can be extended to the modified Stockwell transform \( S^s_\varphi f (b,\xi)= |\xi|^{1/s-1}S_\varphi f (b,\xi)\).
Reviewer: Boris Rubin (Baton Rouge)
MSC:
42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
47G10 | Integral operators |
47G30 | Pseudodifferential operators |
65R10 | Numerical methods for integral transforms |