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Modified Stockwell-transforms. (English) Zbl 1237.42007

If \(1\leq p<\infty\), then \(L^p(\mathbb{R})\) denotes the space of Lebesgue measurable functions on \(\mathbb{R}= (-\infty,\infty)\) defined by \(L^p(\mathbb{R})=\{[f]: \|[f]\|_p=\| f\|_p<\infty\}\), where \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure \(0\), and \[ \| f\|_p= \Biggl(\int^\infty_{-\infty} |f(\xi)|^p\, d\xi\Biggr)^{1/p}. \] The Fourier transform \(\varphi^\wedge={\mathcal F}(\varphi)\) of \(\varphi\) in \(L^1\cap L^2(\mathbb{R})\) is defined by \[ \varphi^\wedge(x)= (2\pi)^{-1/2}\int^\infty_{-\infty} e^{-ix\xi} \varphi(\xi)\,d\xi. \] If \(\varphi\in L^1\cap L^2(\mathbb{R})\) then the Gabor transform \(G_\varphi(f)\) of \(f\in L^2(\mathbb{R})\) is defined by \[ G_\varphi(f)(b,x)= (2\pi)^{-1/2}\int^\infty_{-\infty} e^{-ix\xi} f(\xi)\,\overline \varphi(\xi-b)\,d\xi, \] the Stockwell transform, \(S_\varphi(f)\), is defined by \[ S_\varphi(f)(b,x)= (2\pi)^{-1/2} |x|\int^\infty_{-\infty} e^{-ix\xi} f(\xi)\,\overline \varphi(x(\xi- b))\,d\xi, \] and the modified Stockwell transform, \(S^s_\varphi(f)\), considered in this paper, is defined by \[ S^s_\varphi(f)(b,x)= (2\pi)^{-1/2} |x|^{1/s} \int^\infty_{-\infty} e^{-ix\xi} f(\xi)\,\overline \varphi(x(\xi- b))\,d\xi. \] One of the main theorems of this paper contains a Parseval type identity of the form \[ \int^\infty_{-\infty} f(\xi)\,\overline g(\xi)\,d\xi= (1/c_\varphi) \int^\infty_{-\infty} \int^\infty_{-\infty} (S^s_\varphi(f)(b, \xi)\,(\overline S^s_\varphi(g)(b,\xi)|\xi|^{1-(2/s)}\,db\,d\xi, \] where \[ c_\varphi= \int^\infty_{-\infty} |\xi|^{-1} |\varphi^\wedge(\xi- 1)|^2\,d\xi\text{ and }\int^\infty_{-\infty} \varphi(x)\,dx= 1. \] The representation of the fractional integral as a Fourier transform multiplier operator is indicated as \[ {\mathcal F}^{-1}\Biggl(\int^\infty_{-\infty} JS^s_\varphi(f)(b,.)\,db\Biggr)= I_{1/s'}(f)= \gamma(1/s')(k* f), \] where \[ (k*f)(x)= \int^\infty_{-\infty} |x-y|^{(1/s)- 1}f(y)\,dy. \] Reference is made in the paper of the connection of the integral identities with considerations in signal analysis.
Reviewer’s comment: the constant \(c_\varphi\) of the identities stated in the main theorems does not appear to be finite since \(\varphi^\wedge(x-1)\) is not assumed to be \(0\) at \(x= 0\); A modified definition of \(c_\varphi\) may be applicable.

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
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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