SHAPIRO TYPE INEQUALITIES FOR THE WEINSTEIN AND THE WEINSTEIN-GABOR TRANSFORMS

Year 2017, Volume: 5 Issue: 1, 68 - 76, 01.04.2017

NEJIB Ben Salem Amgad Rashed Nasr

https://izlik.org/JA72ZG69KW

Abstract

The aim of this paper is to prove new uncertainty principles for the Weinstein and the Weinstein-Gabor transforms associated with the Weinstein operator dened on the half  space $\mathbb{R}^d_{+}$ by $\Delta_W =\sum_{i=1}^{d } \frac{\partial}{\partial x_i^2}+ \frac{2\alpha+1}{x_{d}}\frac{\partial}{\partial x_{d-1}};\ \ \ \ \ d\ge2,\ \alpha>-1/2.$ More precisely, we give a Shapiro-type uncertainty inequality for the Weinstein transform that is, for $s>0$ and $\{\phi_n\}_n$ be an orthonormal sequence in $L^2_\alpha(\mathbb{R}^d_{+})$, $\sum_{n=1}^N(\Vert \vert x\vert^s \phi_n\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2}+ \Vert \vert\xi\vert^s \mathcal{F}_W(\phi_n)\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2 })\geq KN^{1+\frac{s}{2\alpha+d+1}},$ where $K$ is a constant which depends only on $d$; $s$ and $\alpha$. Next, we establish an analogous inequality for the Weinstein-Gabor transform

Keywords

Weinstein operator; Shapiro uncertainty inequality , time frequency concentration.

References

• [1] Z. Ben Nahia, N. Ben Salem: Spherical harmonics and applications associated with the Weinstein operator. Potential theoryICPT 94 (Kouty, 1994), 233241, de Gruyter, Berlin, 1996.
• [2] Z. Ben Nahia , N. Ben Salem: On a mean value property associated with the Weinstein operator. Potential theoryICPT 94 (Kouty, 1994), 243253, de Gruyter, Berlin, 1996.
• [3] N. Ben Salem , AR. Nasr: Heisenberg-type inequalities for the Weinstein operator. Integral Transforms Spec. Funct. 26 (2015), no. 9, 700718.
• [4] A. Bonami, B. Demange, Ph. Jaming: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19 (2003), no. 1, 2355.
• [5] M. Brelot: quation de Weinstein et potentiels de Marcel Riesz. (French) Sminaire de Thorie du Potentiel, No. 3 (Paris, 1976/1977), pp. 1838, Lecture Notes in Math., 681, Springer, Berlin, 1978.
• [6] S. Ghobber: Phase space localization of orthonormal sequences in L2 (R+). J. Approx. Theory 189 (2015), 123136.
• [7] S. Ghobber,Ph. Jaming: Uncertainty principles for integral operators. Studia Math. 220 (2014), no. 3, 197220.
• [8] S. Ghobber, S. Omri: Time-frequency concentration of the windowed Hankel transform. Integral Transforms Spec. Funct. 25 (2014), no. 6, 481496.
• [9] I. Gohberg, S. Goldberg and N. Krupnik: Traces and determinants of linear operators. Operator Theory: Advances and Applications, 116. Birkhuser Verlag, Basel, 2000.
• [10] Ph. Jaming, A. Powell: Uncertainty principles for orthonormal sequences. J. Funct. Anal. 243 (2007), no. 2, 611630.
• [11] E. Malinnikova: Orthonormal sequences in L2(Rd) and time frequency localization. J. Fourier Anal. Appl. 16 (2010), no. 6, 9831006.
• [12] H. Mejjaoli, A. Ould Ahmed Salem: Weinstein Gabor transform and applications. Advances in Pure Math J. 2012;2:203-210.
• [13] H. Mejjaoli, M. Salhi: Uncertainty principles for theWeinstein transform. Czechoslovak Math. J. 61(136) (2011), no. 4, 941974.
• [14] A. Weinstein: Singular partial dierential equations and their applications. 1962 Fluid Dynamics and Applied Mathematics (Proc. Sympos., Univ. of Maryland, 1961) pp. 2949 Gordon and Breach, New York.
• [15] H.S. Shapiro: Uncertainty principles for basis in L2(R). Unpublished manuscript.