Abstract

The aim of this paper is to prove new variations of uncertainty principles for Weinstein operator. The first of these results is variation of Heisenberg-type in equality for Weinstein transform that is for s>0. Then, there exists a constant C(α,s), such that for all f∈L\(_{α}^{1}\) (R\(_{+}^{d}\) )∩L\(_{α}^{2}\) (R\(_{+}^{d}\))
\(‖|x|^{2s} f‖_{L_{α}^{1} (R_{+}^{d})} ‖|ξ|^{s} F_{W} (f)‖_{L_{α}^{2} (R_{+}^d)}^{2}≥C(α,s)‖f‖_{L_{α}^{1} (R_{+}^{d})} ‖f‖_{L_{α}^{2} (R_{+}^{d})}^{2}.\)
The second result is variation of Donoho-Strak's uncertainty principle for Weinstein transform, Let S,Σ⊂R\(_{+}^{d}\) and f∈L\(_{α}^{1}\) (R\(_{+}^{d}\))∩L\(_{α}^{2}\) (R\(_{+}^{d}\)). If f is (ε\(_{1}\),α)-timelimited on T and (ε\(_{2}\),α)-bandlimited on Σ, then μ\(_{α}\) (S) μ\(_{α}\) (Σ)≥(1-ε\(_{1}\))\(^{2}\) (1-ε\(_{2}^{2}\)). The third result is variation of the local uncertainty for Weinstein and Weinstein-Gabor transform.