Some uncertainty inequalities for the continuous wavelet transform
In this paper, the authors prove some uncertainty inequalities for the continuous wavelet transform in the multidimensional setting. A nonzero function \(\psi \in L^2(\mathbb{R}^d, \mu_d),\) where \(\mu_d\) is the normalized Lebesgue measure on \(\mathbb{R}^d,\) is said to be a mother wavelet (or an admissible wavelet) if it satisfies the following admissibility condition: \[ 0 < c_{\psi} := \int_0^{\infty} \frac 1a |\widehat{\psi}(a \xi)|^2 \ d a < \infty, \quad \forall \xi \in \mathbb{R}^d \setminus \{0\} : |\xi|=1, \] where \(| \cdot |\) is the usual norm on \(\mathbb{R}^d,\) and \(\widehat{\psi}\) represents the Fourier transform. The daughter wavelet of \(\psi\) is defined by \[ \psi_{a,b}(x) = \frac{1}{\sqrt{c_{\psi}}} \frac{1}{a^{d/2}}\psi\left(\frac{x - a}{b}\right), \quad (a, b) \in \mathbb{U} = (0, \infty) \times \mathbb{R}^d. \] Let \(f \in L^2(\mathbb{R}^d, \mu_d),\) and let \(\mathcal{W}_{\psi} f\) be its wavelet transform. Consider the left invariant Haar measure \(d \mu_{\textrm{aff}}(a,b) = a^{-(d+1)} da \ d_{\mu_d}(b).\) There are three main results in this paper: \par 1) For all \(f \in L^2(\mathbb{R}^d, \mu_d),\) it is shown that \[ \| \ |b| \mathcal{W}_{\psi} f \ \|_{2, \mu_{\textrm{aff}}} \ \| \ |\xi| \widehat{f}\ \|_{2, \mu_{d}} \geq \frac d2\|f\|^2_{2, \mu_{d}}. \] For \(s,\) \(\beta \geq 1\), this gives rise to \[ \| \ |b|^s \mathcal{W}_{\psi} f \ \|^{\beta}_{2, \mu_{\textrm{aff}}} \ \| \ |\xi|^{\beta} \widehat{f}\ \|^{s}_{2, \mu_{d}} \geq \left(\frac d2 \right)^{s \beta} \|f\|^{s + \beta}_{2, \mu_{d}} \] which can be looked upon as a Heisenberg-type uncertainty inequality for the continuous wavelet transform in the multidimensional setting. One can compare the concentration of \(f\) in frequency (given by \(\xi\)) with the concentration of its wavelet transform \(\mathcal{W}_{\psi} f\) in position (given by \(b\)). \par 2) Given a set \(A\), \(PW_{A} = \{f \in L^2(\mathbb{R}^d, \mu_d): \textrm{supp} \widehat{f} \subset A\}\) is the Paley-Wiener subspace of \(L^2(\mathbb{R}^d, \mu_d)\), and \(P_A = \{f \in L^2(\mathbb{R}^d, \mu_d): \forall a > 0,\ \textrm{supp} \mathcal{W}_{\psi} f (a, \cdot) \subset A\}.\) Let \(S\) and \(\Sigma\) be two subsets of \(\mathbb{R}^d\) of finite Lebesgue measure. It is shown that if \(\psi\) is a radial admissible wavelet, then for all \(f \in P_S \cap PW_{\Sigma},\) there is a constant \(c(\psi, s, \beta, d)\) such that \[ \|a^{-\beta} \mathcal{W}_{\psi} f \|^s_{2, \mu_{\textrm{aff}}} \| \ |b|^s \mathcal{W}_{\psi} f \|^{\beta}_{2, \mu_{\textrm{aff}}} \geq \frac{c(\psi, s, \beta, d)}{(|S||\Sigma|)^{\frac{s\beta}{d}}} \|f\|^{s + \beta}_{2, \mu_d},\ \quad 0 < s,\ \beta < \frac d2. \] This is referred to as a local uncertainty inequality. A Heisenberg type uncertainty result like that in 1) can be derived from this. \par 3) For a radial admissible wavelet \(\psi\) and a function \(f\) in the Schwartz space \(\mathcal{S}(\mathbb{R}^d)\), the following Beckner-type uncertainty inequality has been established. \[ \int_{\mathbb{U}} \ln |b| \left| \mathcal{W}_{\psi} f (a,b) \right|^2 \ d\mu_{\textrm{aff}}(a,b) + c_{\psi}\int_{\mathbb{U}} \ln(a^{-1}) \left| \mathcal{W}_{\psi} f (a,b) \right|^2 \ d\mu_{\textrm{aff}}(a,b) \geq C_d(\psi) \| f \|^2_{2, \mu_d}. \] This inequality is not sharp, meaning that there is no \(f \in \mathcal{S}(\mathbb{R}^d)\) that attains equality. A similar inequality for functions in the Sobolev space is also given in which case the inequality is also sharp.
scientific article; zbMATH DE number 7381172
scientific article