Reproducing kernels and variable bandwidth
Given a positive, radial, submultiplicative function \(v\) (i.e., \(v(z_1+z_2)\leq v(z_1)v(z_2)\) on \(\mathbb{R}^2\) such that \(v(0)=1\), a second weight \(m\) is called a \(v\)-moderate weight if \(m(z_1+z_2)\leq C v(z_1)m(z_2)\). Given such a weight, one defines the modulation space \(M_m^2(\mathbb{R})\) in terms of the norm \[ \|f\|_{M_m^2}^2 =\int_{\mathbb{R}}\int_{\mathbb{R}} |V_gf(x,\omega)|^2 m(x,\omega)^2 dx\, d\omega, \] where \[ V_gf(x,\omega)=\int_{\mathbb{R}} f(t)\overline{g(t-x)} e^{-2\pi i \omega t}\, dt \] denotes the short-time Fourier transform of \(f\) with window function \(g\). Set \(M_\omega T_x f(t)=e^{2\pi i t\omega} f(t-x)\) and \(\pi^\ast g(y)(x,\omega)= \overline{M_\omega T_x g(y)}\) with \(x,\omega\in \mathbb{R}\). It is proved that, for a \(v\)-moderate weight \(m\), \(M_m^2\) has reproducing kernel \(\Phi_y=V_g^\ast (m^{-2} \pi^\ast g(y))\) where, for \(y\) fixed, \(\|\Phi_y\|= \|m^{-1}g(y-\cdot)\|_2\). The reproducing property can be expressed as \(f(y)=\langle mV_g f,\, m V_g\Phi_y\rangle_{L^2(\mathbb{R}^2)}\). The space \(M_m^2\) is well-defined for a suitable class of windows \(g\), but the reproducing property applies to a specific choice of \(g\). The reproducing kernel is computed explicitly here for the case of a Sobolev space. Variable bandwidth spaces are defined as modulation spaces with weight \(m_{b,s}(z)=(1+d_b(z))^{s/2}\), where \(d_b(x,\omega)=0\) if \(\omega\leq b(x)\), while \(d_b(x,\omega)=|\omega-b(x)|\) if \(|\omega|>b(x)\). They were introduced in an earlier work of the authors [J. Math. Anal. Appl. 382, No. 1, 275--289 (2011; Zbl 1223.42022)]. Here, \(b(x)>0\) is called a variable bandwidth function. The reproducing kernel for the space \(M_{m_b}^2\) is computed explicitly here.
scientific article; zbMATH DE number 6102735