摘要 : We study pseudospectral and spectral functions for Hamiltonian system Jy '-B(t)=lambda Delta(t)y$Jy<^>{\prime }-B(t)=\lambda \Delta (t)y$ and differential equation l[y]=lambda Delta(t)y$l[y]=\lambda \Delta (t)y$ with matrix-valued... 展开 We study pseudospectral and spectral functions for Hamiltonian system Jy '-B(t)=lambda Delta(t)y$Jy<^>{\prime }-B(t)=\lambda \Delta (t)y$ and differential equation l[y]=lambda Delta(t)y$l[y]=\lambda \Delta (t)y$ with matrix-valued coefficients defined on an interval I=[a,b)$\mathcal {I}=[a,b)$ with the regular endpoint a. It is not assumed that the matrix weight Delta(t)>= 0$\Delta (t)\ge 0$ is invertible a.e. on I$\mathcal {I}$. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization sigma=sigma tau$\sigma =\sigma _\tau$ of all pseudospectral and spectral functions sigma by means of a Nevanlinna parameter tau and single out in terms of tau and boundary conditions the class of functions y for which the inverse Fourier transform y(t)=integral R phi(t,s)d sigma(s)y(s)$y(t)=\int _\mathbb {R}\varphi (t,s)\, d\sigma (s) \widehat{y}(s)$ converges uniformly. We also show that for scalar equation l[y]=lambda Delta(t)y$l[y]=\lambda \Delta (t)y$ the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm-Liouville equation -(p(t)y ')'+q(t)y=lambda Delta(t)y$-(p(t)y<^>{\prime })<^>{\prime }+q(t)y=\lambda \Delta (t) y$ to such equations with arbitrary coefficients p(t)$p(t)$ and q(t)$q(t)$ and arbitrary non trivial weight Delta(t)>= 0$\Delta (t)\ge 0$. 收起