On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

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