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

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