论文标题
在重量变化的微分方程和哈密顿系统的逆傅立叶变换的均匀收敛上
On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight
论文作者
论文摘要
我们研究哈密顿系统系统的伪谱和光谱函数$ jy'-b（t）=λδ（t）y $和微分方程$ l [y] =λδ（t）y $，具有矩阵值的系数，该系数按间隔$ \ mathcal $ \ mathcal {i} = [i} = [a，b）$带有常规终点。不能假定矩阵重量$δ（t）\ geq 0 $是可逆的。在$ \ Mathcal {i} $上。在这种情况下，始终存在伪光谱函数，但是光谱函数的集合可能是空的。我们通过nevanlinna参数$τ$获得所有伪和光谱函数的参数化$σ=σ_τ$ $σ$ $σ$，并以$τ$和边界条件为单一的函数$ y $ clounder y y $ y $，inverseforder forforder y y $ y $ yversefter frustrier y（t）= \ int \ limits = \ limits _ dσ（s）\ widehat y（s）$均匀收敛。我们还表明，对于标量方程$ l [y] =λδ（t）y $，光谱函数集并非空。这使我们能够将Kats-Krein和Atkinson的结果扩展到标量sturm -liouville方程$ - （p（t）y'）'+q（t）y =λδ（t）y $到具有任意系数的等式$ p（t）$和$ q（t）$和$ q（t）$和任意的非小质量$ $ $ $ $ $ $ $ ge（t）$ geq 0。
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 $Δ(t)\geq 0$.
