Abstract

The paper deals with the Sturm–Liouville operator generated on the finite interval \([0,\pi]\) by the differential expression \(-y^{\prime\prime}+q(x)y\), where \(q=u^{\prime}\), \(u\in L_{\varkappa}[0,\pi]\) for some \(\varkappa\geq 2\), and arbitrary regular boundary conditions. Consider two such operators with different potentials but the same boundary conditions. We prove that the difference between spectral decompositions \(S_{m}^{1}(f)-S_{m}^{2}(f)\) of this operators tends to zero as \(m\to\infty\) for any \(f\in L_{\mu}[0,\pi]\) in the norm of the space \(L_{\nu}[0,\pi]\) if the indices satisfy the inequality \(1/\varkappa+1/\mu-1/\nu\leq 1\) (except for the case \(\varkappa=\nu=\infty\), \(\mu=1\)). In particular, in the case of a square summable function \(u\) the uniform equiconvergence on the whole interval \([0,\pi]\) is proved for an arbitrary function \(f\in L_{2}[0,\pi]\)

中文翻译：

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Sturm-Liouville 算子谱分解的等收敛：Lebesgue 空间的三元组

摘要

该论文涉及通过微分表达式 \(-y^{\prime\prime}+q(x)y\)在有限区间\([0,\pi]\)上生成的 Sturm–Liouville 算子，其中\ (q=u^{\prime}\)，\(u\in L_{\varkappa}[0,\pi]\)对于某些 \(\varkappa\geq 2\)和任意规则边界条件。考虑两个具有不同电位但边界条件相同的算子。我们证明了这个算子的谱分解\(S_{m}^{1}(f)-S_{m}^{2}(f)\)之间的差异趋于零为\(m\to\infty\ )对于空间范数\(L_{\nu}[0,\pi]\) 中的任何\(f\in L_{\mu }[0,\pi]\)如果指数满足不等式 \(1/\varkappa+1/\mu-1/\nu\leq 1\)（除了情况 \(\varkappa=\nu=\infty\)，\(\mu= 1\) )。特别地，在一个正方形可累加功能的情况下\（U \）对整个间隔均匀equiconvergence \（[0，\ PI] \）证明对于任意函数在L_ {2} [0 \（F \ ,\pi]\)