Abstract—

We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval [0, π] generated by the differential expressions \({{l}_{1}}(y) = - y'' + {{q}_{1}}(x)y\) and \({{l}_{2}} = - y'' + {{q}_{2}}(x)y\) and the same Birkhoff-regular boundary conditions. The potentials are assumed to be singular in the sense that \({{q}_{j}}(x) = u_{j}^{'}(x)\), \({{u}_{j}} \in {{L}_{\kappa }}[0,\pi ]\) for some \(\kappa \in [2,\infty ]\) (here, the derivatives are understood in the sense of distributions). It is proved that the equiconvergence in the metric of \({{L}_{\nu }}[0,\pi ]\) holds for any function \(f \in {{L}_{\mu }}[0,\pi ]\) if \(\frac{1}{\kappa } + \frac{1}{\mu } - \frac{1}{\nu } \leqslant 1\), \(\mu ,\nu \in [1,\infty ]\), except for the case \(\kappa = \nu = \infty \), \(\mu = 1\).

中文翻译：

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空间尺度上具有分布势的Sturm–Liouville算子的谱分解的等收敛性

摘要-

我们研究了两个Sturm–Liouville算子在由微分表达式\（{{l} _ {1}}（y）=-y''+ {{q}生成的区间[0，π]上的频谱分解的等收敛性_ {1}}（x）y \）和\（{{l} _ {2}} =-y''+ {{q} _ {2}}（x）y \）和相同的Birkhoff-regular边界条件。从\（{{q} _ {j}}（x）= u_ {j} ^ {'}（x）\），\（{{u} _ {j} } \ in {{L} _ {\ kappa}} [0，\ pi] \）对于某些\（\ kappa \ in [2，\ infty] \）（此处，导数是从分布意义上理解的） 。证明了\（{{L} _ {\ nu}} [0，\ pi] \）度量的等价收敛对于任何函数\（f \ in {{L} _ {\ mu}} [ 0，\ pi] \）如果\（\ frac {1} {\ kappa} + \ frac {1} {\ mu}-\ frac {1} {\ nu} \ leqslant 1 \），\（\ mu，\ nu \ in [1， \ infty] \），但\（\ kappa = \ nu = \ infty \），\（\ mu = 1 \）除外。