Abstract The uniform convergence of spectral expansions on the real line $$\mathbb {R} $$ is established for a self-adjoint operator $$\mathcal {A} $$ generated on $$\mathbb {R} $$ by the differential operation $$Au\equiv (-1)^n u^{(2n)}+\sum _{k=0}^{n-1} (q_k(x)u^{(k)})^{(k)}$$ with uniformly locally integrable coefficients. The uniform convergence of the derivatives of these expansions is studied as well. The results obtained are based on a uniform estimate for the increment of the spectral function of the operator $$\mathcal {A} $$ on the diagonal.

中文翻译：

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一般偶数阶微分算子全实线上谱展开的均匀收敛

摘要 在 $$\mathbb {R} $$ 上生成的自伴随算子 $$\mathcal {A} $$ 建立了实线 $$\mathbb {R} $$ 上谱展开的一致收敛微分运算$$Au\equiv (-1)^nu^{(2n)}+\sum _{k=0}^{n-1} (q_k(x)u^{(k)})^{( k)}$$ 具有一致的局部可积系数。还研究了这些扩展的导数的一致收敛。获得的结果基于对运算符 $$\mathcal {A} $$ 在对角线上的谱函数增量的统一估计。