乘法算子的谱逼近：基于渐近（结构化）线性代数方法
The spectral approximation of multiplication operators via asymptotic (structured) linear algebra
AI总结：
针对非紧乘法算子，利用正交基的有限截面逼近其谱，并探讨实正交多项式基与经典Fourier基（Toeplitz情形）的关系，提出基于循环矩阵逼近的构造性算法。
摘要：
通过选择希尔伯特空间的标准正交基，可以用有限截断来逼近希尔伯特空间上的乘法算子。函数$L^2$空间上的非零乘法算子永远不是紧的，因此这种逼近不能在范数拓扑下收敛。相反，我们考虑有限截断的谱在多大程度上逼近乘法算子的谱，该算子的表达式由符号（即乘子）的本质值域简单给出。我们讨论了实正交多项式基的情况，以及与经典Fourier基（其选择导致被充分研究的Toeplitz情形）的关系。循环矩阵逼近的使用也导致了适用于可分离多变量和矩阵值情形的构造性算法。
英文摘要：
multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol (i.e. the multiplier). We discuss the case of real orthogonal polynomial bases and the relations with the classical Fourier basis whose choice leads to well studied Toeplitz case. The use of circulant approximations leads to constructive algorithms working for the separable multivariate and matrix-valued cases as well.
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