Approximating classes of sequences: The Hermitian case

作者：Serra Capizzano Stefano^*; Sesana Debora

来源：Linear Algebra and Its Applications, 2011, 434(4): 1163-1170.

DOI：10.1016/j.laa.2010.10.025

• Approximating class of sequences
• Eigenvalue distribution
• Sparsely vanishing sequence or symbol
• Sparsely unbounded sequence or symbol

Given a sequence (A(n)) of matrices A(n) of increasing dimension d(n) with d(k) > d(q) for k > q, k, q is an element of N, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that {{f (B(n,m))} m is an element of N) is an a.c.s. for {f (A(n))}, if {{B(n,m)} m is an element of N} is an a.c.s. for {A(n)}, {A(n)} is sparsely unbounded, and f is a suitable continuous function defined on R. We also discuss the potential impact and future developments of such a result.

• 出版日期2011-2-15