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Dichotomies for Band Matrices

Carl de Boor
SIAM Journal on Numerical Analysis
Vol. 17, No. 6 (Dec., 1980), pp. 894-907
Stable URL: http://www.jstor.org/stable/2156808
Page Count: 14
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Dichotomies for Band Matrices
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Abstract

The bounded invertibility (as a linear map on l∞, say) of a bounded, strictly m-banded biinfinite matrix A is shown to be equivalent to a dichotomy or splitting of its kernel N (as a map on RZ) into N+ and N-, with N+ containing those which decay exponentially at + ∞, and N- those which decay exponentially at - ∞, together with a certain uniformity (with respect to the sequence index) of this direct sum decomposition. The approximability of the solution of the biinfinite system Ax = b by solutions of finite sections of this system is characterized in terms of linear independence, uniform as I* → - (* ∞), of N over I+ ∪ I-, with I* an integer interval of length dim N*, * = +, -.

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