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Mathematics of Computation

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A note on the semi-infinite programming approach to complex approximation

Authors: Roy L. Streit and Albert H. Nuttall
Journal: Math. Comp. 40 (1983), 599-605
MSC: Primary 49D39; Secondary 30E10, 90C05
MathSciNet review: 689476
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Abstract: Several observations are made about a recently proposed semi-infinite programming (SIP) method for computation of linear Chebyshev approximations to complex-valued functions. A particular discretization of the SIP problem is shown to be equivalent to replacing the usual absolute value of a complex number with related estimates, resulting in a class of quasi-norms on the complex number field $\mathbf {C}$, and consequently a class of quasi-norms on the space $C(Q)$ consisting of all continuous functions defined on $Q \subset {\mathbf {C}}$, Q compact. These quasi-norms on $C(Q)$ are estimates of the ${L_\infty }$ norm on $C(Q)$ and are useful because the best approximation problem in each quasi-norm can be solved by solving (i) an ordinary linear program if Q is finite or (ii) a simplified SIP if Q is not finite.

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Article copyright: © Copyright 1983 American Mathematical Society