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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1983-0689476-0
PII: S 0025-5718(1983)0689476-0
Article copyright: © Copyright 1983 American Mathematical Society