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

Streit, Roy L.; Nuttall, Albert H.

doi:10.1090/S0025-5718-1983-0689476-0

A note on the semi-infinite programming approach to complex approximation
HTML articles powered by AMS MathViewer

by Roy L. Streit and Albert H. Nuttall PDF

Math. Comp. 40 (1983), 599-605 Request permission

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.

References

K. Glashoff and K. Roleff, A new method for Chebyshev approximation of complex-valued functions, Math. Comp. 36 (1981), no. 153, 233–239. MR 595055, DOI 10.1090/S0025-5718-1981-0595055-4
A. Charnes, W. W. Cooper, and K. Kortanek, Duality, Haar programs, and finite sequence spaces, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 783–786. MR 186393, DOI 10.1073/pnas.48.5.783
Sven-Ȧke Gustafson, On semi-infinite programming in numerical analysis, Semi-infinite programming (Proc. Workshop, Bad Honnef, 1978) Lecture Notes in Control and Information Sci., vol. 15, Springer, Berlin-New York, 1979, pp. 137–153. MR 554208
R. L. Streit and A. H. Nuttall, A general Chebyshev complex function approximation procedure and an application to beamforming, J. Acoust. Soc. Amer. 72 (1982), no. 1, 181–190. MR 668082, DOI 10.1121/1.388002
Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York, Inc., New York, 1968. MR 0239384
I. Barrodale and C. Phillips, An improved algorithm for discrete Chebyshev linear approximation, Proceedings of the Fourth Manitoba Conference on Numerical Mathematics (Winnipeg, Man., 1974) Congr. Numer., No. XII, Utilitas Math., Winnipeg, Man., 1975, pp. 177–190. MR 0373585

Numerical Solutions of Systems of Complex Linear Equations with Constraints on the Unknowns

G. Alistair Watson, Approximation theory and numerical methods, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1980. MR 574120