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

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689476-0

MathSciNet review:
689476

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 , and consequently a class of quasi-norms on the space consisting of all continuous functions defined on , *Q* compact. These quasi-norms on are estimates of the norm on 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.

**[1]**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**, https://doi.org/10.1090/S0025-5718-1981-0595055-4**[2]**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**0186393****[3]**Sven-È¦ke 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****[4]**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**, https://doi.org/10.1121/1.388002**[5]**Kôsaku Yosida,*Functional analysis*, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York Inc., New York, 1968. MR**0239384****[6]**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) Utilitas Math., Winnipeg, Man., 1975, pp. 177–190. Congr. Numer., No. XII. MR**0373585****[7]**R. L. Streit,*Numerical Solutions of Systems of Complex Linear Equations with Constraints on the Unknowns*, Stanford University Department of Operations Research SOL Report. (To appear.)**[8]**G. Alistair Watson,*Approximation theory and numerical methods*, John Wiley & Sons, Ltd., Chichester, 1980. A Wiley-Interscience Publication. MR**574120**

Retrieve articles in *Mathematics of Computation*
with MSC:
49D39,
30E10,
90C05

Retrieve articles in all journals with MSC: 49D39, 30E10, 90C05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689476-0

Article copyright:
© Copyright 1983
American Mathematical Society