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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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  • [1] K. Glashoff & K. Roleff, "A new method for Chebyshev approximation of complex-valued functions," Math. Comp., v. 36, 1981, pp. 233-239. MR 595055 (82c:65011)
  • [2] A. Charnes, W. W. Cooper & K. O. Kortanek, "Duality, Haar programs and finite sequence spaces," Proc. Nat. Acad. Sci. U.S.A., v. 48, 1962, pp. 783-786. MR 0186393 (32:3853)
  • [3] S.-A. Gustafson, "On semi-infinite programming in numerical analysis," in Semi-Infinite Programming (R. Hettich, Ed.), Lecture Notes in Control and Information Sciences, Vol. 15, Springer-Verlag, Berlin and New York, 1979, pp. 137-153. MR 554208 (81c:90077)
  • [4] R. L. Streit & A. H. Nuttall, "A general Chebyshev complex function approximation procedure and an application to beamforming," J. Acoust. Soc. Amer., v. 72, 1982, pp. 181-190; Also NUSC Technical Report 6403, 26 February 1981. (Naval Underwater Systems Center, New London, Connecticut, U.S.A.) MR 668082 (83g:78024)
  • [5] K. Yosida, Functional Analysis, 2nd ed., Springer-Verlag, Berlin and New York, 1968. MR 0239384 (39:741)
  • [6] I. Barrodale & C. Phillips, "Solution of an overdetermined system of linear equations in the Chebyshev norm," Algorithm 495, ACM Trans. Math. Software, v. 1, 1975, pp. 264-270. MR 0373585 (51:9785)
  • [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. A. Watson, Approximation Theory and Numerical Methods, Wiley, New York, 1980. MR 574120 (82e:41001)

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Article copyright: © Copyright 1983 American Mathematical Society

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