A Note on the Semi-Infinite Programming Approach to Complex Approximation

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 C, and consequently a class of quasi-norms on the space C(Q) consisting of all continuous functions defined on Q C C, Q compact. These quasi-norms on C(Q) are estimates of the Lx 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. Glashoff and Roleff [1] solve a semi-infinite program (SIP) which is shown to be equivalent to the linear approximation problem for functions in C(Q), where C(Q) is the space of complex-valued continuous functions on a compact (and not necessarily finite) subset Q of the complex plane C and is equipped with the uniform (7^) norm (O ||/||00=max|/(z)|. Their method is a two-step procedure: the first step applies the usual simplex method of linear programming to solve a discrete approximation of the SIP; the second step uses the end result of the first step as the initial starting point in a Newton-Raphson iteration to solve a certain system of nonlinear algebraic equations whose solution (if feasible) is a solution to the linear approximation problem (Problem 1 below). The purpose of this note is to make some observations about the linear program of their discrete first step, which closely connects its solution with the solution of the approximation problem. A knowledge of the SIP definition and solution method is not needed to understand the results presented here. The interested reader is referred to [ 1 ], [2], [3], and to their bibliographies. We point out that Theorems 1 and 2 were first proved in [4], where a method identical to the first step of Glashoff-Roleffs procedure for finite Q was discovered independently of knowledge of [1] and of semi-infinite programming. Readers interested in practical examples and an engineering application of linear complex approximation are also referred to [4]. Received February 24, 1982. 1980 Mathematics Subject Classification. Primary 65DI5, 65E05, 65K.05; Secondary 30C30, 41A50, 30A82. ©1983 American Mathematical Society 0025-5718/82/0000-1068/S02.50 599 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 600 ROY L. STREIT AND ALBERT H. NUTTALL Let hx(z),...,hn(z) and f(z) be given functions in C(Q). For any set of complex parameters a = {ax,.. .,an), define (2) L(a;z)= 2 akhk(z). k=\ Problem 1. Compute a set of complex parameters a* = {a*,..., a*) such that, for all parameter sets a, (3) \\f-L(a^z)\\B0<\\f-L(a;z)\\aB. We set (4) E„(f) = \\f-L(a*;z)\\ 2 be a positive integer. Define the angles 6j = tt(j-1)/p; j = l,2,...,2p, and let Sp = {6j}. Define, for any complex number z, (5). I z L = max (Re(z)cosö + Im(z)sin#■}. p l<7<2/> J ' It may be readily verified that: (i) | z | > 0 and | z \p = 0 if and only if z = 0. (ii) | z + w | < | z | + | w 1^ for all complex z and w. (iii) Given a complex, | az \ = \ a \ ■ \ z | for all z if and only if arg a E Sp. (iv) For a and a„ complex, | -z \p = | z \p, lima^0 I 2, for all complex z. To see (7), it is helpful to visualize the set of all z in C such that | z | = 1 as an equilateral polygon of 2p sides whose inscribed circle is the unit circle \z\= 1. It is easy to verify that (8) H/H^maxl/iz)!, ze<3 is a quasi-norm on C(Q) for each integerp > 2. Further, from (7), (9) U/H, < 11/11. 2. Compute a set of complex parameters a** = {ax**,.. .,a**) such that, for all parameter sets a, (10) \\f-L(a**;z)\\p<\\f-L(a;z)\\p. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SEMI-INFINITE PROGRAMMING IN COMPLEX APPROXIMATION 601 We set (11) Enp(f)=\\f-L(a**;z)\\p. Theorem 1. Enp(f)2, En(f) = 0 if and only if Enp(f) = 0. Corollary 3. 7/£„(/) ^ 0, (12) „2. The numerical solution procedures of Glashoff and Roleff may then be appropriately, and potentially significantly, simplified. On the other hand, if Q is finite, Problem 2 becomes an ordinary linear program, although Problem 1 remains an SIP. The finite Q case is precisely the first step of the Glashoff and Roleff method for solving Problem 1. It is not hard to see that, for Q = (z,... ,zm) C C, Problem 2 may be reformulated as solving an overdetermined system of mp real linear algebraic equations in 2n real unknowns in the usual Chebyshev (lx) norm. Full details for setting up the linear equations can be found in [4]. (This formulation works for any choice of T = {6k} provided only that 6k E Til License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 602 ROY L. STREIT AND ALBERT H. NUTTALL and only if 8k + ir E T.) This real system may be written in the following block-partitioned form: (13) Ticoso, + Ssinf?, ' 7?sinf?, Scos0, 7? cos 8-, + S sin 8-, ¡ R sin 61 — S cos 67 R cos 6p + Sún 0p ' 7? sin 6p Seos 8p ucos ö, + üsinö, u cos 0j + v sin 8-,