Discrete Rational Lp Approximation

In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [a, b] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obtained is used to analyze the convergence of best approximations on discrete subsets of [a, b] to best approximations over the whole interval (as the discrete subsets become dense) in the case that the function approximated is continuous on [a, b] and the rational family consists of quotients of algebraic polynomials. It is found that the convergence is uniform over [a, b] if the function approximated is a so-called normal point. Only L norms with 1 < p < <*> are employed. Introduction. In the application of nonlinear approximation theory one is normally constrained to the calculation of best approximations on finite subsets of the underlying domain. Moreover, the discrete problem may be more difficult than the continuous problem, in the sense that the discretized family may not be (pointwise) closed so that best approximations need not exist. In this paper, these problems will be investigated for a family of generalized rational functions defined on a closed subinterval of the real line. In particular, let P and Q be Haar subspaces of C[a, b] of dimension n and m, respectively, let X = {xl, ■ ■ ■ , xM} C [a, b] with M > m + n + 1 and let RiX) = {plq \pEP,qGQ and q(x) ± 0 for all x E X). Then, if IIII is a norm on B(X) = {/|/is a real-valued function on X] and fE B(X) is given, we seek an r* E R(X) such that 11/r*\\ = infrSÄ,xJl/HI. The case when 11/11 = maxx6A-|/(jr)| has been extensively studied (see [ 1 ] and [2] for example) and we shall consider only norms of the form ll/B( = [S^e^l/tol'l1^ where 1 < t < °°. The techniques that will be used are somewhat different from those useful for uniform approximation since there is no characterization theorem available for these norms. Remark 1. The family R(X) is usually defined by requiring that, if p/q E R(X), then q(x) > 0 for all x E X. However, there seems little to be gained by this requirement since it does not preclude the existence of a pole in [a, b], even at a best approximation, and there is usually no simple way to maintain this condition during computation. Moreover, as will be seen, for sufficiently dense discrete subsets using Received July 30, 1973. AMS (MOS) subject classifications (1970). Primary 41A20; Secondary 65D10. Copyright © 1975. American Mathematical Society 540 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use DISCRETE RATIONAL Lp APPROXIMATION 541 ordinary rational approximation and least-square approximation, say, the condition qix) > 0 for all x E X will hold for a best approximation anyway. Existence of Discrete Best Approximations. We begin by identifying explicitly the pointwise closure R(X) of the set R(X) in B(X). It is clear that R(X) is an existence set in the sense that each / E B(X) has a closest point in R(X) with respect to any norm on B(X). The notation R(Y), where Y is some subset of [a, b], will denote the set {p/q \pEP,qEQ and q(x) ± 0 for all x E Y}. The following example illustrates the "exceptional" types of elements that may appear in R(X). Example 1. Let X = {3, 2, 1, 0, 1, 2, 3} and R(X) = Ri(X) = an + a,x + a .x 2fiA-. O 2 ]b0 +Z>jjc + b2x¿ b0 + btx + b2x2 ¥= 0 for all xEX (a) Let p„(jr) = x2 + l/v and qv(x) = x1 l/v, v = 2, 3, Then pv(x)/qv(x) —> 1 at all x except x = 0 where pv(0)/qv(0) = 1. The limiting function is clearly not an element of R(X) itself. (b) Let pv(x) = l/v and qv(x) = x2 (1 1/u)2. Then pv(x)lqv(x) -»0 except at jt = ±1 where the limiting value is Vi. Again, the limiting function is not in R(X). As will be seen, the types of behavior in (a) and (b) above are the only ones to be dealt with in describing R(X). To clarify the presentation of Theorem 1, we have the following definition. Definition. Let S. denote the set of functions g in B(X) such that there exists some set S C X (depending on g) containing at most k = min(n — 1, m 1) elements and some rational function p/<7 in R(X ~ S) with p(jr) = q(x) = 0 for all jt E S for which g = p/q on X ~ S. Let S2 denote the set of all functions g in B(X) such that g is zero except precisely on some subset TEX (depending on g) having at most m I elements. Remark 2. We should note here that the sets S. and S2 are not disjoint since {0} ES. n S2 and that S. D R(X). In general, neither is a subset of the other. Also, some elements of Sl may not have a unique pair S and p/q E R(X ~ S) corresponding to them. For example, in R\(X), with X as in Example 1, the function g(x) = 1 is "represented" by the empty set and 1/1, but also by S = {0} and x/x E R(X ~ {0}), etc. This ambiguity will not affect what follows. In Example 1 above, the limiting function in (a) is in S.. That is, the function equals p/q = x2\x2 E RiX ~ {0}), except on S = {0} where its value is 1. Note also that p = q = 0 on S. Similarly, in (b) the limiting function is in S2, since it vanishes except at precisely 2 (= m 1) points where its value is M. 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