Simultaneous Approximation of a Set of Bounded Real Functions

The problem of simultaneous Chebyshev approximation of a set F of uniformly bounded, real-valued functions on a compact interval / by a set P of continuous functions is equivalent to the problem of simultaneous approximation of two real-valued functions F+ (x), F~ (x), with F~ (x) ¿ F+ (x), for all x in I, where F~ is lower semicontinuous and F+ is upper semicontinuous. 1. Formulation of the Approximation Problem. In this introductory section, which consists of nine "points," the "general problem of the simultaneous approximation of a family of functions" is formulated (see, in particular, point 4). Besides, a "heuristic derivation" of the basic equation (equation (T2) of point 8) is given. 1. Let jbea (finite) real-valued function defined for all real numbers x on the finite-closed real number interval [a, b] = [x\a ^ x ^ b}, where a < b. The "norm," \[g\\, is defined to be ||Sf|| = sup \gix)\ • a¿x¿b If g is not bounded in absolute value on [a, b], then ||^|| = + oo ; otherwise, \\g\\ is a nonnegative number. 2. Let fbea nonempty set ("family") of real-valued functions /, defined on [a, b]. The set of functions F is supposed to be uniformly bounded (in absolute value) on [a, b]; i.e., there is a nonnegative number M such that [fix)\ ^ M for any function/ G F and any number x (E [a, b]. Clearly, 11/11 = M for any/ Ç F. (In the "general approximation problem" of point 4 below, the family F is "the set of functions being approximated.") 3. Let P be a nonempty set ("class") of real-valued, continuous functions p, defined on [a, b]. (In the "general approximation problem," the class P is "the set of approximating functions"; usually, for n a nonnegative integer, the class P consists of all real polynomials of degree á »•) 4. For the purposes of the present paper, the "general problem of the simultaneous approximation of the family F by means of functions from the class P" consists in the determination of the number inf sup ||/p\\ . pGp /-Si(The formulation of this "general approximation problem," as given here, was suggested by the "problem of simultaneous approximation of two bounded functions Received October 21, 1968. This paper was presented to the American Mathematical Society at the August 1968 meeting at Madison, Wisconsin (see Abstract 658-200, Notices Amer. Math. Soc, v. 15, 1968, p. 778). 583 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 584 J. B. DIAZ AND H. W. MCLAUGHLIN !?i is s72, with gi lower semicontinuous and g2 upper semicontinuous," of C. B. Dunham [1, p. 472]; this problem of Dunham will be discussed more fully under point 5 below.) 5. Consider, in particular, the very special case in which the family F consists of a single function/, which is bounded in absolute value on [a, b]. In this case, the "general problem of the (simultaneous) approximation of the bounded function/by means of functions from the class P" consists in the determination of the number inf ||/p\\ . (Suppose, for the moment, that the function / were allowed to be unbounded in absolute value, that is to say, ||/|| = + ■». Then, for every continuous approximating function p one would have that ||/ — p|| = + , and hence infj,ep ||/ — p|| = + °° also. Therefore, the restriction that the function / being approximated be bounded in absolute value is a natural requirement to make.) Dunham [1, specially p. 476] showed that this "approximation problem of a single bounded function /" is equivalent to the "simultaneous approximation problem of a certain pair of bounded functions /~ ^ /+, where /~ is lower semicontinuous, and/+ is upper semicontinuous, on [a, b]" (the set of approximating functions P used in [1] is "unisolvent of degree n on [a, b]," and includes, as an important special case, the case when P is the class of real polynomials of degree ^ n). Specifically, this "equivalence" result of [1] can be formulated as follows: Let the function /+ be defined by f+ix) = inf sup fiy) , S>0 0£\x-yl0 0S\x-y\0 0¿\x-y\<¡ /fcF for a Sx S b; and let the function F~ be defined by F~ix) = sup inf inf fiy) , S>0 Og\x-yl / II \/Gf / License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 586 J. B. DIAZ AND H. W. MCLAUGHLIN Viewing equation (T2) purely formally, as an equality between two products, and then applying the "cancellation law" (by simply "cancelling infpGp on both sides of the equation"), leads one to suspect that it may be true that, for any p£P, (C) sup /Gp |/p|| = sup \^Ff)-p'\\i£f) ~p (After one has recovered from the initial shock of this "deduction" of (C) from (T2), it is readily realized that all that is being said is that, if (C) holds for any p in P, then, upon taking the inf over P, equation (T2) will follow.) Returning to (C), since it presumably holds for any continuous function p (because P is just any set of continuous functions on [a, b]), it perhaps is valid when p is just the identically zero function, which is a very special continuous function on [a, b]. That is, one is led to consider the equality (TO sup 11/11 = sup /Gf ( /Yllllf-f^ and to conjecture that (Ti) implies (C). However, instead of (Ti), it is rather easy to show directly that (Li) sup /Gf sup sup/ /Gf inf/ /Gf and hence, upon comparing the right-hand sides of equations (Li) and (Ti), it is clear that the "missing link," in order to complete the chain of reasoning in a proof, "by retracing the steps," of the desired equation (T2), is simply sup (L2) sup sup/ /Gf inf/ /Gf ( ,VII \\(-, ,Y The structure of the formal proof of (T2), given in Section 2, obtained by retracing the preceding heuristic steps, is then clear, and can be expressed as follows: (Li) is the conclusion of Lemma 1; (L2) is the conclusion of Lemma 2; (Ti) is the conclusion of Theorem 1; (C) is the conclusion of the Corollary; and, finally, (T2) is the conclusion of Theorem 2. 9. It is evident that, in this paper, the finite interval [a, b] may be replaced throughout by a closed and bounded (i.e., compact) subset of the real numbers; and, in fact, even by a nonempty compact metric space, with only minor changes in the text. 2. Equivalence of the General Approximation Problem to a Simpler Approximation Problem. This section contains the formal proof of the basic equation (T2), as outlined in point 8 of Section 1. Lemma 1. sup /Gf sup sup /||, /Gf inf / I/Gf Proof. To save writing, denote (1) uF = sup / , /Gf If = inf / ; /Gf License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SIMULTANEOUS APPROXIMATIONS 587 then the equality to be proved is sup 11/11 = max {||mf||, ||Zf||} . /Gf It will be shown first that (2) sup 11/11 Ú max \\\uF\\,\\lF\\\ . /Gp This inequality follows from the fact that, for all / £ F, Ipix) ^ fix) S uF0x) , a ¿ x S b . Therefore \fOx)\ g max {|wf(:t)|, \IfÍx)\\ , g max {||mp||, ||Zp||} , which yields 11/11 Z max {\\uf\\, Ml, and hence (2) follows. It remains to show that (3) sup||/|| è max flKII, llallí ; /Gf and, in the first part of this argument, it will be proved that (4) sup D/11 g IMI . /Gf Let {xk}^, be a sequence of numbers, with a ^ xk ^ b for k = 1, 2, • ■ -, such that lim |«f(:Cí:)| = ||iíf|| . k—*oo In the sequence of numbers {uFÍxk)\TM=1, there are either infinitely many numbers ^ 0, or else infinitely many < 0; in the first case, it follows that there is an infinite subsequence {uFix„k)\^v such that lim uFixnk) = ||wp|| , &-+00 while, in the second case, there is a subsequence {uFixnk)}TM=v such that — lim UpOXnk) = \\uF\\ . k—*ao In either case, letting {xk}TM=i denote the chosen subsequence, if necessary, one has that lim IwfOcOI = lim Mf(^a) = ||mp|| • k—»oo I k—»00 Now, let {fk)1c be a sequence of functions from F such that upOxk) — 1/k S fkixk) S upOxk) , for k = 1, 2, • • • (the existence of such a sequence of functions follows from the definition of uf). Then, one has License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 588 J. B. DIAZ AND H. W. MCLAUGHLIN Ihn fkixk) = Hm uFOxk) , k—»oo k~»x