APPROXIMATION FROM VARISOLVENT AND UNISOLVENT FAMILIES WHOSE MEMBERS HAVE RESTRICTED RANGES THESIS FOR THE DEGREE. 0F Ph. D. MICHIGAN STATE UNIVERSWY J. EDWARD TORNGA - 1971 LIBP 'W Michigan State University xfit'w”! This is to certify that the thesis entitled "Approximation From Varisolvent and Unisolvent Families Whose Members Have Restricted Ranges". presented by J. Edward Tornga has been accepted towards fulfillment of the requirements for Ph. D. degree in Mathematics flay/U D‘ M 7 Major professor 47 Date February 15. 1971 0-7639 7. Jr. P. I: u ABSTRACT APPROXIMATION FROM VARISOLVENT AND UNISOLVENT FAMILIES WHOSE MEMBERS HAVE RESTRICTED RANGES By J. Edward Tornga We consider the questions of existence, charac- terization, and uniqueness for the following approxi- mating problem. Approximate in the uniform norm a real valued function f e C(X), where X is a compact set contained in the real closed interval [a,b], from a subset of a certain family of continuous real valued functions defined on [a,b]. The subset con- sidered is the subset of the family lying between two curves u and 2, where Ll> 2 . Our family is a varisolvent family in Chapter l. We also look at the constant error curve difficulty in the characterization theorem for a varisolvent family. In Chapter 2 we consider a family which is unisolvent. Adding a Haar subspace condition to a varisolvent family gives us strong uniqueness and continuity of the best approximation operator theorems in Chapter 3. Finally in Chapter 4 we consider con- tinuous generalized weight function approximation J. Edward Tornga where local solvency of a varisolvent family is replaced by property A. Our theorems are the natural extension of the theorems for the same problem considered by G. D. Taylor for a linear family of functions. APPROXIMATION FROM VARISOLVENT AND UNISOLVENT FAMILIES WHOSE MEMBERS HAVE RESTRICTED RANGES By J. Edward Tornga A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1971 " / ‘I /'. ACKNOWLEDGEMENTS My sincere appreciation goes to my major Professor Gerald D. Taylor for his guidance. I am especially indebted to him for his time, his many suggestions, and his encouragement. I would also like to thank my wife Sondra for her moral support and patience during this time. ii Chapter TABLE OF CONTENTS INTRODUCTION ......................... APPROXIMATION FROM MEMBERS OF A VARISOLVENT FAMILY HAVING RESTRICTED RANGES .................... Section l-l: Introduction and Definitions ............ Section 1-2: Existence of Best Approximations ......... Section l-3: Characterization of Best Approximations.... Section 1-4: The Constant Error Curve Condition in the Characterization Theorem ................ Section l-S: Uniqueness of Best Approximations ......... APPROXIMATION FROM MEMBERS OF A UNISOLVENT FAMILY HAVING RESTRICTED RANGES .................... Section Z-l: Introduction and Definitions ............ Section 2-2: Existence, Charac- terization, and Uniqueness of Best Approximations ......... APPROXIMATION FROM MEMBERS OF A VARISOLVENT FAMILY HAVING RESTRICTED RANGES WITH THE ADDITIONAL HYPOTHESIS OF A HAAR SUBSPACE EXISTING FOR EACH MEMBER OF THE FAMILY ........................ Section 3-1: Introduction and Definitions ............ iii Page 1 20 3O 34 34 35 41 41 Page Chapter Section 3-2: Strong Unicity and Continuity of the Best Approximation Operator Theorems ...... ST 4 GENERALIZED WEIGHT FUNCTION APPROXIMATION WITH PROPERTY A AND RESTRICTION BETWEEN THE CURVES u AND 2 WHERE u > B ........... 59 Section 4-l: Introduction and Definitions ............ 59 Section 4-2: Existence of Best Approximations ......... 63 Section 4-3: Uniqueness and Characterization of Best Approximations.... 66 BIBLIOGRAPHY ......................... 77 iv INTRODUCTION Beginning with Tchebyshev (see pg. 224-227 of [5] for specific details), the following problem was consid- ered: approximate a continuous function f(x) on [a,b], a,b real finite numbers, from a set of continuous linear functions {P(A,x)} on [a,b] where A = (a1....,an)e: En, Euclidean n space, and the {P(A,x)} formsa Tchebyshev set of degree n I . That is. for P(A],x) t P(A,x), both belonging to {P(A,x)}, P(A],x) - P(A,x) can have at most n - 1 zeros. By 1918 existence, uniqueness, and charac- terization of the best approximation to f on [a,b] from {P(A,x)} were determined. In 1949 Motzkin [16] and in 1950 Tornheim [26], extended this type of theory to a non-linear case. Their approximating family was a set of unisolvent functions. That is. a set of continuous functions {U(A,x)}, A = (a1....,an)e: En’ of degree n such that given points (xi: xi < x1+], i = 1,...,n} on [a,b] and real numbers {y1}?=1 , there exists a unique U(A],x) belonging to {U(A,x)} which interpolates the yi at the xi. Existence, uniqueness, and characterization of the best approximation i: all functions considered in this paper are real- valued functions were discussed. Since many useful non-linear approximating families, such as the rational and exponential functions were not unisolvent, J. R. Rice [20] in 1961 weakened the hypoth- eses of a unisolvent family to a locally uniSolvent family of continuous functions {F(A,x)] having variable degree. Rice called this family a varisolvent family. Exponentials and rationals as well as other non-linear functions have been shown to be varisolvent families (see table 7.1, page 40 of [20]). Questions of exist- ence, uniqueness, and characterization of best approxi- mations from a varisolvent family were examined by Rice. In 1967 C. Dunham noticed that the proofs of Tornheim's and Rice's characterizationtheorems for the unisolvent and varisolvent families were incomplete. The unisolvent characterization proof by Tornheim has been completed (see R. Barrar and H. Loeb [1]), whereas the varisolvent characterization proof is only partially completed (see D. Braess [4] and R. Barrar and H. Loeb [1]). By assuming additional conditions on the vari- solvent family, however, the characterization theorem has been completed. R. Barrar and H. Loeb [3] gave one such additional condition. They assumed the existence of a Haar subspace for each member of the approximating family in order to complete the proof (see Chapter 3). G. D. Taylor [22], [23], and [24] among others, has examined the situation of the first paragraph in a more general setting along with certain restrictions on the approximating family {P(A,x)}. In particular, the problem was considered when the only acceptable approxi- mations were members of {P(A,x)} whose ranges were between two curves. Questions concerning existence, uniqueness, and characterization of best approximations were studied in this restricted setting. In this paper, we will examine non-linear approxi- mations in the restricted setting of Taylor. Restricted varisolvent families will be considered in Chapter 1. We wi11 also look at the completeness of Rice’s charac- terization theorem when additional hypotheses are assumed. Chapter 2 will consist of approximation by unisolvent families in our restricted setting. A strong uniqueness theorem and a continuity of the best approxi- mation operator theorem will comprise Chapter 3. Our last chapter, Chapter 4, will consider generalized weight function approximation with property A in our restricted setting. Throughout the paper, we will of course assume that our approximating families are non-empty. CHAPTER I APPROXIMATION FROM MEMBERS OF A VARISOLVENT FAMILY HAVING RESTRICTED RANGES Section 1-1: Introduction and Definitions. Let En represent Euclidean n space. For Ai 5 En. let Ai = (aIi,a21,...,ani). Let R be the set of real numbers. Let X be a compact subset of R with XC [a,b] where a and b are finite and fix P a subset of En' Let V be the set of functions {F(A,x): Px[a,b] ~ R} where (1-1) F(A,x) is continuous in the sense that, given A0 5 P, xo e [a,b], and 6 > 0, there exists a 6 > 0 such that A e P, x e[a,b] , and ”A0 - A” + |x - xol < 6 imply that IF (Ao,xo) - F(A,x)l <'e where HAO - A” = max la.° - a. I lSiS n 1 1 (1-2) F(A,x) is locally solvent of degree m(A),i.e., suppose we are given a set of points S = {sz a S x1 < x2~-°< xm(A) S b} and an 6 > 0. Then there exists a 6 = 6(S,F(A,x), E) > 0. such that lyj - F(A,xj)l < 5 j = l,...,m(A) implies that there exists anF(A],x) e V such that both F(A1.xj) = yj j = 1.....m(A) and ”F(A.X) - F(A].x)H < 6 hold where ”F(A,X) ' F(A19XIN = max lF(A,X) ' F(A19X)l xe[a,b (1-3) F(A,x) has property Z of degree m(A) on [a,b], i.e. for any F(A*,x) ¢ F(A,x), F(A*,x) e V, we have that F(A*,x) - F(A,x) has at most m(A) - 1 zeros on [a,b] (we assume m(A) 2 1). We will assume that m(A) is uniformly bounded for all A e P and that X has more points (in number) than any m(A) for A e P. It should be noted that the degrees of two different members of V may not be the same. The family of functions V defined above was called a varisolvent family by J. R. Rice [20]. As was men- tioned in the introduction to this paper, Rice considered the questions of existence, uniqueness and characteriza- tion of best approximations from a family V. We will examine a generalization of Rice's work. Suppose an additional assumption that only members of V lying between two curves are to be used in the approxi- mating problem. Do we then obtain comparable existence, uniqueness and characterization results? Let ff’ be the subset of V where (M) F(A,x) e 51’ if £(x) s F(A,x) s u(x) for all x e x where u(x) and £(x) are defined below. Thusfif is a subset of V bounded between two curves u(x) and 2(x). We will consider the existence, characterization, and uniqueness of approximating a given real-valued continuous function f onXin the uniform norm by members of ft, That is to say, can we determine an F(A*,x) 23’ such that "F(A..x) - f(x)» = inf (”F(A.x) - f(x)“: F(A,x) as!) where again (and for the entire paper), ”F(A.x) - f(X)” = max |F(A.X) - f(Xll xeX First we require a few more definitions. Since we want ”F(A,x) - f(x)” = K < .. for some F(A,x) efc’, we must ensure that u does not assume the value - w and 2 the value + w. Also when u and 2 assume finite values, we will wish to examine the distance, between u and members of’éf, and between I and members offlf’. To ensure the existence of maxima and minima (of u(x) - F(A,x) or 2(x) - F(A,x), F(A,x) cf), we require that u and 9. are continuous on closed subsets of X. Therefore, let u and t be defined on x such that (i) I may assume the value - s, but never + w. (ii) u may assume the value + a, but never - 0. (iii) X_; {x: 2(x) = —m I and X+°= {xz u(x) = + coI are open subsets of X. (iv) 2 is continuous on x - X_co and u is continuous on X - X+° . (v) A < u for all x e X. Note that (i) + (v) imply that inf [u(x) - 2(x): x c X] = d > O. The reason for (v) will be discussed in the remark at the end of Section 1-5. J. R. Rice assumes that A i A* implies that F(A,x) s F(A*,x). Instead of this, we will use R. B. Barrar and H. L. Loeb's notation in [3]. Let N be the maximal value of m(A) for A e P. A function f e c[a,b] (i.e. continuous on [a,b]) will be called a normal point in C[a,b] if it has a best approximation F(A*,x) which has the property that m(A*) = N. Note that if F(A,x) an: and m(A,) = N, then F(A,x) itself is a normal point in C [a,b]. We say that An-is equivalent to An' if F(An,x) a F(An',x). Also the sequence {An} is equivalent to the sequencevan'} if An is equivalent to An' for each n. We will require this concept of equivalence for Theorem 3-2. The above definitions, unless specifically changed, will apply throughout this paper. Section 1-2: Existence of Best Approximations. The existence of a best approximation to f on X fromjf is not assured from our definitions about57’. He will parallel the discussion on page 9 in [20] by J. R. Rice to obtain one criterion for the existence of a best approximation. On page 6 of [20], the following theorem is given: Let {fa} be a uniformly bounded infinite set of functions continuous on [0,1] with property Z of fixed degree. Then {fa} contains a pointwise convergent subsequence. Noting that the above theorem holds for our family 71’, on [a,b] we have Theorem 1-1: If the limit of every pointwise convergent sequence of members ofi4fbelong to;fi!, then a best approxi- mation to f on X exists fromf. Efigflii There always exists a sequence {F(An.x)Iefi7’ such that lim ”F(An,x) - f(x)” = inf l (A,x) - f(x)” n+m F(A,X)€ By taking subsequences, since m(A) is uniformly bounded, there exists a subsequence {F(As,x)} of [F(An,x)} with property 2 of fixed degree such that lim “F(As,x) - f(x)” = inf ”F(A,x) - f(x)” s+w F(A,x 5 Applying the theorem of J. R. Rice mentioned in the first paragraph gives a pointwise convergent subsequence {F(At,x)] of {F(As,x)] such that lim ”F(At,x) - f(x)" = inf ”F(A,x) - f(x)” t+m F(A,X)E Therefore, by our hypothesis, the limit of this pointwise convergent sequence belongs to)?’and we have a best approxi- mation to f on x fromfl’. As Rice mentions (page 9 [20]), if there is conver- gence by functions infi’to functions not infi’one usually enlarges the familyfiito include these functions or some type of equivalence is set up between the limit function (not infi’) and a function from 9’. As examples of this, see pages 42-44 of [20] for exponential families and pages 77-84 of [20] for rational families. Section 1-3: Characterization of Best Approximations. The standard Tchebyshev characterization of a best approximation by the number of alternations of the error curve (given by Rice [20] for the unrestricted varisolvent approximating family) extends to our familyl’f”, defining alternations as G. D. Taylor does in [23]. A zero xO of g(x) e C [a,b] is said to be a simple zero if g(x) changes sign at x and a double zero if 0 g(x) does not change sign at x0. Property 2 limits the number of distinct zeros a member offii’may have. He now wish to limit the number of simple and double zeros a member offi’may have. The following Lemma 1-1 is Lemma 7.1 of [20] in our restricted setting. Lemma l-l: Let F(A,x) 2,9’. Then for F(A*,x) ej?’ such that F(A*,x) ¥ F(A,x), we have that F(A*,x)-F(A,x) can have at most m(A) - l zeros, counting a simple zero once and a double zero twice. £3221; Lemma 1-1 is just Lemma 7.1 of [20] with the added condition of (1-4). Therefore F(A*,x) and F(A,x), considered as unrestricted varisolvent functions, satisfy Lemma 7.1 which implies that F(A*,x) - F(A,x) can have at most m(A) - 1 zeros, counting a simple zero once and a double zero twice. 1 10 Fix f(x) 5 C (x), and let F(A,x) e;¥’ be given. Then we define, following Taylor [23], x+1 = {x e x: f(x) - F(A,x) = ”f(t) - F(A,t)H} X.] = {x e X: f(X) - F(A.X) = - ”f(t) - F(A.t)H} x+2 = {x e x: F(A,x) = £(x)} x_2 = {x e x: F(A,x) = u(x)} xA = x+1 u x+2 u x_1 u x_2 The set XA is said to be the set of critical points of f(x) - F(A,x) on X. As is done in [23], we will divide our problem into two possibilities. Lemma 1-2 and Example l-l below are from Taylor [24] and [22]. Lemma 1-2: (x u x+2) n (x._1 U X_2) # ¢ +T implies that F(A,x) is a best approximation to f on X from 5’. Proof: Case 1: X+1 n X_1 % 6 implies F(A,x) 5 f(x). Case 2: X+1 n X2 # 6 or X_1 n X+2 s 0 implies that at some critical point, f is a distance ”f(x) - F(A,x)” above the curve u or below the curve 2. Therefore any other member of6?’must be a greater or equal distance from f at this critical point. There- fore F(A,x) is a best approximation. Since X n X_2 = o, the proof is complete. +2 11 Best approximations satisfying the hypothesis of Lemma 1-2 need not be characterized by alternations as in the case where (X+1 U X+2) n (X_] U X_2) = 6 . The following example is one illustration of this fact. 2 Example l-l: Let f(x) = x on [-l,l] = x. Letji = (F(A,x): F(A,x) = ax2 + bx + c, where O 2 F(A,x) 2 -1]. Any member of?! passing through 0 for x = - l and x = + 1 is a best approximation. The example above also shows that for (X+] U x+2) n (x_] U x_2) # ¢ 9 a best approximation fromjgfmay not be unique. For the remainder of this section we will consider the case (X+1 U X+2) O (X_1 U X_2) = ¢ Definition: F(A,x) - f(x) is said to alternate K times on X if there exist k + 1 critical points {xi} in X where a S x1 < x2 --- < xk+1 S b and such that x. e X_1 U X_2 implies x. 1+] 8 X+1 U X +2 OY‘ x. s X U X U X 1 +1 +2 implies x. 1+] 6 X_ l -2 for i = l,2,...,k These k + 1 points are said to form an alternant of length k. 12 A standard type Tchebyshev characterization is as follows: Let X_2 = X+2 = 6 in the above definition. Then g(x) is a best approximation to f on X if and only if the error curve f - g alternates n times on X (n depends upon the approximating family). As was mentioned in the introduction to this paper, Tchebyshev and his contemporaries examined a continuous linear family of functions {F(A,x)} which formed a Tchebyshev set. The above type of characterization of the best approximation was discovered. In 1949 and 1950 Motzkin and Tornheim noted the same type of characterization theorem for their unisolvent families. In 1961 J. R. Rice exhibited a similar type of characterization theorem for his more general non-linear family, the varisolvent family. In 1967 C. Dunham [6] noted that the characteriza- tion proofs of Tornheim and Rice for unisolvent and varisolvent families were incomplete. They both neglected to consider the possibility of a constant error curve (i.e. f - g E c). R. Barrar and H. Loeb [1] showed that the constant error curve could not exist for unisolvent families in Tornheim's proof (this was already known since a characterization proof for unisolvent families, different from Tornheim's existed - see Novodvorskfi and Pinsker [18]). Barrar and Loeb in the same paper also showed that for m(A) S 3, a constant error curve could not exist for varisolvent families. 13 D. Braess [4] has shown that if in an<3 neighborhood of the best approximation from V (V is a varisolvent family), all the members of V have the same degree as the best approximation, then the best approximation must alternate. Since the degree is an upper semi- continuous function (see Theorem 2 of Rice [20]), it then follows that best approximationsof maximal degree must alternate. At this time, however, it is not known in general whether a constant error curve for the best approximation from a varisolvent family can exist. By adding additional hypotheses, the possibility of a constant error curve can be eliminated. As was mentioned, a Haar condition can be added (see Chapter 3). By using property A (see Chapter 4) in lieu of local solvency, this difficulty can also be overcome. A third possibility would be to assume that the varisolvent family is extendable to a larger interval (see Corollary 3 of Theorem 1-2). A fourth might be to assume closure offflfunder pointwise convergence (see Corollary 4 of Theorem l-2). An obvious fifth possibility would be to assume that each member of V intersects f at some point of X. The proof of the next theorem follows the standard approaches used by both Tornheim and Rice in their characterization proofs. 14 Theorem 1-2: Let f e C(X). F(A,x) 5.2F , and assume (X+1 U X+2) n (X_1 U X_2) = 4 , where X is a compact set contained in [a,b]. (1) If F(A,x) is a best approximation to f from/f! and F(A,x) - f(x) i c, c a non-zero constant, then F(A,x) - f(x) alternates at least m(A) times on X. (2) If F(A,x) - f(x) alternates at least m(A) times on X, then F(A,x) is a best approximation to f from! on X . Proof: (of (2)) Assume F(A,x) - f(x) alternates at least m(A) times on X. Assume F(A,x) is not a best approximation to f from.§{. Then there exists an F(A*,x) inff’such that ”F(A*,X) ' f(X)” < ”F(A,X) ' f(X)” .: ' = ..., + , . . , . Let [xJ J 1,2, m(A) l xJ < xJ+1 xJ e X} be an alternant of length m(A) for F(A,x) - f(x) on X. These must be distinct since (X+1 U X+2) n (X_1 U X_2) = 6. Now at these critical points we have F(A*,Xj)'F(A,Xj) = (F(A*,xj)-f(xj))-(F(A,xj)-f(xj)). He assert that F(A*,x) - F(A,x) has at least m (A), zeros on [a,b], counting simple zeros once and double zeros twice, which will contradict Lemma 1-1. We will match a counting zero (a double zero has two counting zeros, a simple zero one) to one and only one interval x., . (JXJ As Then F( better Case 1: Case 2: 15 +1) for j = l,2,...,m(A) + l. sume x1 5 X__1 U X__2 for F(A,x). A*,x]) - F(A,xl) s 0 since F(A*,x) elf’ is a approximation to f(x) than F(A,x). F(A*,x1) - F(A,xI) = 0 . Associate the zero x1 with the interval (x],x2). F(A*,x1) - F(A,xl) < 0. Two possibilities can occur. (a) F(A*,x2) - F(A,xz) 2 O for some x£e(x],x2). Since F(A*,x) - F(A,x) is a continuous function on [a,b], it has a zero on (x],x2) . Associate a zero on (x],x2) with the interval (x],x2). (b) F(A*,x) - F(A,x) < O for all x e (x],x2). Since x2 6 X+1 U X+2 and F(A*,x) e.fif is a better approximation than F(A,x), there exists a zero at x2. Associate this zero with (x],x2). We have associated a zero in [x],x2] with (x],x2). Now x2 6 X+1 U X for F(A,x) and therefore +2 F(A,,x2) - F(A,xz) 2 o. If F(A*,x) - F(A,x) changes sign in (x2,x3), associate a zero in (x2,x3) with (x2,x3). If not, consider cases. Case 1: F(A*,x) - F(A,x) > O on (x2,x3). Since x3 6 X_1 U X_2 for F(A,x), a zero exists T6 at x3. Associate this zero with (x2,x3). Case 2: F(A*,x) - F(A,x) < O on (x2,x3). For a sufficiently small 6 > 0, two possibilities can occur, (a') F(A*,x) - F(A,x) 2 O on (x2 -6, x2) in which case a zero exists at x2 which was not associated with (x1,x2). Associate this zero with (x2,x3). (b') F(A*,x) - F(A,x) < O on (x2- 6, x2) in which case a double zero exists at x2 that was used at most once on (x1,x2). Associate an unused counting zero at x2 with (x2,x3). We have associated a zero in [x2,x3] with (x2,x3) which was not associated with (x],x2). Proceed in the same manner with the remaining inter- vals (xj,xj+]), j = 3,4,...,m(A) as was done for (x2,x3). For x1 5 X+1 U X+2 , the argument would be similar. In either case, F(A*,x) - F(A,x) has at least m(A) zeros on [a,b], counting double zeros twice and simple zeros once. This is the desired contradiction of Lemma 1-1 and (2) of Theorem 1-2 is proved. (of (1)) Assume F(A,x) is a best approximation to f from? and F(A,x) - f(x) 9! c, c a non-zero constant. Further assume that F(A,x) - f(x) alternates exactly 5 < m(A) times at the points 17 5+] S b, xj e X}. {sz a S x1 < x2 < --- < x (Note that it is possible for s to be zero here). Case 1: Assume that a and b are not critical points for every critical point set of (s + 1) points of F(A,x) — f(x). (Note that a and/or b may not even belong to X).‘ If a (or b) is not a critical point, select a 61 > 0 such that a + 61 is less than the first possible critical point (or (b - 6]) is greater than the last possible critical point). For concreteness. assume a is not a critical point. Determine a 6(60) where 61 > 6 > 0, such that for some 60 > 0 (sufficiently small), we have max {£(x) - f(x), - ”F(A,x) - f(x)H] + 60 < F(A,x) - f(x) < min {u(x) - f(x), HF(A),x) - f(x)”I - 60 for all x e [a,a + 6] n X. This is possible by contin- uity, compactness, and the fact that u(x) > £(x) on X. Now, the facts of the previous sentence and (X+1 U X+2) n (X_] U X_2) = 6 imply that we can select the following points a = x0 < xm(A)-s < xm(A)-s+l < --- < xm(A) = b which divide X into s + 1 subsets so that for 62 > 0 sufficiently small, (a) F(A,x) - f(x) alternates exactly once on any two adjacent subsets, but does not alternate on any 18 one subset. (b) If (X+1 U X+2) # 6 on a subset, then F(A.x) - f(X) < min TU(X) - f(X). "F(A.x) - f(X)H} - 6 for all x e (the subset O X) 2 If (X_1 U X_2) # 4 on a subset, then F(A,x) - f(x) > max {2(x) - f(x), - "F(A,x) — f(x)”) + 52 for all x e (the subset n X) (c) max {£(x) - f(X). - ”F(A.X) - f(X)Hl +62 < F(A,x)-f(x) < min {u(x) - f(x), ”F(A,x) - f(x)N} - 62 for all 1 points in the set {x , x ,...,x m(A)-s m(A)-5+1 m(A) Choose m(A) - s - 1 distinct points {sz J = l,2,...,m(A) - s - 1, xj < xj+11 in [a,a + 5]. 62 92 Let 6 = min( —7 ,1—5 ) > O and x' denote a point in X above where “F(A,x) - f(x)" is assumed. Since F(A,x) is varisolvent of degree m(A), there exists an F(A*,x) efi’ where (a) F(A*,xj) - F(A,xj) = 0 j = l,2,...,m(A) - 1 (b) [F(A*,x') - f(x')] < | F(A,x') - f(x') I, and (C) ”F(A.X) - F(A*.x)H < 6 Now by Lemma 1-1. (a) and (b), max|F(A*,x) - f(x)] < max|F(A,x) - f(x)| for all x e ([a + 6,b] n X) while (c) implies max|F(A*,x) - f(x)] < maxlF(A,x) “f(XII 19 for all x e [a,a + 6] n X. Therefore “F(A*.X) - f(X)" < "F(A.X) - f(X)H (for all x e X) and F(A*,x) 2,}? is a better approxi- mation to f on X than F(A,x). This is a contradiction. Case 2: Assume both a and b are critical points. Replace [a,a + 6] by a similar interval [xm(A)-s '6’ xm(A)-s] where [sz j = O,m(A) - s,...,m(A)],6, 60, 62, and 6 are defined analogously to the previous construction. For m(A) - s - 1 even, choose m(A) - s - 1 points {sz xm(A)-s '6 s xl < x2 -" < xm(A)-s-l < xm(A)-s} and determine anF(A*,x) iii! such that (a') For x e {a,b}, x e x+1 u x+2 implies that F(A*,x) - f(x) > F(A,x) - f(x), while x e x_1 u x_2 implies F(A*,x) - f(x) > F(A,x) - f(x). (b') ”F(A*,x) - F(A,x)”< e, and F(A,xj) - F(A*,xj) = o for j = l, 2, ... , m(A) - l. The evenness of m(A) - s - l and our construction imply that the two conditions in (a') are in reality only one restriction on our varisolvent function, so that indeed, an F(A*,x) e Jf’does exist. Then ”F(A*.X) - f(X)H < ”F(A.X) - f(x)” which is a contradiction. 20 For m(A) - s - 2 even, choose m(A) - s - 2 points {sz xm(A)-s -6 < x2 < x3 --- < xm(A)-s-1 < xm(A)-S} and determine an F(A*,x) e f’such that (a') and (b') above hold and (C') F(Asxj) ' F(A*sxj) = 0 j: 2,3,...,m(A) " I. If F(A,x) - F(A*,x) has a zero in addition to those in (c'),then (a') implies that this is a double zero, which is impossible by Lemma l-l. Therefore IIF(A..X) - f(x)” < llF(A.X) - f(X)” which is a contradiction. Now for all possibilities we have constructed a better approximation to f than F(A,x). This contradicts the fact that F(A,x) can alternate at most 5 < m(A) times. Therefore (l)of Theorem 1-2 is proven. ' Section 1-4: The Constant Error Curve Condition in the Characterization Theorem. It would be very desirable to omit the condition of the constant error curve in (l) of Theorem l-?. He would then have an if and only if statement for the characteri- zation of best approximations in terms of alternations. This section will consist of corollaries to Theorem 1-2 where the constant error condition is eliminated. One such case with the added hypothesis of a Haar subspace, will be deferred until Chapter 3. 21 R. Barrar and H. Loeb [1] have shown that for m(A) S 3, a constant error curve for the best approxi- mation cannot exist. With the addition of our condition on the critical point sets, their reasoning applies for our family,§f Corollary 1: For m(A) S 3, the constant error curve condition in (1) of Theorem 1-2 can be omitted. Proof: For concreteness, assume that f(x) - F(A,x) E c > 0 (if c < O, a similar argument holds). Assume m(A) = 1. Since (X+1 U X+2) n (X_1 U X_2) = 6, max (u(x) - F(A,x))= 51 > 0. Then for 51 > E] > 0, xeX there exists a 61(61) > O and an F(A],x) c.5fl such that (i) ”F(A,x) - F(A],x)H < G], (ii) F(A],x) doesn't intersect F(A,x), and (iii) F(A],a) - F(A,a) = 61 > O, a! since,# is a varisolvent family. But then F(A],x) is a better approximation to f from’flfthan F(A,x) which is a contradiction. Assume m(A) = Again since 2. (X+1 U X+2) n (X_1 U X_2) = 6, max (u(x) - F(A,x)) = $2 > O. xeX Then for $2 > 62 > 0, there exists a 62 (62) > 0 and an F(A2,x) e ad’such that (i) ”F(A,x) - F(A2,x)H < 62, and (ii) F(A ) - F(A,a) = 62 > O,F(A2,b) - F(A,b) = 62 > 2'3 O. 22 Now, F(A2,x) - F(A,x) has at most one zero (m(A) = 2), but (ii) implies that if a zero occurs, it is a double zero or another simple zero exists. This is impossible, therefore .F(A2,x) > F(A,x) for all x 5 [a,b] and by construction, F(A2,x) is a better qpproximation to f from.figihan F(A,x). This contradicts our hypothesis. Assume m(A) = 3. There exists an 63 > O constructed as above such that for 53 > 63 > 0, there exists by solving a 63 (63) > O and an F(A3,x) eff such that (i) F(A3.a) = F(A,a), F 0 neighborhood of the best approximation F(A,x) e V, such that any member of V lying entirely in this neighborhood 23 has the same degree as F(A,x). J. R. Rice (pages 5 and 6 of [20]) showed that for F(A],x) e V, there existed a 6 > 0 neighborhood of F(A],x), such that if F(A,x) (belonging to V), was entirely in this neighborhood, then m(A) 2 m(AI). Braess used both these results to arrive at the following corollary (which is placed in our setting). Corollary 2: If F(A,x) is the best approximation to f on X fromf’such that there existsno member of V (V is the varisolvent family where if: V) with degree greater than m(A) lying entirely in some 6 > O neighborhood of F(A,x), then the constant error curve in (l) of Theorem 1-2 can not occur. One result, noted by Braess, is that if the best approximation has maximal degree, (see Corollary 2 above), a constant error curve can not occur. The following corollary is due to G. Meinardus and G. D. Taylor (oral communication). Corollary 3: If for each F(A,x) e ‘17: there exists an extension [a],b]] of [a,b] (either - w < a] < a or a > b1 > b - possibly both) and a varisolvent family V' on [a,b] such that V'IEa b] = f! and for some F(A*,x) e V', we have that F(A*,x) E F(A,x) on [a,b], we can then omit the constant error curve condition in (l) of Theorem 1-2. 24 Proof: As in the proof of Corollary 1, for concreteness, assume f(x) - F(A,x) E c > 0 and F(A,x) is the best approximation to f fromjfifon X. Then there exists an s > 0 such that max (u(x) - F(A,x)) = 5 since xeX (X+1 U X+2) n (X_1 U X_2) = 6. Using our hypothesis, we then have an F(A*,x) belonging to V' such that F(A*,x) E F(A,x) on [a,b]. Select m(A*) - 1 distinct points on [a],b]] which are not in [a,b]. Now by our solvency condition, there exists an F(A1,x) e V' such that F(A],x) equals F(A*,x) at the above m(A*) - 1 points and for some point in [a,b], F(A],x) - F(A*,x) = S/2. Now v.l[a,b] =j'implies F(A],x) e f, . But by our construction F(A],x) e}?’ is a better approximation to f than F(A,x). This contradicts our hypothesis. ‘ The next corollary to Theorem 1-2 eliminates the constant error curve whenJF'is closed under pointwise convergence. This does not appear to be that strong a condition, since some form of compactness on {1’ is usually required to ensure existence of a best approxi- mation (see Section 1-2). We first require some preliminary results due to C. B. Dunham [7]. Definition: A family G of functions is dense compact on X, a compact space, if every bounded sequence of elements of G has a subsequence converging pointwise on 25 a dense subset Y of X to an element 9 of G and for x t Y, lim inf g(y) S g(x) S lim sup g(y) for y c Y. Note that if G c C(X), then the above inequality can be omitted in the definition of dense compactness. Lemma 1-3: Let G be dense compact. Let {fk} c C(X) converge uniformly to f e C(X) and gk be a best approxi- mation to fk. Then {9k} has a subsequence [9k I con- verging pointwise on a dense subset of X to a best approximation to f. The theorem mentioned in the first paragraph of Section 1-2 gives us a pointwise convergent sequence for a bounded sequence of functions from fifli Therefore ,I if we assume that,f is closed under pointwise convergence, 5f will be dense compact. Corollary 4: If gris closed under pointwise convergence, and the number of points of X is at least twice the uniform bound of m(A), then the constant error curve condition of (l) in Theorem 1-2 can be omitted. Proof: For concreteness, assume f(x) - F(A,x) E c.> 0 (a similar argument holds of c < 0). Let {x14 i =i,2,...,m(A)l and {yi: i =1,2,...,m(A) - 1] be a set of points in X such that xi < yi < xi+1 Let 9n 5 C(X) be defined as follows, 26 f(x) - c/n for x l,2,...,m(A) ll X ..a II gn(X) = f(x) for x = yi i l,2,...,m(A) - 1 any continuous curve h(x) on X connecting gn(x) at the xi's and yi's such that f(x) 2 h(x) 2 f(x) - c/n on {xi, xi+]} for i = l,2,...,m(A) - 1. Now since gn is continuous on X, let F(An,x) be the best approximation to gn on X from,fl{. If "9,,(X) - F(An.X)H 2 C. then F(A,x) is a best approximation to 9n and by con- struction F(A,x) is not parallel to 9". Applying (2) of Theorem 1-2 gives us a better approximation to gn than F(A,x) (or F(An,x)). Therefore 119nm - F(An.x)u < c. We assert that F(An,x) is not parallel to 9". Assume it is, i.e. gn(x) - F(An,x) E C". We first note that if cn = 0, by construction F(An,x) would be a better approxi- mation to f than F(A,x). Therefore cn # 0. Next, assume cn < 0. Then F(An,x) is above 9", but F(A,x) is within c of 9". Therefore since F(A,x) S F(An,x), (gn is not parallel to f), |F(An,x) - f(x)] S c. But F(An,x) is a 27 best approximation to f which is not parallel to f on X. Applying (2) of Theorem 1-2 gives us a better approxi- mation to f fromfi’than F(An,x) (or F(A,x)). This is not possible, hence cn > 0. We should further note that |cn - cl < c/n, for if not by the construction of 9n and the fact that gn(x) - F(An,x) cn > O, F(An,x) would be a better approximation to f than F(A,x). Now, cn > 0 implies that f(x) > F(An,x) and gn(x) > F(An,x). If F(A,x) and F(An,x) do not inter- sect, then one is above the other and is a best approxi- mation to both f and 9n° This is a contradiction since one function cannot be parallel to both f and gn. There- fore assume F(A,x) and F(An,x) do intersect. Then by construction, at the xi, F(A,x) - F(An,x) 2 0 while at the yi, F(A,x) - F(An,x) < 0 since cn < c. Now F(A,x) and F(An,x) both being continuous on [a,b] implies that F(A,x) - F(An,x) has at least m(A) zeros on [a,b] counting double zeros twice and simple zeros once, because of our construction. This contradicts Lemma 1-1. Therefore the best approximation F(An,x) to 9n from fcannot be parallel to 9". Since it is not parallel, Theorem 1-2 says that F(An,x) - gn(x) alternates at least m(An) times on X. A further property of our construction is that there exists 28 a point x' c X such that F(An,x') - F(A,x‘) < O (or at x', F(An,x) is below F(A,x)). For, if not, F(An,x) would be a best approximation to f which is not parallel to f. Again apply (2) of Theorem 1-2 to arrive at a contradiction. We now have for each 9", n > 1, a non-parallel best approximation from 5?: F(An,x) which has a point of X below F(A,x). By construction gn tends uniformly to f as n goes to w . Since we have assumed thatjg’is dense compact (see the paragraph immediately preceeding Corollary 4), Lemma 1-3 may be applied to give us some subsequence of [F(An,x)]co converging pointwise to a n=l best approximation F(A*,x) of f fromj?’. Each member of this pointwise convergent (sub) sequence alternates at least once on X and at one critical point is below F(A,x). Therefore a cluster point of critical points of the pointwise convergent sequence exists, which is on F(A,x). Likewise a cluster point of the form X_1 U X_2 exists since each best approximation in the sequence alternates at least once. Therefore since the limiting function F(A*,x) is a continuous function (it belongs to;§’by our closure hypothesis), it must alter- nate at least once on X. Then F(A*,x) must alternate m(A*) times on X or we could apply (2) of Theorem l-2 to obtain a contradiction. But HF f(x) > £(x) for all x c X, minimin(u(x) -f(x)), min(f(x) -£(x))} < ”F(A,x) -f(x)H xeX xeX Remark: An open question which I plan to look at later, is whether the previous sections can be generalized to the point where u(x) = 2(x) at a finite number of points (see G. D. Taylor [24] for the linear case and K. Taylor [25] for rational families). R. Barrar and A. Loeb [2] examined the question of approximating a function f from a varisolvent family which interpolates f at a finite number of points. Extending G. D. Taylor's paper to our setting would give us interpolation and restricted range approximation at the same time. The non-linear case of h(x) S u(x) appears, however, 33 to require much stronger hypotheses. First, a certain amount of differentiability for functions of ,TflHs needed at the points where u(x) = £(x) in order to generalize Taylor's paper. Although part (2) of theorem 1-2 can be shown to be true, in part (1) a difficulty occurs in constructing a better approximating function from 5;? around the points where u(x) = 2(x). It appears that more than this added differentiability is necessary to obtain (1) of Theorem 1-2. We could assume a uniform bound on a certain order derivative of members of ¢where u(x) = 2(x). This would remove the difficulty around the points u(x) = £(x) and allow us to complete part (1) of Theorem 1-2. CHAPTER 2 APPROXIMATION FROM MEMBERS OF A UNISOLVENT FAMILY HAVING RESTRICTED RANGES Section 2-1: Introduction and Definitions. Let U be the set of functions {G(A,x)z P x [a,b] + R} which satisfy (2-1) and (2-2) below. (2-1) G(A,x) is a continuous function on [a,b]. (2-2) Given the set (xi: a S x1 < x2 ---< xn S b] of n distinct points and n arbitrary real numbers {inL1 , then there exists a unique G(A*,x)c_U such that G(A*,xi) = y, for i = l,2,...,n. U is called a unisolvent family of degree n 0n [a,b]. As in Chapter 1, we will generalize our problem to where we consideronly members of U between our two curves 2 and u. That is to say, our family of approxi- mating functionsd' is a subset of U such that for G(A,x) e,AF , (2-3) u(x) 2 G(A,x) 2 £(x). We will look at existence, characterization, and unique- ness of approximating a given real-valued continuous function f on X in the uniform norm with members of! . 34 35 We should first note that each family xy'is also a family g’of Chapter 1 (i.e., G(A,x) being unisolvent of degree n implies G(A,x) is varisolvent of degree n). Each member of J is continuous, satisfies property Z of fixed degree n, and is between the curves u and t . Each member of 49 is locally solvent by its solvency property ((2-2)), and Theorem 5, page 460 of Tornheim [26]. We will at times call,& a restricted unisolvent family, even though,d’, itself, may not be a unisolvent family. Section 2-2: Existence, Characterization, and Uniqueness of Best Approximations. Although we do not have existence of best approxi- mations from a varisolvent family, for.d9 (our subset of a unisolvent family), best approximations always exist. Tornheim's Theorem 7 [26] gives existence of best approximations for a unisolvent family. Since X is compact and our 2 and u are sufficiently nice the argument of Theorem 7 applies for our restricted family 5 as well. We include it here for completeness. Theorem 2-1: There exists a best approximation to f on X fromfl . Proof: There exists a sequence {G(An,x)} in AV such that lim llG(An.x) - f(X)” = inf ”G(A.X) - f(XHI . n+w G(A,x)c 36 Choose n distinct points x],x2,...,xn in X. By unisolvency, there exists a one to one correspondence between the functions G(A,x) of,” and the set of values y],...,yn taken by the functions G(A,x) at x‘,...,xn. Let G(An’xi) = yni' Then there is a sub- sequence G(An ,xi) of G(An’xi) for which yn i j j is convergent for i = l,2,...,n converging to yi'. By the nature of u and 2 , £(x) S yn . S u(x) implies J1 h(x) S yi' s u(x). Let G(A*,xi) = yi'. Then by Tornheim's Theorem 5, page 460 in [26], G(An ,x) J converges uniformly to G(A*,x) on X C [a,b]. But again 2(x) S G(An ,x) S u(x) for all x c X implies £(x) S G(A*,x) S u(x) for all x c X. Therefore G(A*,x) e,AV and is a best approximation to f on X fromAfi . ' i For the characterization of best approximations from AV , we must consider, as in Chapter 1, the place- ment of critical points. The following lemma is Lemma 1-2 for 47, since,&? is also a family‘jf’from Chapter 1. Lemma 2-1: For G(A,x) e ,1 . if (x+1 u x+2) n (x_1 u x_2) if 6, then G(A,x) is a best approximation to f on X fromIA? . 37 Therefore we need only consider the case when (X+1 U X+2) n (X_1 U X_2) = 6. Theorem 1-2 and Corollary 4 to Theorem l-2 apply ford? (considered as an jzrfamily of Chapter 1). However we can combine these two results for,A9 if we note that xg'is closed under pointwise convergence. Theorem 5 of Tornheim [26] tells us that pointwise convergence gives us uniform convergence and by our selection of X, u, and t , uniform convergence gives us the closure of 19 . This argument is essentially that of Theorem 2-1 and gives us Theorem 2-2: Let G(A,x) e 49', (X+1 U X+2) O (X_] U X_2) = ¢ and assume that X contains at least 2n points. Then G(A,x) is a best approximation to f on X if and only if G(A,x) - f(x) alternates n times on X. R. Barrar and H. Loeb [26] proved Theorem 2-2 for an unrestricted unisolvent family omitting the fact that X must contain at least 2n points (we of course always assume X contains at least n points). Their proof extends directly to our setting. Theorem 2-3: Let G(A,x) c AV and (X+]U X+2) n (X_1 U X_2) = 6. Then G(A,x) is a best approximation to f on X if and only if G(A,x) - f(x) alternates n times on X. 38 3529:: We need only show that a constant error curve for the best approximation can not occur, since Theorem 1-2 applies (.AV being an jV’family). We will show that a constant error curve can not exist for any n (the degree of the unisolvent family U) by induction on n. For n = 1, Corollary 1 of Theorem 1-2 applies and a constant error curve for the best approximation from a restricted unisolvent family of degree 1 can not occur. Assume that a constant error curve can not occur for restricted unisolvent families of degree n S k - 1. Letfl* be a restricted unisolvent family of degree k. Let G (A*,x) be the best approximation to f on X c [a,b] from 49‘*. Assume without loss of generality that f(x) - G(A*,x) E c > O on X. Let A At: {G(A,x) e A! *= G(A.x1> = G(A..x,) where x1 = max {x} I. xeX If x < b, the behavior of a member of; or 1* on (x],b] T will not affect its norm with f on X. Therefore let .A A A 4 .A g=fl'[a,x]] . Then forO<€ c. But for small 6, x2 2 [a,x],-€] which implies that |f(x) - G€(A,x)| < c. Taking the limit as 6 + w gives us a contradiction. A But by the nature of G(A,x), A f(X) ' G(A,X) ¥ C] 9 c1 a constant. Corollary 1 of Theorem 1-4 then gives us a contradiction. Therefore constant error curve best approximations can not occur from restricted unisolvent families. | Example 1-1 shows that the best approximation to f 40 on X from a need not be unique when (x+1 u x+2) n (x_1 u x_2) # 6. But when (X+1 U X+2) n (X_1 U X_2) = 6, we have Theorem 2-4: (X+1 U X+2) O (X_1 U X_2) = 6 implies there exists a unique best approximation to f on X from A . Proof: The proof follows directly from Theorem 2—3 using the standard uniqueness argument. I CHAPTER 3 APPROXIMATION FROM MEMBERS OF A VARISOLVENT FAMILY HAVING RESTRICTED RANGES WITH THE ADDITIONAL HYPOTHESIS OF A HAAR SUBSPACE EXISTING FOR EACH MEMBER OF THE FAMILY Section 3-1: Introduction and Definitions. In this chapter we will give §T’(of Chapter 1) an additional hypothesis, namely a Haar subspace. This will eliminate the constant error curve condition in the characterization theorem for}?’ and allow us to give strong unicity and continuity of the best approxi- mation operator theorems. R. Barrar and H. Loeb [3] did this for an unrestricted family. We will gener- alize their paper to our setting. We first require some definitions. Let P be an open subset of En. Let 27* be the family of functions {F(A,x)z Px[a,b] + R] where,for A = (3],...,an) c P, each éfiiflell for Bai i = l,2,...,n as well as F(A,x) is continuous in A and x. Definition: Let 91’92""’9n be a set of continuous functions on [a,b]. Then {9i}?=l generates a Haar 41 42 subspace of dimension n if 91’92"°"9n forms a lin- early independent set of functions and the only linear combination of 91’92"'°’9n having n or more zeros is the zero function. (For more information on Haar systems see [5], [9] and [13]). The Haar subspace condition that we will use is w aFA. " (3—1) For F(A,x) 5 4g? * ’ {’ 3:1 X) i = T generates a Haar subspace (Haar system) of dimension m(A). We further require that for F(A,x) c /gV *‘, F(A,x) has property 2 of degree m(A). Property Z (from Chapter 1) for members of/fl“ is (1-3)' F(A,x) cg * will have property 2 of degree m(A) on [a,b], i.e., for any F(A*sX) f F(A,x), F(A*.x) 8/5’* , we have that F(A*,x) - F(A,x) has at most m(A) - 1 zeros on [a,b] (m(A) 2 1). Again as in the previous chapters, our approximating family fi7’* will be the subset of ,5? * lying between the two curves t and u (defined in Chapter 1). Our family j§’* differs from,5flpof Chapter 1 in that we have replaced local solvency (or 1-2)) by the Haar subspace condition (3-1). That (3-1) is at least as strong a condition as (1-2) will be shown by Lemma 3-1 43 where we find that members of fr!” satisfy (1-2). R. Barrar and H. Loeb [3] note that exponential and rational families satisfy this Haar subspace condition (or (3-1)). The next two lemmas are found in [3]. They apply for A!" . We include their proofs for completeness. .Lemma 3-1: Let A, = (a]*,...,a *) and m(A*) = q. n Further, let x]....,xq be distinct points in X such that F(A,“ X1.) = C1. 1:1,2,...,q Then for sufficiently small 6 > 0, there exists a 6(6) > 0 such that the equations (1) F(A,xi) =31. i=1,2,...,q where lci - Gil S 6 have a solution A = (a],...,an) e P where a, = a1* for n 2 i 2 q + l and (2) ”A - 4*" S G Proof: Let f1.(a],...,aq , C1,...,Cq) a 'k * F(a],...,aq, aq+],...,angxi) ‘ Ci a F(A*,x) for i = l,2,...,q. Since the i = l,2,...,q a a. 1 form a Haar subspace, for 6(6) > O sufficiently small (or |c - Gil S 6), we may apply the implicit function i theorem to the f, system of equations in order to solve 44 for a],...,aq (where Ci = 3i for i = l,2,...,q). The implicit function theorem can be invoked since the Haar subspace implies that the Jacobian of the trans- formation is non-zero. Therefore there exist i = 1,...,q which solve the system * ,Xi) -30 =0 T=I,...,q * F(a]....,a ,aq+],...,an 1 q * where la. - ai Is 6 for i = l,2,...,q. Since P is open, 1 6 and consequently 6 can both be taken small enough to ensure that A = (a],...,an) c P. This proves the lemma.‘ For the next lemma we should recall the definition of a nOrmal point of,gy *. F(A*,x) 5.5? * is called a normal point if m(A*) = N = maximal value of m(A) for A c P. Lemma 3-2: For Lemma 3-1, F(A*,x) being a normal point implies that there is a unique F(A,x) satisfying (1) of Lemma 3-1. Egggf: Since for each A c P, m(A) S N = m(A*), it follows that if both F(A,x) and F(A],x) satisfy (1), they agree at N points. (l-3)‘ then implies that F(A,x) a F(A],x). ‘ We are now ready to state and prove the lemma which will assure that a constant error curve for the best approximation from55(* can not occur. It is Lemma 3 45 of [3], modified for our setting. Before we state and prove the lemma, however, we should note that Lemma 3—1 implies that the members of'a8’* are locally solvent (or satisfy (1-2) ). Therefore the results for 5?’ apply for j?’* (since A?’* is a varisolvent family V of Chapter 1). Lemma 3-3: If F(A*,x) e ,Zr; is a best approximation to f on X from j§’* such that (x+1 u x+2) n (x_1 u x_2) = 6 , then a constant error curve cannot exist. Proof: Without loss of generality, assume F(A*,x) - f(x) 5 c > 0 on X. Now a Haar subspace always has a strictly negative function (see [9]). Therefore 2 there exist scalars fail] such that 2 2') a. a F(A1;x) i l 1 a a < 0‘ i Let A = (a],a2,...,a£,0,0,...) 5 En' Then 1 O > E a. a F(A*,x) i=1 ‘ ** = a a, f: a a F(A”: + my) I ,__ d F(AL+ tA,x) l ._ 1 * 1-1 6(ai + t ai) t=O dt t=O for all x c [a,b]. By the mean value theorem, for 46 sufficiently small t, we have that F(A, + t A, x) < F(A*,x) for all x 5 [a,b]. Now F(A*,x) is continuous with respect to A, and P is open. Therefore there exists a t > t > 0 such that l ”F(A, + t A,x) - f(X)H < ”F(A*.X) - f(X)H. Because (X+1 U X+2) n (X_1 U X_2) = 6 , t can also be chosen so small that F(A* + t A,x) c fi4¥z since f(x) < F(A* + t A,x) < F(A*,x) for t sufficiently small. I Combining Lemma 3-3 and Theorem l-2, we have Theorem 3-1: Let F(A,x) e j?’* satisfy (x+1 U x+2) n (x_1 U x_2) = 6 . Then F(A,x) is a best approximation to f on X from,§#,* if and only if F(A,x) - f(x) alternates at least m(A) times on X. Uniqueness of best approximating functions from 55’* for (x u x+2) n (x_] u x_2) = 6 +1 follows from Lemma 3-3 and the fact that members of j§{* are members ofjgf. 47 We will now reproduce Theorem 2 of [3] in our setting since it will be necessary in the proofs of the next section. 5f, Theorem 3-2: Let F(A*,x) 8 2T * be a best approxi- mation to f on X where (X+1 U X+2) n (X_] U X_2) = 6 and F(A*,x) is normal. Let the sequence {F(Asooi. F(As.x) e A“ be such that lim ”F(As.x) - f(x)” = ”F(A..x) - f(x)” S+w and assume (3-2) for 6 > 0, there exists an M(€) > 0 such that for all s > M and all x c X, £(x) - 6 S F(As,x) S u(x) + 6 then we can find a sequence {Ap'l C P such that lim ”A, - A 'H = o, p... P where the sequence {Ap'} is equivalent to a subsequence of the {AS}, and the last n - N (recall that m(A*) = N) components of each Ap' agree with the corresponding components of A* . O on X. Proof: Without loss of generality, assume f(x) Let {xiz a S x1 < x2 < ... < XN+1’ xi 5 X} be a 48 critical point set for F(A*,x). Let {F(Ap,x)} be a subsequence of (F(As,x)} which converges at these critical points. Call the limits at these N + 1 points F(xj). Now by (3-2) and the assumption that ”F(As,x)H tends to ”F(A*.x)N. (3-3) max (2(x3).- "F(A,.X)H) S F(xj) 5 min (U(Xj)s "F(A*9X)N)g We wish to show that F(xj) = F(A*,xj) for j = l,2,...,N + 1. If this is true, Lemma 3-1 and Lemma 3-2 imply that a sequence {Ap'} equivalent to a subsequence of [AS] can be found such that lim "A* - Ap'H = o p+m and the last n - N components of each Ap' agree with the corresponding components of A*, proving the theorem. Assume for concreteness that x] e X_1 U X_2 . Also assume that F(xj) r F(A*,xj) for some xj. There- fore let F(xN+1) # F(A*,xN+]) (the method will apply for any other xj). Let c = lF(A,,xN+1) - F(xN+])| > 0. By means of a construction we will apply the unrestricted form of Lemma l-l and arrive at a contradiction. Let a = 1S?;N+l [|F(A*,xj) - F(xj)|: F(A*,xj) # F(xj)}. We will construct-a function F(A,x) belonging to 137* 49 from our local solvency property in the following way. For 0 < E < c/2, there exists a 6 sufficiently small where O < 6 < min (a,€) and an A c P such that (by Lemma 3-1), (3-4) F(A*,xj) + (-1)j+‘(6/2) where F(A*,xj) # F(xj) (a) F(A,xj) = F(A*,xj) + (-l)j(6/2) where F(A*,Xj) = F(Xj) for j = l,2,...,N and (b) ”F(A.X) - F(A..x)H S 6 < c/2 We will consider only p so large that (3-5) [F(Ap,xj) - F(xj)| S 6/4 j = l,2,...,N + 1 Now we have constructed F(A,x) so that F(A,x) - F(Ap,x) changes sign at the xj , j = l,2,...,N. i.e., (3’6) 59" (F(Asxj) ' F(Ap’xj) ) = - sgn (F(A,xj_]) -F(Ap,xj_]) ) for j = 2,...,N 50 To see this, for concreteness assume that j = k and xk e X+1 U X+2 where F(xk) = F(A*,xk). Then by our construction F(xk) = F(A*.xk) = F(A.xj) + 6/2 Since IF(Xk) ' F(Ap’xk)l S 6/4 we have F(Ap’xk) > F(A,xk) There ex1st two cases for xk+1 Case 1: F(xk+]) = F(A*,xk+]) . By (3-4) (a) we have F(A,xk+]) = F(A*,xk+]) + 6/2 > F(xk+]) + 6/4 or F(Ap’xk+l) < F(Asxk+1) Case 2: F(xk+]) # F(A*,xk+]) . By (3-4) (a) we have F(A*,xk+]) = F(A,xk+1) + 6/2 . But F(A*,xk+]) - F(xk+]) 2 a implies that F(A*,xk+]) e a + F(xk+]) 2 a + F(xk+]). Then F(xk+]) + 6/2 S F(Asxk+]) or F(Ap,xk+]) < F(A,xk+]) 51 Therefore sgn (F(Asxk) -F(Ap9xk) ) = T sgn (F(Asxk+]) ‘F(Apsxk+]) ) when xk c X U X+2 and F(xk) = F(A*,xk). +1 If xk c X_1 U X_2 or F(xk) # F(A*,xk), the argument is similar. Now (3-6) holds for j = 2,...,N. But it also holds for j = N + l by (3-3), (3-4)(b) and (3-5). But F(A,x) and F(Ap,x) both belong to .fl* and by the unrestricted form of Lemma l-l, F(A,X) ' F(Apsx) ¥ 0! (F(Ast+]) % F(Apst+]))s can have at most N - l zeros on [a,b], since N = max m(A) for A e P. (3-6) contradicts this and therefore F(A*,x = F(x n+1) n+1) This is the desired result. Section 3-2: Strong Unicity and Continuity of the Best Approximation Operator Theorems. In this section we will extend Theorems 3 and 4 of [3] to our setting Theorem 3-3: If F(A*,x) is a best approximation to f on X from .T *, (x+1 u x+2) n (x_] u x_2) = 6 , and F(A*,x) is normal, then there exists an a > O 52 a”? such that for each F(A,x) e,'f * , ”f(X) - F(A.X)H 2 ”f(X) - F(A*.X)H+ a” F(A.X) - F(A*.X)H fl Proof: Assume f ¢‘/7 * . If Theorem 3-3 is false, a 6%? sequence {F(An,x)} C,z‘ * and a sequence {an} , an > O can be found such that lim an = 0 so that new F(An,x) s: F(A*,x) and such that (3-7) ”f(X) - F(An.X)H = “f(X) - F(A*’X)N + an ”F(An.x) - F(A..X)H We claim {HF(An,x)”} is bounded. Consider (3-8) ”F(A..X) - F(An.X)H - ”f(X) - F(A..X)H S ”f(X) - F(A.,X)N + an ”F(A*,X) - F(An.x)H which is true by ”F(A..x) - F(An.X)H - ”f(X) - F(A*,X)H S ”F(A.,X) - F(An.X)) - (f(x) - F(A.,X))H = ”f(x) - F(An.x)H and (3-7). Now ”F(A*sx) - F(Ansx)” f 0 since F(An,x) S F(A*,x). Divide both sides of (3-8) by ”F(A*,x) — F(An,x)H. We have 53 l - ”f(X) - F(A..x)H ”f(X) - F(A,.x)H ”F(A*,X) " F(AnaxH' “F(A*,X) ' F(Ansx)N I'l If (”F(An,x)Hl is not bounded, then 1 s an as n +w , since ”f(x) - F(A*,x)H is bounded, which is a contradiction. Therefore assume (”F(An,x)N} is bounded. Then by (3-7), (3-9) iim ”f(x) - F(An.x)H = ”f(x) - F(A,.x)H. n+oo Apply Theorem 3-2 to (3-9), i.e., there exists a sequence {Bk} c P converging to A* where the sequence is equivalent to a subsequence of {An} and the last n - N components of each Bk agree with the corresponding components of A*. We should note two things. First F(Bk,x) c 9’* since F(An,x) c ff’h Second (3-7) remains valid for {Bk} Lat -1 if x e x_ u x 1 -2 C(X) = 1 if x e x+1 u x+2 By the assumption that (X+1 U X+2) O (X_1 U X_2) = ¢ 9 o is well defined. Now, we wish to show that (3-10) ak ”F(Bk,x) - F(A*,x)H 2 max C(X) (F(A*,x))- F(Bk,x) chA 'k 54 We have that (3-7) is valid for {Bk} . Therefore ak ”F(Bksx) -F(A*,X)” = ”F(X) 'F(Bksx)” ' ”f(X) 'F(-A*9X)” Now ak ”F(Bk,x) - F(A*,x)H > 0 for each k since F(A*,x) is the best approximation to f from 5#'* and F(Bk,x) 6 Case 1: Case 2: Case 3: r9 . tT *. We now con51der cases Let x] c XA* such that F(A*,x]) = 2(x1) (i.e., 0(x1) = l) . Then since ,4 F(Bkax) 5 ’r *9 F(Bksx1) ‘ F(A*,X]) 2 0 or 0(x1) (F(A*,x]) - F(Bk,x])) S 0 Let x c X such that F(A*,x]) = u(x1) l A,r (l-e-s G(x1) = - l) . Then since F(Bkox) 6)! * 9 F(Bk’xT) ' F(A*,X-l) S 0 or o(x1)(F(A*.x]) - F(Bk.x]) S 0 Let x1 9 XA* such that f(xT) -F(A*,X-l) = : ”f(X) -F(A*,X)” ° Then ak ”F(Bksx) ' F(A*9X)N = ”f(x) -F(Bk.x)u - ”f(x) -F(A..x>u G(x1) (f(x'l) 'F(Bksx])) "' ”f(X) “F(A*,X)” IV 0(X17 (f(X1) ‘F(Bk,X]) ‘ G(X])(f(x]) "F(A*9X])) G(X1) (F(A*,X]) 'F(Bk,X])) 55 We therefore have that for all x1 6 XA , * ak ”F(Bk,X) ' F(A*,X)” 2 am) (F(A..x,) - F(Bk.x])) Therefore, ak “F(Bksx) ' F(A*,X)” 2 max C(X) (F(A*,x) - F(Bk,x)) xeXA * or (3-10) is shown to be true. We now wish to show that there exists a y > 0 such that for all k, (3-11) max {C(X) (F(A*,x) - F(Bk.X))} 2 YHBk- A*N chA * Assume (3-11) is false. Then there exists a sequence of positive {Yk} such that Yk tends to O and a subsequence of {Bk} such that (3-12) max 0(x) (F(A*,X) - F(Bk.x)) chA* “A* _ Bk” S yk By the mean value theorem for large k, 3F(Ak(x),x) (ai*- bki) 1 3G1 NA* ' B iitmz (3-13) max 0(x) xeXA i * ku where Bk = (bkl""’bkn) , N = m(A*) and Ak(x) c P is on the line between Bk and A, (k large enough and P 56 A*-B open says that Ak(x) e P). Set ck = k ”A1: - Bk” Since ”ck” = l for all k, and we are on a compact set, we have a convergent subsequence (which we will not relabel) where this subsequence c converges to k c = (c1...,cn) and ”c” = 1. Using this subsequence in (3-13) and taking limits, we have N (3-14) max C(X) 2 Ci §£153451- S O chA* i=1 aai N aFA x Now 2 Ci is a non-zero function because i=1 3ai of linear independence and the fact that ”c“ = 1 By (3-14) and Theorem 3-1, 8F(A*,x) Bai has at least N zeros which contradicts our Haar sub- space hypothesis. Therefore (3-11) is proven. When we combine (3-10) and(3-ll), we have that (3'15) (1k ”F(Bksx) " F(A*,X)” 2 Y ”Bk ' A~k” But since Bk + A* , by the mean value theorem, there is a D > 0 such that for sufficiently large k, (3'16) ”F(Bksx) ' F(A*,X)” S D "Bk " Ag.” 57 From (3-15) and (3-16) we have that I, ak 2 D > 0 which says that ak + 0 is impossible. This contradicts our original assumption that the theorem was false. ' It is obvious that F(A,x) must belong to ,9f* and not just to A9* in TheOrem 3-3 as the following example shows. Example 3-1: Let X = [-l,l], f(x) = x2, u(x) = x2 + l/4. £(x) = - l and {F(A,x): F(A,x) = ax + b} = 65y* . Then F(A*,x) E 1/4 is the best approximation to f(x) on X fromthS, but for F(A],x) 2 1/2 6 ,JV * , we have that ”f(X) ‘F(A]sx)ll i N f(X) -F(A*,X)” + Y ”F(A*9X) 'F(A]9X)N for y > 0 since 1/2 2 3/4 + v . l/4 for y > O. The last theorem of this chapter is a Continuity of the best approximation operator theorem. For A9'* , the theorem is Theorem 4 of [3]. The proof is based on the strong uniqueness theorem and compactness. Their proof applies in our setting and we have the following theorem. Theorem 3-4: Let F(A*,x) be a best approximation to f 58 on X from J?’* such that F(A*,x) is normal and (x+1 u x+2) n (x_1 u x_2) = 6 . Then, (1) There exists a y > 0 such that ”f(x) - g(x)H< y implies g has a best approximation on X from’fifg , say F(Ag,x) (2) There exists a A > 0 such that for all g(x) of (1) above which have a best approximation F(Agsx)s ”F(A*,X) ' F(Ag,X)" S l ”9(X) - f(X)” Theorem 3-4 is a local continuity theorem. (1) of Theorem 3-4 says that 9 must be sufficiently close to f to apply the theorem. As an example of a global con- tinuity theorem, see C. Dunham [7]. C. Dunham's theorem, however, assumes that pointwise convergence is uniform convergence (see the paragraphs following Corollary 3 of Theorem 1-2) CHAPTER 4 GENERALIZED WEIGHT FUNCTION APPROXIMATION WITH PROPERTY A AND RESTRICTION BETWEEN THE CURVES u AND 2 WHERE u > 2 Section 4-1: Introduction and Definitions D. Moursund [14], D. Moursund and G. D. Taylor [15], G. D. Taylor [22], I. Ninomiya [17], L. Wuytack [27], and C. Dunham [8] among others looked at a general- ized weight function for an approximating family of functions. G. D. Taylor [22] examined a generalized weight function for polynomials in our restricted setting, while C. Dunham did likewise except that his approximating family was non-linear in a non-restricted setting. In this chapter we will combine these last two papers and look at a generalized weight function for a class of non-linear functions bounded between the curves u and 2 . Our generalized weight function will not be the least restrictive generalized weight function (see [14]). A function W(x,y) mapping W X R into R will be called a generalized weight function if (4-1) (i) sgn W(x,y) = sgn y (ii) W is continuous 59 60 (iii) for each x e X, W is a strictly monotone increasing function of y with lim|W(X.y)| = °° lyl-mo For our family’9!(of Chapter 1), we say that F(A*,x) 59’ is a best generalized approximation to f with respect to W and 9’if sup |W[x,f(x) - F(A*,x)]| S xeX . w s ' s F(AT:)efl’()S(:X l [x f(X) F(A X)]l ) Our first observation is that if W(x,y) = y, our problem is the problem of Chapter 1. Even if W(x,y) # y we will show that a form of Chapter 1 applies for our generalized approximation with respect to W and Er. We now require a theorem of D. Moursund and G. D. Taylor [15]. Theorem 4-1: If W is continuous and F(A,x) is vari- solvent (of degree m(A)), then W is varisolvent (of degree m(A)). We also require a slightly altered definition of alternation. (see [22]). Definition: For F(A,X)!SJ??, we define sgn*(f(x)-F(A,x)) by u(x) £(x) -1 if F(A,x) sgn* (f(xi) -F(A,xi)) = N 1 if F(A,x) sgn(f(x) -F(A,x))otherwise 61 Definition: For F(A,x) c j?’, the error curve W [x.f(X) - F(A,x)] is said to alternate n + 1 times on X if there exist n + 2 points x1 < x2 < ... < xn+2 in X such that sgn* (f(xl) T F(Asxi)) T T sgn* (f(XT+T) T F(Asxi+])) s i = 1,2, ... , n + l and at least one of the following conditions is satisfied by each xi: (1) INEXT’f(XI) T F(Aaxi)]l T max |W[x,f(x) - F(A,x)]l xeX (ii) F(A.x,.) m1.) (iii) F(A,xi) u(xi) As before,the xi are called critical points. An xi is a positive critical point if sgn* (f(xi) - F(A,xil) II _I and a negative critical point if sgn* (f(xi) T F(A,X1)) -1 Using Theorem 4-1 and our revised definition of alternation we could develop the results of Chapter 1. Rather than this however, we will examine a perhaps more general problem. J. R. Rice has shown (page 18 of [20]) that the weakest hypothesis with property Z and continuity which 62 will ensure a Tchebyshev type of characterization theorem is property A (defined below). The difference between the above conditions and varisolvency is essentially the solvency condition (not considering the constant error curve possibility). On page 22 of [20], Rice gives two examples of families satisfying property A which are not (varisolvent. It has not been shown yet whether varisol- vency implies property A because of the constant error curve difficulty. We will consider generalized weight function approxi- mation using a family in our restricted setting, where the family has property A in place of local solvency. First, a theorem comparable to Theorem 4-1 will be given for a family with property A (instead of local solvency). We will then examine the questions of existence, unique- ness and characterization for our (property A) family with respect to W. Let (g : be the family of functions {F*(A.x)= P x [a,b] + R} where each F*(A,x) 6,19 : satisfies (l-l), property 2 (or (1-3))s and property A. Our approximating family 55’* “g * . . . 1 will be the subset of 1 satisfying (1-4) (i.e., * members of AV 1 lying between the curves u and t ). As for our familylgf, F*(A,x) e §FP:' is a best gen- * eralized approximation to f with respect to W and jg’] if 63 (4-2) sup IW[x.f(X) -F*(a.X)]| S xeX inf (sup |W[x.f(X) -F*(B.x)]|) = e F*(B,x)cfiF: XEX We will use our new definition of alternation for ?‘1 . Thus there remains only the definition of property A. Definition: F*(A,x) has property A of degree m(A) if for any integer m < m(A), any sequence {x1,...,xm} with a = x0 < x1 < --- < xm+1 = b any sign 0 , and any real 6 with O < E < min {xj+1 - xj: j = O,...,m}, * there exists an F*(B,x) 6 ’5] such that HF*(A.x) -F*(B.X)H < 6 sgn (F*(Asx) -F*(B.X)) = 0 for a S x < x1 - E c (-l)J for xi + 6 < x < xj+1 - E o(-l)m forxm+€ O and F*(B,x) c ,éf : such that ”F*(A.x) -F*(B.x)H = ”[f(x) -F*(A.X)] - [f(X) -F*(B,x)] H < 61 where sgn (F*(A,x) -F*(B,x))= a for a S x < x1 - 6] o (-l)J for xi + 61 < x < x3],1 - E] a (-1)m for xm + €1< x S b Now E < 6 can be chosen sufficiently small so that 1 by the continuity of W, “W TX.f(X) -F*(A.X)] - NEX.f(X) - F*(B.X)]H <'e But since sgn W(x,y) = sgn y, sgn (“[X.f(X) -F*(A.X)l - N [X.f(X) -F*(B.X)]) = o for a S x < x1 - E - j - c ( l) for xi + E < x < xj+1 6 o (-l)m for xm + E < x g b 65 Therefore W has property A of degree m(A). If W had property A of degree m(A) + l, we would have a contradiction of property 2. This concludes the proof.‘ We now consider the existence question. Our discussion will parallel that of Chapter 1. We first note that existence of best approximations is not assured under ordinary approximation, and therefore will not be assured under generalized approximation. An example of the above non-existence of a best approximation is as follows. Example: Let P = [ala is rational] AV; = [F*(A,x) = a}, and f be defined by f(x) = J2 . Then a best approxi- * mation to f from ,5] does not exist. (see page 22 of [20]). Therefore, as in Chapter 1, in this section we will . r'*. . add the hypothe51s that /x 1 15 closed under p01nt- wise convergence. This will allow us to obtain existence of a best approximation. For the e defined in (4-2), let (4-3) lim sup |W[x,f(x) -F*(Ai,x)]| = e i+w xeX Our existence question will be answered in the affir- * mative if we can find an F*(Ar,x) E: d] such that sup! th.f(x) - F*(Ar.x)] i = e xeX 66 Now, since m(A) is bounded for all A, we can find a subsequence of {F*(Ai,x)} of fixed degree such that (4-3) holds (we will not relabel). Applying Theorem 7.2 of Rice [20] to our subsequence {F*(Ai,x)} satisfying (4-3) gives us a pointwise convergent sub- sequence satisfying (4-3). But our closure assumption * a on )T 1 then gives us that there eXists an which is the pointwise limit of the . r“ F (AS’X) EX] pointwise convergent subsequence satsifying (4.3). We must therefore have that sup |W[x,f(x) -F*(As,x)]| = xeX max|W[x,f(x) -F*(As,x)]| = e < w xeX since W is continuous on a compact set. Therefore F*(As.x) is a best approximation to f with respect to (’1' W and gt 1. We have just proved the following theorem. Theorem 4-3: For a generalized weight function W, there exists a best generalized approximation to f on . ,1'* X with respect to W andJ 1. Section 4-3: Uniqueness and Characterization of Best Approximations. Our first observation is that Examplel-land Lemmal-2 apply in our setting so that we need consider only the 67 = 6 . For the case where (X U X+2) n (X._1 U X +1 -2) remainder of this section we therefore assume that (x+1 u x+2) n (x_1 u x_2) = 6. Uniqueness will follow directly from the character- ization theorem in the usual manner (see the proof of Theorem 1-4 ). For the characterization theorem we require a lemma which C. Dunham [8] gives without proof for the unrestricted setting. For completeness we will give a proof of the lemma in our setting. Lemma 4-1: For F*(A,x) e :5! : , F*(A,x) -F*(B,x) can have at most m(A) - l zeros, counting double zeros twice, for F*(A,x) S F*(B,x) e j?! : Proof: The proof will consist of constructing an F*(C,x) 2: fl 1; such that F*(C,x) -F*(A,x) can have as many distinct zeros as F*(B,x) -F*(A,x) has zeros, counting double zeros twice. If xj is a double zero of F*(B,x) -F*(A,x) such that F*(B,x) > F*(A,x) in a small neighborhood of Xj (not including xj of course), F*(C,x) will be constructed close enough to F*(B,x) where F*(B,x) > F*(C,x) in the neighborhood of xj. This will ensure two distinct zeros of F*(C,x) -F*(A,x) 68 associated only with xj i.e., FT( '3) X) FTCCJX) Flex For F*(B,x) < F*(A,x) in a neighborhood of a double zero, F*(C,x) will be above F*(B,x) in a manner analogous to the above. Each simple zero of F*(B,x) -F*(A,x) will also have a zero of F*(C,x) -F*(A,x) associated only with it. We now proceed with the construction. Let {xi}? be the set of zeros of F*(A,x) -F*(B,x). Let p = min max |F*(B,x) -F*(A,x)| > O 0S1Sk xe[xi,xi+] xi 0. Select an OSiSk 1+1 1 xi<"i 1 appropriate 6 > 0 such that a/2 > 6 > O and 69 0/4 > E > 0. We then construct a set of points 9. {Xi'Iiz1 ,2 S k in the following way. Case 1: Let x1 be a double zero (a = x0 = x1 or xk = xk+1 = b implies a simple zero). Select an x]' such that x0 + E < x]' < x1 - 6 (i.e., x]' is between x0 and x1 and an 6 distance away from both). Proceed to the first simple zero after x](if it exists) and select xz' as follows, i.e. if x5 is the first simple zero after x],x2' is selected such that ' - Xs-l + 6 < x2 < xS 6 Proceed to the next zero, xs+]. Select x3' where xS + 6 < x ' < x + E. If x is a simple zero, 3 5+1 s+l proceed to x5+2 and do the same for x5+2 as for xs+1. If x is a double zero, proceed to the next simple s+1 zero after xs+1 as in selecting xz'. Continue through all the zeros of F*(A,x) -F*(B,x). Case 2: Let x1 be a simple zero where x1 = x0. Proceed to x2. If x2 is a double zero, proceed as in Case 1. If x is a simple zero, proceed as if for x] 2 in Case 3 below. Case 3: Let x1 be a simple zero where x1 # x0. Let x1 be such that x0 + E < x1 < x1 - 6. Let xz' be 70 such that x1 + E < x2' < x2 - E (naturally only if x2 exists). If x2 is simple proceed and repeat for x3. If x2 is double, proceed as if x2 is the double zero x in Case 1 above. s+l Letxd be the first double zero. Suppose, for concreteness, that near xj, F*(B,x) > F*(A,x). Select a in property A below so that sgn (F*(B,xd) -F*(C.xd)) = + 1 (If no double zero exists, property 2 proves the lemma). Now, for the points {xi'}.f 1 1 and our 6 > 0, there exists (by property A) an F*(C,x) such that sgn (F*(B,x) -F*(C,x)) = a for a S x < x]' - E I _ _ J - _ - 0 ( l) for xj + 6 < x < xj+1 6 o(-l)£forx£+€ 0 (ii) 2(x1) + Si < F*(A,xi) < u(xi) - 51, where s > 51 > 0 Now by continuity, there exists an 61 > 0 such that on [x. — G. 1 1, x1 + 61], h(x) + 51/2 < F*(A,x) < u(x) - 51/2 and |W[x,f(x) - F*(A,x)]l < p* - 51/2 for all x. + 6.]. If X n (y',y") = (I) ’ Tat xi = D' +yn) 2 Then there exists an 61 > 0, since y' < y" such that X O [x1 - 61, x1 + 61] = 6. Construct x1 and 61 in a similar manner for each i = l,2,...,k, letting x0 = inf Ty: y c X} and xk+1 = sup Iy: y e X} Let 6k = min (61} > O and sk = min {51/2} > 0. Then TSiSk lSiSk k for x c U (X O [x. - 6k, x. + 6k]) , we have P i=1 1 I that |W[xp, f(xp) = F*(A,xp)]l < p* - sk , k and £(xD) + s < F*(A,xp) < u(xp) - sk. Now,by k construction {X O [x1,x1+1]}i=0 15 a set of k + 1 intervals in X each of which contains no alternations 74 and such that W[x,f(x) -F*(A,x)] alternates exactly once on any two intervals. Each X O [x1,x1+1] , i = O,...,k contains a critical point, namely y1+1 and no critical point of an opposite sign. Without loss of generality assume y1 is a positive critical point. Then there exists a 61 > 0 such that W[x,f(x) -F*(A,x)] >-p* + 61 and F*(A,x) < u(x) - 61 for all x c X O [xo,x1], since X is compact and all functions involved are continuous and u(x) > £(x). Now since W is a continuous and strictly monotone function of f(x) -F*(B,x) for F*(B,x) c :9}: , there exists an 0 < 61 < min (sk, 61/2) so small that if [F*(A,x) -F*(A1,x)| < 6 and f(x) -F*(A,x)> f(x) -F*(A1,x) for all x c X O [x0,x1], then (4-4) (W[x,f(X) -F*(A.X)] -W[x.f(X) -F*(A1.x)]l S sk/3 . W[x,f(x) -F*(A1,x)] > - p* + sk/3 , and F*(A19X) S u(x) - 61 for all x c X O [x0,x1]. Repeat this process, i.e., on X O [x1,x2], there exists a 62 > 0 such that F*(A,x) > £(x) + 62 and W[x,f(x) -F*(A,x)] < p* - 62 for all x c X O [x1,x2]. Then there exists an O<:€2 < min (sk, 62/2 ) so small 75 so that if |F*(A,x) -F*(A2,x)l < 62 and f(x) -F*(A,x) < f(x) -F*(A2,x) for all x c X O [x1,x2], we have (4-5) thx.f(x) -F*(A.x)]-WLx.f(x) -F*(A2.X)]I s sk/s , W[x,f(x) -F*(A2,x)] < p* - sk/3 , and F*(A2,x) 2 £(x) - 62 for all x c X O [x1,x2]. Continue with X O [x2,x3] etc. Let ER = min {61/2 , 62/2,..., ek/2} > 0. Let 6 = min (€1,6k) . Now apply property A to {x1,...,xk} , k < m(A) with e > o and 0 = - 1. Th f th ' t F*(A ) 579* h th t ere ore ere EXlS S an S,X E 1 SUC a ”F*(A,x) -F*(AS.X)H < G and sgn (F*(A,x) -F*(As,x))= -l on xo S x < x1 - E (-l)J+] on Xj + E < x < Xj+1 - E _ k+l - (-l) on xk + E < x S xk+1 By construction, property A and strict monotonicity give us that F*(As,x) is a better approximation on the intervals X O [x1 + E, x1+1 -6] i = 0,1, ..,k , and (4-4) and (4-5) along with our selected 6 > 0 give us 76 that (W[x,f(x) -F*(As,x)]| < p* and £(x) S F*(As,x) S u(x) on X O [x1 - €,x. + E] .1 for i = l,2,...,k. Therefore max [W[x,f(x) -F*(AS.X)]| xeX < max |W[x,f(x) - F*(A,x)]l = 0* XeX which is a contradiction. Therefore the best generalized approximation F*(A,x) must alternate at least m(A) times and the theorem is proven. Uniqueness follows directly from the first part of the proof of Theorem 4-4 where the first '<' is replaced by ISI Theorem 4-5: The best approximation of Theorem 4-4 is unique. 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