PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 70, Number 2, July 1978
ANOTHER EXTREMAL PROPERTY OF PERFECT SPLINES
T. N. T. GOODMAN AND S. L. LEE
Abstract. Let t = {<,}, /' = 1, 2,..., n + k, be a given sequence in [a, b], /0 e WkJa, b], A > H/o^lL, and F = {/ 6 Wkx[a, b]: f\t = /0|t and ll/(/c)lloo< A }• We show that F contains precisely two perfect splines g, h of degree k with | g(*>|= \h^\ = /4 and n interior nodes, and for all / e F, min(g(x), h(x)) < f(x) < max(g(x), ä(jc)); Vx E [a, b].
1. Introduction. Let t denote a sequence ¿z = tx < i2 < • • • < tn+k = b (n, k > 1) such that /, < fi+fcfor all / = 1, 2,..., n, and define Wkaa[a,b] = [f E Ck~x[a,b]:fk-X) absolutely continuous andf'eljfl.ô]}. (1.1) Take a function/0 E W^a, b], and set F(t,/0) = {/ E W* [fl, /3]:/|t =/0|t). (1.2) Recently De Boor [1] and Karlin [2] showed the existence of a perfect spline p E F(t,/0) of degree k with less than n interior knots, i.e. a function of the form fc-lk-\ f T-\ p(x) = 2 aix' + c xk + 2^(-iy(x-i)k+ (1.3) i = 0 1=1 for some real constants a0, ¿7,, . . . , ak_ „ c and for a < £, < ■ • • < £r_, < b (r < n), and that this perfect spline is a solution of the problem of minimising ll/^lloo over au / m F(t,/0). A special case of this result was earlier considered by Schoenberg [7], who attributed his main results to Glaeser [3] and Louboutin [5]. Now, let A be a positive number larger than H/o^U,»,and let F = F(t,UA) = {/ £ F(t,/0): ll/X < A). (1.4) Our main result is the following, which generalises some work of McClure [6]. Theorem 1. There are precisely two perfect splines g, h in F with |gw| = lA^I = A and less than (n + 1) interior nodes. For any f £ F min(g(x), h(x)) < f(x) < max(,j(;c), h(x)), Vx E[a, b}. (1.5)
Received by the editors April 12, 1977. AMS (MOS) subjectclassifications (1970). Primary 41A05, 41A15; Secondary 26A87. Key words and phrases. Perfect splines.
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Furthermore g (or h) has exactly n interior nodes a, < a2 < • • • < an which are the unique set of points such that Ma) - 2*(a,) + 2t(a2)-+ 2(- !)>(«„) + (-1)"+V(¿>)
= A '/aVV (or-^'/Vf) (1.6) /or ail t// w/'î/it|/' e V(t). Here F(t) denotes the class of all spline functions of degree k — I, with knots at t, which vanish outside [a, b\. The above result persists when the derivatives are replaced by a differential operator of the form J_ d_ 1 d_ _L EJ — I fa Wk(x)„/ /„\ dxA„ Wk_x(x)11/ /„\ ' dx^v Wx(x)H/ /„\ M' where W¡(x) E Ck[a, b] and W¡(x) > 0, / = 1, 2, . . . , k, on [a, b]. A function/ E rV^[a, b] is called a.perfect Lk-spline if \LJ\ = c, c constant. Let Bj(x), i = 1, 2, . . ., n be the 5-splines associated with the differential operator Lk, with nodes /„ ti+x, .. ., tj+k, and let V(Lk, t) be the space generated by B¡(x), i = 1, 2, ..., n. Then V(Lk, t) consists of functions S(x) which vanish outside [a, b] and such that (LkS)(x) = 0 whenever xít. Now, let A be a positive number larger than HL^/olL and let F(Lk, t,f0,A) = {/ E Wkx[a, b]:f\t =/0|t and yL*/^ < A). (1.8) Then the following result holds: Theorem 2. 77iere are precisely two perfect Lk-splines g, h in F(Lk, t,/0, A) with \Lkg\ = \Lkh\ = /I and less than n + 1 interior nodes. For any f E F(Lk,t,f0,A), min(g(x), h(x)) < f(x) < max(g(x), h(x)) VxE[a,b]. (1.9) Furthermore g (or h) has exactly n interior nodes a, < a2 < • • • < a„ which are the unique set of points such that *(a) - 2*(ttl) + 2tK«2)+ • • • + 2(-iy>fo) + (-1)"+SK6) = A-xf*LJ0V (or-^-'jfVo^) (LIO)
/or a//1// w/7A»//' E ^(L^, t). Our proof of Theorem 1 in §3 depends on an extremal problem in Banach space, stated in §2, combined with an argument of De Boor [1].
2. Auxiliary results. Our main result depends on Theorem 3 below, which is a straightforward deduction from the Hahn-Banach Extension Theorem. A proof and some further applications were given in [4]. Theorem 3. Suppose V is a closed subspace of a real normed space W and k E W - V. Suppose <#>0E W*, the continuous dual of W, and ||d>0||< A.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ANOTHER EXTREMALPROPERTY OF PERFECT SPLINES 131
Then for w E V, A\\k + w||-
Remark. Observe that (2.4) is equivalent <¡>(k+ w) = A\k + w\. (2.5) If the set of zeros of (* + w) has measure zero, then (2.5) is equivalent to >= A sgn(* + w). (2.6) We shall also require the following simple property of spline functions.
Lemma 5. If a,, a2, . . . , ar are distinct points in (a, b) and r < n, then 3 a nonzero
Proof. We fix * > 1 and prove by induction on n. If n = 1, any nonzero element of V(t) has no changes of sign and so the result is true. Suppose the result is true for n < N and take t with n = N and a„ a2, . . . , ar with r < N. If r < N — 1, we may apply the hypothesis to t2 < t3 < ■ ■ ■ < tN+k or tx < t2 < ■ ■ ■ < tN+k_x to obtain <í>E V(t) which changes sign precisely at a,, a2, . . ., ar. Suppose then that r = N — I. Let 7?„ B2, . . . , BN be the ß-splines for t. Choose a„ a2, .. ., aN, not all zero such that
¿7,t3, (a,) + ¿727i2(a,.) + • • ■ + aNBN(a) = 0, V/ = 1, 2, . . ., tV - 1, (2.7) and let
First suppose <#>is oscillating in [/,, tN+k\. Since ¿>has at most N - 1 zeros in (/,, tN+k), a,, a2, . . . , aN_x are simple zeros of tb and so <¡>changes sign precisely at these points. On the other hand if
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 132 T. N. T. GOODMAN AND S. L. LEE
then apply the inductive hypothesis to each of these segments to obtain the required
f(x)-f0(x) = \\k(x, t) + (0|dt - (\k(x, t) + supf-^rV^o +^oi^-fVi^o +^o^c)*]. <(>eI^(t) ( Ja Ja I (3.4) Since V(t) is finite dimensional, 3>,E F(t) for which the infimum in (3.3) is attained. By Corollary 4, 3y E Lx[a, b] with Hyll«,= ;4 and J*d>y= fbMk\ \f \\k(x, t) + <¡>x(t)}y(t)dt= A fb\K(x, t) + License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ANOTHER EXTREMAL PROPERTY OF PERFECT SPLINES 133 gx(x) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 134 T. N. T. GOODMAN AND S. L. LEE [%g= f \tik) v¿>e v(t). Ja Ja Then there is a perfect spline f in F with \f(k)\ = A and interior nodes at ax < a2 < • • • < a„ if and only if ±a\ /%-/%+••• +(-l)"fV] = fVoU), V^F(t), (3.9) I a Ja, Ja„ j J a from which (1.6) follows. □ Let B = inf(||/w||OD: / E F(t,/0)}. Theorem 1 can be applied for all A > B. Letting A converge to B we have the following result. Corollary 6. There are perfect splines g, h in F(t,f0) with \g(k)\ = |AW| = B and less than (n + 1) interior nodes such that (1.5) holds for any f E F(t,/0) with ||/W|L = B. In this case g and h may have less than n interior nodes. Indeed g may equal h, so that g is the only function in F(t, f0) with || g^H«, = B, and by [1], g has less than n interior nodes. As a special case of Theorem 1, take n = k, tx = t2= • • • = tn = —1, tn+\ = tn+2 = • • • = t2„ = I,/, = 0 and.4 > 0. Then F= {/ E W"x[-l, l]:/'>(-l) «/'>(1) = 0, r = 0, 1,. . . , n - l,and ¡f^W^ < A). (3.10) Theorem 1 tells us there is a perfect spline h E F with HA^H^, = A and n interior nodes, and ± h are the only such spline functions in F. Moreover for any/ E F, \f(x)\ <\h(x)\, VxE[-l, 1]. (3.11) The nodes of h are the unique set of points ax, a2, . . . , an for which *(-l) - 2tK«,) + • • • + 2(- lYV(a„) + (- 1)"+1^(1)= 0 (3.12) for all polynomials \p of degree n. It can be shown (for instance by using Lemma 1 of Schoenberg [7]) that (3.12) is satisfied if av = -cos(i^r/(n + 1)), v = 1, 2, ...,«, the zeros of Chebyshev polynomial of the second kind. Next consider the differential operator L = D(D2 + I) • • • (D2 + n2) and the sequence t defined by r, = t2 = • ■ • = t2n+x = 0, t2n+2 = t2n+3 -••.•- i4„+2 = 2tt. For A > 0, define F = F(L, t, 0, A) = {/ E WS,[0,2*]: /w(0) = /W(2tt) = 0, r = 0, 1,.... In, and ||Z/Hoo< ^}- The operator L is not of the form (1.7) on [0, 2tr\ and so we cannot apply Theorem 2 directly. However it can be put in the form (1.7) on any proper subinterval of [0, 2tt\. It follows that there is a perfect L-spline h E F with HLAH^= A and knots at vu/(n + 1), v ■» 1,2,..., 2« + 1. We can deduce that for any/ E F, |/(x)| <|A(jc)|, VxE[0,2tt]. (3.13) By the same argument as in §3, it can be shown that ± h are the only License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ANOTHER EXTREMAL PROPERTY OF PERFECT SPLINES 135 perfect L-splines in F with \Lh\ = A and less than 2« + 2 interior nodes. References 1. Carl De Boor, A remark on perfect splines, Bull. Amer. Math. Soc. 80 (1974),724-727. 2. S. Karlin, Some variational problems on certain Sobolev spaces and perfect splines, Bull. Amer. Math. Soc. 79(1973), 124-128. 3. G. Glaeser, Prolongement extremal de fonctions differentiables d'une variable, J. Approx- imation Theory 8 (1973),249-261. 4. T. N. T. Goodman and S. L. Lee, Some extremal problems involving perfect splines, Comment. Math, (to appear). 5. R. Louboutin, Sur une "bonne" partition de l'unité, Le Prolongateur de Whitney, G. Glaeser (ed.), Vol. II, 1967. 6. D. E. McClure, Perfect spline solutions of Lœ extremal problems by control methods, J. Approximation Theory 15 (1975), 226-242. 7. I. J. Schoenberg, The perfect B-splines and a time-optimal control problem, Israel J. Math. 10 (1971),261-275. Department of Mathematics, University of Science of Malaysia, Penang, Malaysia License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use