spline solutions to l1 extremal problems in one and several variables

7/30/2019 Spline Solutions to L1 Extremal Problems in One and Several Variables

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JOURNAL OF APPROXIMATION THEORY 13, 73-83 (1975)

Spline Solutions to L1 Extremal Problems inOne and Several Variables

S. D. FISHER AND J. W. JEROME*

DEDICATED TO PROFESSOR G. G. LORENTZ ON THEOCCASlON OF HIS SIXTY-FIFTH BIRTHDAY

1. JNTRODUCTION

Let a = x0 < s1 < . -< x,,, = h be a fixed partition of the closedinterval [a, 61 and let 0 be the closed flat in the Sobolcv space P,*(a, b),1 :g n :: 171I- 1, defined by the interpolation to specified values ri at thepoints xi , i = 0, I ,..., ITT. f

then the minimization problem (I) does not, in general, have an interpolatingsolution in the class Wn~l(a, b). More generally, if U is a closed flat in aBanach space A, and R is a continuous linear mapping of X onto Y withfinite-dimensional null space, then it is possible that inf{i/ Ru I( Y: ZIE Uj is notattained in U; for example, if Y is not reflexive. In his recent paper [6], Holmeshas discussed, with a both a literature survey and new results, the techniqueof embedding such optimization problems in dual spaces. By considering theextended problem of minimizing (1R**p iiy.* over the flat JU in X* *. whereJ denotes the natural injection of X into X**, Holmes has shown that asolution p exists in JU, achieving the same norm extremal value as in theoriginal problem under a natural poisedness hypothesis. Thus, the problemhas a solution in the sense of roffe and Tihomirov [7].In this paper we shall discuss concrete ways in which such problems can beextended to possess natural solutions. In particular, for the problem (l),we expand the class JP,l(a, b) to include functions whose nth derivatives

* Research sponsored by National Science Foundation Grant GP 321 I673

Copright 8, 1975 by Academic Press, Inc..\I1 rights nf reproduction in any form reserved


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L' EXTREMAL PROBLEMS 75

THEOREM I. Let L, ,..., L,,, be linearly independent lir~ear ltncriottals 017Hc~f the ,fortn

L,(h, P) = [ F,dh + I,(P),Xwhere F, is confinuous on X, i = O,..., ttz and (Ii]:& is a set of linear fLtnctionalsdt$ned art N suck that P L= 0 if P t N and I,(P) y 0 for each i = O,..., ttt. LetI,, . . . I, be compact itttercals in R, each possibly cotwisting of a single point,atid let

U = {(A, P) E H: L,(h, P) E Ii , i = 0, I . . . nz).

ac= inf{i! X ~1:A, P) E U). (1.1)Then U contains at least one pair (h, P),for lvhiclt ii h 11== a; the set S qf suchpairs is convex and compact in the kveak topology. The extreme poinlts qf S areall qf the,form (C, c,6, , P) where r .Y 111, jj is the unif point mass at the f, E X,forj == 0 . . r and Ci / c, / = z.

ProoJ Let {g,, = (A,, P,)f be a sequence in U with 11 ,, I L Y. SinceL,(g$,) t Zj for all v and since J-r F, dAv s also uniformly bounded for all v and i,we find thatI WV); c, i = O,..., n2 and v = 1. 2,... . (1.2)

Hence, by the completeness of {/,}c it follows from ( I .2) that the sequence P,,is bounded in norm. Thus, there is a subsequence of {P,). denoted {) + I,(P,,) as .j 4 GO for i 7: 0 . . . tn. LikewIse. themeasures {hYj} have a weak accumulation point A, with /I A,, ~, < o(. It is easyto check that (A,, P,,) E I/ and hence I( A,, I = a. Thus we have shown that Sis nonempty. The convexity of S follows from the convexity of U and thedefinition of 01.Now the convex set T of measures determined as the set offirst components of S is clearly weak* closed in IV(X) since it is bounded andcontains all its weak* sequential limit points (since X is compact metric,C(X) is separable [4, p. 276, Theorem 14-9.151 and hence the closed unit ballof IV(X) is metrizable in the weak- * topology [I, p. 4261). Now by theKrein-Milman theorem let (A, P) be any extreme point of S and suppose thereare tn -I- 2 disjoint Baire sets E,, ,..., E,,, 1 in X which have positive X-measure.Let Ai be the restriction of X to Ei for i O,..., m -t I and let 11~e the vectorin (m L- I)-space whose ,jth coordinate is Li(X,, P). ,i : O,..., 177 andi = O,..., tn + I. The vectors z10..., c,,$+,must be linearly dependent in R+l?Tt+,and hence there are scalars a,, ,.... a,,,: 1 , not all zero, with 1, a,vi == 0. Let


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76 FISHEK AND JEKOMEcL -_ -g- l a,h, so that p is not the zero measure. Then we have. for 0 x: Esufficiently small,

,,t 1 ,I& -I~ h tp ~ -- , x, X-,E C (I tui) A, 1 ! c 1 u, Ii A, 1.0 0

Now, if Cl; r u, ); hi ,I + 0, then some choice of E gives a pair (A ~+ CCL, ) inU with 11 + EP 1 < r, a contradiction. Hence. C:: a, j! A, I/ -~-:0 so thateach of (h + EP, P), (h --~EP, P) lies in S. The convex representation

then contradicts the choice of (A, P) as an extreme point of S. It follows thatthere are at most 111 - I disjoint subsets of A of positive h measure and thetheorem follows.

2. APPLICATIONS(I) Univariate Generalized @lines

Let I =: [a, b] be a closed interval in R and let L be a nonsingular lineardifferential operator of order n on I of the form

where a, E C(f),j = 0 . . n - I. If,ft W~(I) then the representation

holds, where P in the null space N, of L is defined by DjP(a) ==D,f(a),0 5; ,j < II -- 1 and where the function 8(.. [) E N[. is defined for eacht E [a, 61 by

and 0 is given by

Here W7fJ(I) is the real Sobolev class of,JE C+(I) such that D-!fis abso-lutely continuous and Dnf E Ll(I). In this application we shall have need of


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L EXTREMAL PROBLERlS 77the larger class QTz(J) r-~- Ji LJE M(Z): w h ere Lf is taken in the weak senseand M(I) is the space of real Baire measures on I. Equivalently,

f(x) = P(s) -1 .I: i&Y, 5) d/L((), w-9for p E M(f) and P t NL ifft Q2(Z). We remark that

Now let A :~=-A,, ,..., A,> be a linearly independent set of linear functionalson C-(Z) for which the Peano representations

hold for every h E A andfE fVl,r(I) with {A&., %)jnEna linearly independentset of continuous functions in 5 on I where P is given by (2.1). Since Lr(Z)is weak dense in M(Z), (2.3) holds over the larger class en(Z). Let I, ,..., I,,be compact intervals in H, which are possibly single points, and let

U = (.fE W*(Z): h;(f) E Zi , i = 0 ,...) m].THEOREM 2.1. Let 01~-=nf{(j I,fl;LIr,~:, f E U). Then there are distivict pointst, < ... < t, in I, r -< m, real mrmhers a,, ,..., a, and a fzinction P E .iv, suchthat the generalized spline fhction

s(x) = jj a,&x, tj) + P(s),i=Osatisfies X,s E Z; , i =- 0 . . m and x;.-. ; aj 1 Y, i.e., I Ls I;,,r[rl == 01.s sohesthe extended minimization problem ix = iI Ls Ii,tIrf~ -= inf{ll Lfl!.bf[,l:,f E 0:.Here 0 3 U consists qf allf e en(Z) satisfi-ing Ai,fE I, , i L O,..., m. Finally,

Ls = 1 a,@., tj), (2.4)where a(., t) represents the Dirac delta jknctional at t.

Proof: Wn-*(Z) is algebraically isomorphic to L1(I) @ NL under (2. I). NowL1(Z) 0 NL is weak* dense in M(Z) 0 NL so thatd = inf{l Lf I,,,r,I:,f E 0) = 01.

Indeed, if pLo= LfO is chosen so that 11 -LO1= Z, ,fn E 0, choose y, -+* p.such that !PVE L1(Z), v == 1, 2,... and II ul, lIL~rl] < II p. ixf[,l We show how640/13/I-6


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78 ~HER AND JEROMI:to adjust u/, slightly so as to obtain an element which, upon integration,is in U.

Thus, we consider the map K: L(I) + RI: given by

wheret;Z(Q = A,&., E), i = 0 ..., LIZ.K is clearly onto by the linear indepen-dence of the Fi and the quotient space L(Z)/ker K is algebraically andtopologically isomorphic to R(j:l. Thus, there exist sequences E, ---f 0 and{yy} C L1(Z) such that i/ yy 11 < E, and y, + PVE LU, v = I,...Since

n: < liT,$f 11 V im qy ~LI(II = lilz,jnf , Y, ,!LI(,) i Ii PO /M(I) == c-,it follows that (Y = &. The result now follows from Theorem 1.1 and therepresentation (2.2).

COROLLARY 2.2. Consider the extremal problem (1) of the introduction.Then there is a polynomial spline firnction of degree n - 1 in Crl-2[a, b] with atmost m $ 1 knots in (a, b) satisfying Var Dnpl.s : : a and s(xi) = r: ,i == O,..., m. Here n > 2.(II) Multivariate Generalized Spline Functions

Let SJ be a bounded domain in R, I 2, and let L be a linear differentialoperator of order II. Our fundamental assumption concerning L is that themappingL: wJ(Q) + L(Q),

is continuous and surjective. Here PVi~l(Q) is the Sobolev space of functionsfwith distribution derivatives Daf~ Ll(Q), 1a 1 < n, with norm

We shall further assume that there is a closed linear subspace F of W7L*1(Q)such that the restriction of L to F admits a unique inverse representation ofthe formf(x) = J, G(s, 0 -U(f) dt> f t K (2.6)

where we explicitly assume that G(x, .) t C(Q) for each x E Q.We have in mind, of course, the specific application where L is a uniformly


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L1 EXTREMAL PROBLEMS 79elliptic operator of even order IZ = 2k, in which case, if n > I, the abovehypotheses are satisfied for sufficiently smooth boundary 8Q, for the choiceF = W,2k*(Q) n IV. Then there are distinctpoints t, ,..., t, in 0, r :S m, and real numbers a,, ,..., a, such that the multi-variate generalized spline function

s(x) = i ajG(x, tj)j=O

satisjies his E Zi , i = 0 ,.+., m and xi==, 1aj 1 = 01, .e., 11 S !i,ll[n] == 01.s solvesan extended minimization problem as before and satisjes the relation (2.4).(III) A Mathematical Programming Application

Discrete spline functions were introduced by Mangasarian and Schumaker[lo] as minimizing a general forward difference operator L: Rz --+ Rz-n+lsubject to affine constraints in the Zp norm, 1 < p < co. Methods ofmathematical programming were employed in the existence theory to deducethe closure of certain sets. We shall show here that, in the special case p = 1,there is a solution s such that Ls has support confined to a subspace of Rz ofdimension m + 1 if there are m + 1 constraint functionals.Specifically, let L be of the form

(LX)j = i a,xv+j-l , ,j = I )...) .? - 11 f 1, (2.7)iI=1


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80 bISHER AND JEROMEwhere a, f 0. Then L maps Rz onto R/m?1 and, by the rank-nullity theorem,the null space N of L is of dimension n 1. The complement A4 of N in Rzis of dimension 1 - n -/- 1 and L maps M bijectively onto R1-71+m1.hus, ifA = {A, )...) h,,,) .S a linearly independent set of linear functionals on Rz ofthe form

his t u,,-\, , .~ E R, (2.X>I 1i-0 ,..> HZ,such that NC span,,:-i17,, Q,~ . . airl. and if I,, . . . I,,, are compactintervals in R and

U --- {x E R: h,.x F I, . i O,..., t?z!then we have the following consequence of Theorem 1. I.

THEOREM 2.4. Let (Y inf{xiI:-l (LX)j 1 I ~~~~.u, ,..., 92) t U). Thetzthere exisfs a vector s E I/ .satisfJVng Gil: I1 l(LS)j / : IY ad, tnoreorer, thesupport of Ls is conjined fo at most 171$- 1 componenfs.

Proof. Take X to be 1 - 11 : 1 distinct points with the discrete topology.

3. APPROXIMATIONTHEOREM 3. I. Lerf E Wdsl(Z), I a compacl hlterual in R, ,z.iflt : Difl tJlr,l I

and Djf (a) 0,for 0 :I ,j c t? ~- I. Let a partifiotl A: a _=m,, .: ... -:. s,,, bof I he spec$ed ntith tn _ : II - I s14cI1 thaf A cotztaim a fixed .set ( Y, ,..., Y,,]of n poitits. Let s be the spline,firncriotl, guaranteed hi> Corollar>~ 2.2, agreeingtllith ,f on A. Then there exists a positbe conslanf C, idependeut qf,f atId A,such fliat[I D(f .- .s), L,(,, C/tl+lmi, 0 _ j l7 I, for I r _ mxa. (3.1)Moreover. the order gitlen 11~3 3.1) is sharp for n 2, j 0, r z=1 am!I := [0, I]: For each suficiently mall E .I 0.

where s,,,(i,m) &i/m)+, i 0 ,..., tn; here s,,, is the splittc ittterpolunt ofCorollary 2.2, nz = 1, 2 ... . and ? = (1 ml- e)-.

LEMMA 3.2. Let f; A and s he ,giuetl as iu Theorem 3.1. There exists ctpositiae constanf C, independettt off and A, mch that


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st. EXTREMAL PROBLEMS 81Proof. Since,

it follows that 11DIL-lfJiLm[,l < 1. Since

it sufftces to show that 1 Dpls(a+)l is bounded independently of .f and A.Now s may be written

s(x) = P(x) -t 1 Uj(.Y - tp, (3.3)I- 0where xl ==,, aj j -< 1, P is a polynomial of degree n - 1 and a < f, .f(t) dtCLimplies that jl,fijc[n,bl -g (b - a)ln ! and. clearly, the function

g(x)= c a& - tJ-j=O


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L1 EXTREMAL PROBLEMS 83Now the functionf(x) = E.++ satisfiesfE FV*J(O, 1) with :\f lr~~l,,,l~ : 1 andf(0) --f(O) :z 0. Thus, choosing Y, = 0 and Y, -: 1 we see that ,f and Asatisfy the hypotheses of the first part of Theorem 3.1. This concludes theproof of the theorem.

Renmrk. The reader will observe that the order of approximation in (3. I),which is seento be bestpossible inagenericsensefor this upproximatio~? process,is of order one less than that achievable by optimal linear approximationprocesses (cf. the linear approximation process, defined for splines of degree11 - I, constructed in [IS] and valid for Wnxl functions). The explanationfor this appears to be that the approximation process defined by Theorem 3.1is actually intended for the larger class of functions whose nth derivatives.in the sense of measures, have variation (as measures) not exceeding one.

ACKNOWLEDGMENTThe authors acknowledge helpful comments from Prof. K. Schcrer.

REFERENCES1. N. DUNFORDAND J. T. SCHWARTZ, Linear Operators, Part 1. Wiley, Interscience,New York, 1958.2. S. D. FISHER AND J. W. JEROME,The existence, characterization and essential uniquenessof solutions of L extrcmal problems, Ttw7s. Amer. Mu/h. Sot. 187 (1974), 391-404.3. S. D. FISHERAND J. W. JTRO~IE, lliptic variational problems in L? and Lm, I&icrnrr .1.

Muth. 23 (I 974), 685.-698.4. A. GLEASON, Fundamentals of Abstract Analysis, Addison-Wesley, Rea~;ling, MA,1966.

5. M. GOLOMB, H,-extensions by HI,-splints, J. Approuimrrtioif Throry 5 (1972),238-275.

6. R. B. HOLMES,R-splines in Banach spaces. 1, J. Muth. Anal. AppI. 40 (19721, 574-593.7. A. IOFFEAND V. TIHOMIROV,Extension of variational problems, Trnrn. .Mosr~o~~Math.Sue. 18 (1968), 207-773.X. J. W. JEROME, Minimization problems and linear and nonlinear spline functions. 11.Convergence, SIAM J. Numw. Awl. 10 (1973). 826830.9. 0. L. MANGASAKIA~: hD L. L. SCWMAKER, Splines via optimal control, i/z Ap-proximations with Special Emphasis on Spline Functions (I. J. Schoenberg, Ed.),pp. lI9-156, Academic Press, New York, 1969.10. 0. L. MAhCAsARlAN A?.D L. L. SCHUMAKFR,Discrete sphnes via mathematical pro-gramming, SIAM J. Control 9 (1971), 174-183.1 I. M. H. SCHULTZAND R. S. VARGA, L-splines, humer. Math. 10 (1967), 345S369.12. I. W. SMITH, W,*-splines, Dissertation, Purdue University, Lafayette, IN, 1972.13. K. YOSIDA, Functional Analysis, Academic Press, New York, 1965.14. J. FAVARD, Sur lintcrpolation, J. Muth. Pure.? Appl. 19 (1940), X-306.15. J. JEROME, On uniform approximation by certain generalized spline functions,.I. Approximation Throry 7 (1973), 143-I 54.

￼spline solutions to l1 extremal problems in one and several variables

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spline solutions to l1 extremal problems in one and several variables