Splines. Kuwait University. P.O. Box 5969, Safat Kuwait. Abstract

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1 A Bound on the Approximation Order of Surfae Splines M. J. Johnson Dept. Mathematis & Computer Siene Kuwait University P.O. Box 5969, Safat Kuwait Abstrat The funtions m := jj m,d if d is odd, and m := jj m,d log jjif d is even, are known as surfae splines, and are ommonly used in the interpolation or approximation of smooth funtions. We show that if one's domain is the unit ball in R d, then the approximation order of the translates of m is at most m. This is in ontrast to the ase when the domain is all of R d where it is known that the approximation order is exatly m. 1 Introdution The area of Radial Basis Funtion Approximation is motivated by the pratial problem of approximating a smooth funtion f : R d! C d whih isknown only at nitely many sattered points R d. A standard tehnique for approximation is that one starts with a radially symmetri funtion C(R d ), and then seeks an interpolant off j from the spae spanned by the -translates of. In other words, one seeks s spanf(,) : g suh that s() =f() for all. Of ourse there is, in general, no guarantee that suh a funtion s exists, but onditions on whih ensure the existene of suh ans are known. However, for manyinteresting hoies of the funtion, the above setup needs to be modied slightly in order to guarantee the existene of the interpolant s. Denition 1 Let C(R d ) and m + := f0; 1; ;:::g. For nite pointsets R d,we denote by S(;;m) the olletion of all funtions s of the form s = (,)+p; 1

2 where p m,1 := fpolynomials of deg m, 1g and the oeients f g satisfy q() =0for all q m,1. Mihelli [8] has given general onditions on whih ensure that for all f, there exists a unique s S(;;m) suh that s j = f j. Examples of suitable are the surfae splines given by m := m,d if d is odd m,d log if d is even, m>d=: Here, : R d! R is given by (x) :=jxj, x R d. The surfae splines, m, arise naturally in the solution of a variational problem. Duhon [6] showed that of all funtions whih interpolate f j, the one whih minimizes a ertain semi-norm is the one whih belongs to S( m ;;m). The results of the present work apply primarily to this example. In order to raise the issue of approximation, let us suppose that we desire to approximate f on an open bounded domain R d. One hopes that the interpolant s approximates f better and better as the pointset beomes `dense' in. To make this notion preise, let us measure the density of in by := (; ) := sup x minjx, j: In other words, is the smallest value for whih +B, where B := fx R d : jxj < 1g is the open unit ball in R d.the notion of `approximation order' is typially used to desribe the asymptoti rate at whih the error deays when f, the funtion to be approximated, is smooth. Let us say that interpolation from S(; ;m) provides approximation of order in if kf, sk L 1() = O( ) as! 0 for all suiently smooth funtions f, where s S(;;m) is the unique interpolant tof j. A basi problem is that of determining, for a given, the largest suh ; that is, determining the approximation order of interpolation from S(;;m). In the literature there are two rather distint approahes to this problem. One of the approahes starts by writing the interpolant s S(;;m) as the solution of a variational problem. The error an then be estimated by areful onsideration of the variational problem. In the ase of the surfae spline m, what is known is that if is a ompat subset of R d satisfying a uniform interior one ondition, then interpolation from S( m ;;m) provides approximation of order m, d= (f. [1], [11], [10]). That is kf, sk L 1() = O(m,d= )as := (; )! 0;

3 where s S( m ;;m)interpolates f at. The other approah started under the simplifying assumptions = R d and = h d. Of ourse these assumptions violate our initial setup, but it was hoped that the results and insights gained under these assumptions would shed light on the original problem. The reader is referred to the surveys [4],[1],[9] for desriptions of these works. For the sake of the present disussion, we mention only that for the surfae spline, m,itisknown that interpolation from S( m ; h d ; 0) 1 provides approximation of order m in = R d. Later, Buhmann, Dyn and Levin [] were able to dismiss the assumption = h d, while still retaining the assumption = R d (see also [5],[3]). By and large they obtained the same approximation orders as were obtained under the simplifying assumption = h d. It should be pointed our that their work was done not in the ontext of interpolation from S(;; 0), but rather in the ontext of approximation from S(;; 0). The dierene being that one doesn't insist that the hosen s S(;; 0) atually interpolate f j. With regard to the surfae spline, m, they showed that one an approximate from S( m ;; 0) with approximation of order m in = R d.from their results, one an derive the following onerning the ase when is bounded: If e is a ompat subset of (neessarily at a positive distane from the boundary of ), then as beomes dense in, one an approximate from S( m ;; 0) with approximation of order m in e. In other words, it is possible to hoose s S(m ;; 0) so that kf, sk L 1( e ) = O(m )as := (; )! 0 for all suiently smooth f. This of ourse leaves open the question as to how well f an be approximated near the boundary of. The purpose of the present work is to examine this boundary question primarily in the ontext of the surfae spline m. Sine we intend only to give an upper bound on the approximation order, we will onsider only the nie ase when the domain is taken as the unit ball B. We will show that the approximation order of interpolation from S( m ;;m) is at most m. Note that this lies stritly between the values m, d= and m mentioned above. Our bound on the approximation order applies not just to the error of interpolation from S( m ;;m), but in fat to the error of best approximation from S( m ;;m) (or more generally, from S( m ;;`), ` =0; 1;:::;m). The arguments also apply in ase one measures the error in the L p (B) norm. Here is the result: Theorem 1 For all 1 p 1and ` =0; 1;:::;m, there exists f C 1 (R d ) suh that E(f;S( m ;;`); L p (B)) 6= o( m+1=p ) as! 0: Here, E(f;V ; ) denotes the error of best approximation, measured in the norm, from the spae V to the funtion f: E(f;V ; ) := inf vv kf, vk. 1 S(;; 0) is often used in plae of S(;;m) when = R d or when one is not interpolating f. 3

4 Throughout this paper we use standard multi-index notation: The Laplaian operator is denoted by D 1 ; d d +. d + @x d For d +, we dene jj := d, while for x R d we dene jxj := p x 1 + x + + x d. The natural basis for Rd is denoted as fe 1 ;e ;:::;e d g.we employ the funtion r :(0::1)! R dened by r(t) :=t. For t R, the greatest integer t is denoted by bt. A redution of the problem Our plan for proving Theorem 1 is to exhibit (the existene of) a funtion f C 1 (R d ) and a family of pointsets f h g h(0::1=] suh that ( h ; B) =O(h) ash! 0, but E(f;S( m ; h ;`); L p (B)) 6= o(h m+1=p )ash! 0: (1) Let f h g h(0::1=] be any family of nite pointsets satisfying h (1, h)b and ( h ; B) = O(h) ash! 0: As hinted in the introdution, when examining an approximation to f from S( m ; h ;`), we will fous our attention not on the entire domain B, but rather on just the boundary layer Bn(1, h)b. So rather than (1), we will in fat be showing E(f;S( m ; h ;`); L p (Bn(1, h)b)) 6= o(h m+1=p )ash! 0: We an further redue our fous to just the interval [1, h::1] by employing the spherial averaging operator R : C(R d )! C([0::1)) given by Ru(t) := 1 u(tx) d(x), where := 1 d(x): Here, is the measure assoiated with spherial oordinates (see [7]). In words, Ru(t) is the average of u over the sphere of radius t with enter at the origin. 4

5 Lemma Let C(R d ). If there exists g C 1 (R) suh that E(g;R[S(; h ;`)]; L p [1, h::1]) 6= o(h k+1=p ) as h! 0; () then there exists f C 1 (R d ) suh that E(f;S(; h ;`); L p (B)) 6= o(h k+1=p ) as h! 0: Proof. Let 1 p 1, and assume that for all f C 1 (R d ), E(f;S(; h ;`); L p (B)) = o(h k+1=p ) as h! 0: Let g C 1 (R). Sine we are only onerned with g on the interval [1=::1], we may assume WLOG that g is identially 0 on a neighborhood of 0. Hene, f := g C 1 (R d ). For the ase 1 p<1, we note that if s S(; h ;`), then kg, Rsk p L p[1,h::1] = R 1 1,h R (f(tx), s(tx)) 1 d(x) p dt R 1 1,h R jf(tx), s(tx)jp 1d(x)dt; by Jensen's inequality, d,1 R 1 0 Taking p th roots proves R jf(tx), s(tx)jp t d,1 1 d,1 d(x)dt = kf, sk p L p(b): kg, Rsk Lp[1,h::1] (d,1)=p 1=p kf, sk Lp(B) (3) for the ase 1 p<1. That (3) holds as well for the ase p = 1, an be seen by taking the limit as p!1. The proof of the lemma isnow ompleted by taking, in (3), the inmum over all s S(; h ;m). Although S( m ; h ;`) has a rather ompliated struture, the struture of R[S( m ; h ;`)] (the spherial averages of the funtions in S( m ; h ;`)) is surprisingly simple. We will show that there is an m-dimensional spae V m;` (independent of ) whose restrition to the interval [1, h::1] always ontains the restrition of R[S( m ; h ;`)] to the same interval. The upshot is that we an replae the (apparantly h-dependent) spae R[S( m ; h ;`)] with the (h-independent) spae V m;`. This allows us to show, in a rather simple fashion, that there exists g C 1 (R) suh that () holds with k = m. Theorem 3 Let C(R d ). If there exists a k-dimensional spae V C[1=::1] suh that then there exists f C 1 (R d ) suh that R[S(; h ;`)] j[1,h::1] = V j[1,h::1] for all h (0::1=], E(f;S(; h ;`); L p (B)) 6= o(h k+1=p ) as h! 0: 5

6 Proof. Assume to the ontrary that Then by Lemma, E(f;S(; h ;`); L p (B)) = o(h k+1=p ) for all f C 1 (R d ). E(g;R[S(; h ;`)]; L p [1, h::1]) = o(h k+1=p ) for all g C 1 (R). Sine R[S(; h ;`)] j[1,h::1] = V j[1,h::1], it follows that E(g; e V ; Lp [0::h]) = o(h k+1=p ) for all g C 1 (R), where e V := fv(1,):v V g. Let vn;h e V be suh that kr n, v n;h k Lp[0::h] = o(hk+1=p )ash! 0; for n =0; 1;:::;k. Let p 0 be the exponent onjugate to p. Note that even if p>1, we still have kr n, v n;h k L1 k1k [0::h] Lp 0 [0::h] krn, v n;h k Lp[0::h] ; by Holder's inequality, = h 1=p0 kr n, v n;h k Lp[0::h] = o(hk+1=p+1=p0 )=o(h k+1 ). Sine dim e V < k+ 1, there exists salars n;h suh that k n;h v n;h = 0 and max 0nk j n;hj =1; 8 h (0::1]: Hene, k n;h r n L1[0::h] = k k n;h (r n, v n;h ) L 1 [0::h] j n;h jkr n, v n;h k L1 [0::h] = o(hk+1 ): Dene, M := min E r n ; spanf1;r;:::;r n,1 ;r n+1 ;:::;r k g; L 1 [0::1] : 0nk Sine f1;r;:::;r k g is linearly independent inl 1 [0::1], it follows that M>0. Note that inf 8 < k : a n r n L 1 [0::1] 9 = : max ja nj1 0nk ; M 6

7 Hene, k n;h r n L1[0::h] k = h n;h (hr) n L1[0::1] k = h k+1 h n,k n;h r n L 1 [0::1] h k+1 M beause max 0nk h n,k j n;h j1. Thus we have a ontradition. 3 The spae Vm;` Denition For n =0; 1;:::;m, dene v n :(0::1)! R by v n (t) :=( n m )(te 1 ): For 0 ` m, let V m;` be the subspae ofc 1 (0::1) given by V m;` := spanfv n : ` n m, g + spanf1;r ;r 4 ;:::;r b(`,1)= g: That dim V m;` m an be easily seen. Indeed, if ` is even, then while if ` is odd, then dimv m;` #fn : `= n m, 1g +#f0; 1;:::;b(`, 1)=g = (m, `=) + (`=) = m; dimv m;` #fn : ` n m, g +#f0; 1;:::;b(`, 1)=g = #fn :(` +1)= n m, 1g +#f0; 1;:::;(`, 1)=g = (m, (` +1)=) + (` +1)= =m: Now, in view of Theorem 3, in order to prove Theorem 1 it sues to show the following Theorem 4 For all h (0::1=], R[S( m ; h ;`)] j[1,h::1] = V m;`j [1,h::1] : In order to prove 4,we need the following lemmata: 7

8 Lemma 5 Let 0 a<b1. If g C m (a::b), then m (g ) =0 on the annulus fx R d : a<jxj <bg if and only if + g spanf1;r ; :::; r m, g spanfr,d ;r 4,d ; :::; r m,d g if d is odd spanfr,d ;r 4,d ; :::; r, ; log r;r log r; :::; r m,d log rg if d is even Proof. It is a straightforward matter to verify rstly that (g ) =g 00 ()+ d, 1 g0 (); and seondly that m (g ) =g (m) ()+p m,1 ()g (m,1) ()++ p 0 ()g(), for some funtions p 0 ;:::;p m,1 C 1 (0::1). Hene, we an appeal to the lassial dierential equations theory to onlude that the solution spae of the linear DEQ g (m) + p m,1 g (m,1) + + p 0 g =0 has dimension m. Sine the proposed spae of solutions has dimension m, it sues to simply show that these are solutions of the equation. For m = 1, this an be veried by inspetion. Continuing by indution, assume the Theorem is true for all m 0 <m. Consider m: It sues to show that 1) ( m, ) spanf (m,1), g ) If d isoddorm, d<0, then ( m,d ) spanf (m,1),d g 3) If d is even and m = d, then ( m,d log ) spanf (m,1),d g 4) If d is even and m, d>0, then ( m,d log ) spanf (m,1),d log ; (m,1), g Eah of these an be easily veried. Lemma 6 Let u C(R d ), and for t>0 dene f t : R d! R by If u C 1 (R d n0), then f t C 1 (tb) and for all d +. D f t = (,1)jj f t () :=R[u(,)](t): (D u)(tx,) d(x) on tb (4) 8

9 Proof. We begin by noting that f t () =R[u(,)](t) = 1 u(tx, ) d(x): Clearly f t C(tB) and (4) holds for =0. Proeeding by indution assume that f t C k (tb) and (4) holds for all jj k. Let ; d + be suh that jj = k and jj = 1. Let y tb. Then for h of suiently small magnitude, D f t (y + h), D f t (y) h = (,1)k! (,1)k+1 D u(tx, y, h), D u(tx, y) d(x) h D + u(tx, y) d(x) ash! 0; by the Dominated Convergene Theorem. Therefore, D D f t = (,1)j+j (D + u)(tx,) d(x) ontb: Note that D D f t is ontinuous on tb. Therefore, f t C k+1 (tb) and (4) holds for all jj k +1. Lemma 7 If p k, then Rp spanf1;r ;r 4 ;:::;r bk= g. Proof. Sine k is spanned by homogeneous polynomials and sine R is linear, it sues to show that if q k is homogeneous of order n, then Rp spanfr n g if n is even and Rp =0 if n is odd. So let q k be homogeneous of order n and onsider rst the ase n even. If t>0, then Rq(t) = 1 q(tx) d(x)=t n 1 Hene Rq spanfr n g. On the other hand, if n is odd, then Rq(t) = 1 = 1 =, 1 q(tx) d(x) u(x) d(x) =t n Rq(1): q(t(,x)) d(x), sine is a symmetri measure, q(tx) d(x) =,Rq(t): 9

10 Hene Rq =0. Proof. (of Theorem 4) Let s S( m ; h ;`), say s = h for some p `,1 and oeients satisfying m (,)+p Then Rs = P h R[ m (,)] + Rp. By Lemma 7, q() = 0 for all q `,1. (5) Rp spanf1;r ;r 4 ;:::;r b(`,1)= g: (6) We turn now to analysing R[ m (,)]. As in Lemma 6, for t>0, dene f t () :=R[ m (,)](t); R d. Sine m C 1 (R d n0), we know, by Lemma 6, that f t C 1 (tb) and D f t = (,1)jj (D m )(tx,) d(x) ontb for all +. Hene, sine m m =0onR d n0, it follows that m f t =0ontB. Therefore, by Lemma 5 (note that f t C 1 (tb) has no singularity at 0), f t spanf1; ;:::; m, g. Hene there exist 0 (t); 1 (t);:::; m,1 (t) suh that f t = m,1 n (t) n on tb. (7) The oeients n (t) an be determined in a straightforward fashion: First, one shows that 0 if k 6= n ( k n )(0) = a n if k = n ; where a n := Q n,1 j=0 (n, j)((n, j)+d, ) is positive for n +. Applying n to (7) and evaluating at 0 yields ( n f t )(0) = a n n (t): Hene n (t) = 1 a n ( n f t )(0) = 1 a n ( n m )(tx) d(x) = 1 a n n m (te 1 ), sine n m is radially symmetri, = 1 a n v n (t). 10

11 Therefore, R[ m (,)](t) = m,1 Now, if t [1, h::1], then jj <tfor all h. Hene, h R[ m (,)](t) = = m,1 1 a n v n (t)jj n, jj <t: 1 a n v n (t) h jj n `nm, 1 a n v n (t) h jj n,by (5). Thus P h R[ m (,)] j[1,h::1] spanfv n j [1,h::1] : ` n m, g whih, in view of (6), ompletes the proof. Referenes [1] Buhmann, M.D., New developments in the theory of radial basis funtion interpolation, Multivariate Approximation: From CAGD to Wavelets (K. Jetter, F.I. Utreras eds.), World Sienti, Singapore, 1993, pp. 35{75. [] Buhmann, M.D., N. Dyn and D. Levin, On quasi-interpolation by radial basis funtions with sattered entres Constr. Approx. 11 (1995), no., pp. 39{54. [3] Buhmann, M.D. and A. Ron, Radial basis funtions: Lp-Approximation orders with sattered entres, Wavelets, images, and surfae tting (Chamonix-Mont-Blan, 1993), A K Peters, Wellesley, MA, 1994, pp. 93{11. [4] Dyn, N., Interpolation and approximation by radial and related funtions, Approximation Theory VI (C.K. Chui, L.L. Shumaker, J.D. Ward eds.), Aademi Press, New York, 1989, pp. 11{34. [5] Dyn, N. and A. Ron, Radial basis funtion approximation: from gridded entres to sattered entres, Pro. London Math. So. (3) 71 (1995) no. 1, pp. 76{108. [6] Duhon, J., Splines minimizing rotation-invariant seminorms in Sobolev spaes, Construtive Theory of Funtions of Several Variables, Leture Notes in Mathematis 571, (W. Shempp, K. eller eds.) Springer-Verlag, Berlin, 1977, pp. 85{100. [7] Folland, G., Real Analysis: Modern Tehniques and their Appliations, Wiley, New York,

12 [8] Mihelli, C.A., Interpolation of sattered data: distane matries and onditionally positive denite funtions, Constr. Approx. (1986), 11{. [9] Powell, M.J.D., The theory of radial basis funtion approximation in 1990, Advanes in Numerial Analysis II: Wavelets, Subdivision, and Radial Funtions (W.A. Light ed.), Oxford University Press, Oxford, 199, 105{10. [10] Powell, M.J.D., The uniform onvergene of thin plate spline interpolation in two dimensions, Numer. Math. 68 (1994), 107{18. [11] Shabak, R., Approximation by Radial Basis Funtions with Finitely Many Centers, Constr. Approx. 1 (1996), 331{340. [1] Wu,. and R. Shabak, Loal error estimates for radial basis funtion interpolation of sattered data, IMA J. Numer. Anal. 13 (1993), 13{7. 1