CHAPTER 9 Approximation theory and stability

Transcription

1 CHAPTER 9 Approxmaton theory and stablty Polynomals of degree d have d+1 degrees of freedom, namely the d+1 coeffcents relatve to some polynomal bass. It turns out that each of these degrees of freedom can be utlsed to gan approxmaton power so that the possble rate of approxmaton by polynomals of degree d s h d+1, see Secton 9.1. The meanng of ths s that when a smooth functon s approxmated by a polynomal of degree d on an nterval of length h, the error s bounded by Ch d+1, where C s a constant that s ndependent of h. The exponent d + 1 therefore controls how fast the error tends to zero wth h. When several polynomals are lnked smoothly together to form a splne, each polynomal pece has d + 1 coeffcents, but some of these are ted up n satsfyng the smoothness condtons. It therefore comes as a nce surprse that the approxmaton power of splnes of degree d s the same as for polynomals, namely h d+1, where h s now the largest dstance between two adjacent knots. In passng from polynomals to splnes we have therefore ganed flexblty wthout sacrfcng approxmaton power. We prove ths n Secton 9.2, by makng use of some of the smple quas-nterpolants that we constructed n Chapter 8; t turns out that these produce splne approxmatons wth the requred accuracy. The quas-nterpolants also allow us to establsh two mportant propertes of B-splnes. The frst s that B-splnes form a stable bass for splnes, see Secton 9.3. Ths means that small perturbatons of the B-splne coeffcents can only lead to small perturbatons n the splne, whch s of fundamental mportance for numercal computatons. An mportant consequence of the stablty of the B-splne bass s that the control polygon of a splne converges to the splne as the knot spacng tends to zero; ths s proved n Secton The dstance to polynomals We start by determnng how well a gven a real valued functon f defned on an nterval [a, b] can be approxmated by a polynomal of degree d. We measure the error n the approxmaton wth the unform norm whch for a bounded functon g defned on an nterval [a, b] s defned by g,[a,b] = sup g(x). a x b Whenever we have an approxmaton p to f we can use the norm and measure the error by f p,[a,b]. There are many possble approxmatons to f by polynomals of degree d, and 181

2 182 CHAPTER 9. APPROXIMATION THEORY AND STABILITY the approxmaton that makes the error as small as possble s of course of specal nterest. Ths approxmaton s referred to as the best approxmaton and the correspondng error s referred to as the dstance from f to the space π d of polynomals of degree d. Ths s defned formally as dst,[a,b] (f, π d ) = nf p π d f p,[a,b]. In order to bound ths approxmaton error, we have to place some restrctons on the functons that we approxmate, and we wll only consder functons wth pecewse contnuous dervatves. Such functons le n a space that we denote C k [a, b] for some nteger k 0. A functon f les n ths space f t has k 1 contnuous dervatves on the nterval [a, b], and the kth dervatve D k f s contnuous everywhere except for a fnte number of ponts n the nteror (a, b), gven by = (z j ). At the ponts of dscontnuty the lmts from the left and rght, gven by D k f(z j +) and D k f(z j ), should exst so all the jumps are fnte. If there are no contnuous dervatves we wrte C [a, b] = C 0 [a, b]. Note that we wll often refer to these spaces wthout statng explctly what the sngulartes are. It s qute smple to gve an upper bound for the dstance of f to polynomals of degree d by choosng a partcular approxmaton, namely Taylor expanson. Theorem 9.1. Gven a polynomal degree d and a functon f n C d+1 [a, b], then dst,[a,b] (f, π d ) C d h d+1 D d+1 f,[a,b], where h = b a and the constant K d only depends on d, C d = 1 2 d+1 (d + 1)!. Proof. Consder the truncated Taylor seres of f at the mdpont m = (a + b)/2 of [a, b], d (x m) k T d f(x) = D k f(m), k! k=0 for x [a, b]. Snce T d f s a polynomal of degree d we clearly have dst,[a,b] (f, π d ) f T d f,[a,b]. (9.1) The error s gven by the ntegral form of the remander n the Taylor expanson, f(x) T d f(x) = 1 d! x m (x y) d D d+1 f(y)dy, whch s vald for any x [a, b]. If we restrct x to the nterval [m, b] we obtan The ntegral s gven by 1 d! x f(x) T d f(x) D d+1 f,[a,b] 1 d! m (x y) d dy = x 1 (d + 1)! (x m)d+1 m (x y) d dy. 1 (d + 1)! ( ) h d+1, 2

3 9.2. THE DISTANCE TO SPLINES 183 so for x m we have f(x) Td f(x) 1 2 d+1 (d + 1)! hd+1 D d+1 f,[a,b]. By symmetry ths estmate must also hold for x m. Combnng the estmate wth (9.1) completes the proof. It s n fact possble to compute the best possble constant C d. It turns out that for each f C d+1 [a, b] there s a pont ξ [a, b] such that dst,[a,b] (f, π d ) = 2 4 d+1 (d + 1)! hd+1 D d+1 f(ξ) Applyng ths formula to the functon f(x) = x d+1 we see that the exponent d + 1 n h d+1 s best possble. 9.2 The dstance to splnes Just as we defned the dstance from a functon f to the space of polynomals of degree d we can defne the dstance from f to a splne space. Our am s to show that on one knot nterval, the dstance from f to a splne space of degree d s essentally the same as the dstance from f to the space of polynomals of degree d on a slghtly larger nterval, see Theorem 9.2 and Corollary Our strategy s to consder the cases d = 0, 1 and 2 separately and then generalse to degree d. The man ngredent n the proof s to construct a smple but good approxmaton method that we can use n the same way that Taylor expanson was used n the polynomal case above. Some of the quas-nterpolants that we constructed n Chapter 8 wll do ths job very ncely. We consder a splne space S d,t where d s a nonnegatve nteger and t = (t ) n+d+1 s a d + 1 regular knot vector and set a = t 1, b = t n+d+1, h j = t j+1 t j, h = max 1 j n h j. Gven a functon f we consder the dstance from f to S d,t defned by We want to show the followng. dst,[a,b] (f, S d,t ) = nf g S d,t f g,[a,b]. Theorem 9.2. Let the polynomal degree d and the functon f n C d+1 [a, b] be gven. The dstance between f and the splne space S d,t s bounded by dst,[a,b] (f, S d,t ) D d h d+1 D d+1 f,[a,b], (9.2) where the constant D d depends on d, but not on f or t. We wll prove ths theorem by constructng a splne P d f such that f(x) P d f(x) D d h d+1 D d+1 f,[a,b], x [a, b] (9.3)

4 184 CHAPTER 9. APPROXIMATION THEORY AND STABILITY for a constant D d that depends only on d. The approxmaton P d f wll be a quasnterpolant on the form P d f = λ (f)b,d where λ s a rule for computng the th B-splne coeffcent. We wll restrct ourselves to rules λ lke d λ (f) = w,k f(x,k ) k=0 where the ponts (x,k ) d k=0 all le n one knot nterval and (w,k) d k=0 are sutable coeffcents The constant and lnear cases We frst prove Theorem 9.2 n the smplest cases d = 0 and d = 1. For d = 0 the knots form a partton a = t 1 < < t n+1 = b of [a, b] and the B-splne B,0 s the characterstc functon of the nterval [t, t +1 ) for = 1,..., n 1, whle B n,0 s the characterstc functon of the closed nterval [t n, t n+1 ]. We consder the step functon g(x) = P 0 f(x) = f(t +1/2 )B,0 (x), (9.4) where t +1/2 = (t + t +1 )/2. Fx x [a, b] and let µ be an nteger such that t µ x < t µ+1. We then have so f(x) P 0 f(x) = f(x) f(t µ+1/2 ) = x t µ+1/2 Df(y)dy f(x) P0 f(x) x tµ+1/2 Df,[tµ,t µ+1 ] h 2 Df,[a,b]. In ths way we obtan (9.2) wth D 0 = 1/2. In the lnear case d = 1 we defne P 1 f to be the pecewse lnear nterpolant to f on t defned by g = P 1 f = f(t +1 )B,1. (9.5) Proposton 5.2 gves an estmate of the error n lnear nterpolaton and by applyng ths result on each nterval we obtan whch s (9.2) wth D 1 = 1/8. f P 1 f,[a,b] h2 8 D2 f,[a,b] The quadratc case The quadratc case d = 2 s more nvolved. We shall approxmate f by the quasnterpolant P 2 f that we constructed n Secton and then estmate the error. The relevant propertes of P 2 are summarsed n the followng lemma.

5 9.2. THE DISTANCE TO SPLINES 185 Lemma 9.3. Suppose t = (t ) n+3 s a knot vector wth t +3 > t for = 1,..., n and set t +3/2 = (t +1 + t +2 )/2. The operator P 2 f = λ (f)b,2,t wth λ (f) = 1 2 f(t +1) + 2f(t +3/2 ) 1 2 f(t +2) (9.6) s lnear and satsfes P 2 f = f for all f S 2,t. Note that snce the knot vector s 3-regular we have λ 1 (f) = f(t 2 ) and λ n (f) = f(t n+1 ). We also note that snce P 2 reproduces all splnes n S d,t t certanly reproduces all quadratc polynomal. Ths fact that wll be useful n the proof of Lemma 9.6. Our am s to show that (9.3) holds for d = 2 and we are gong to do ths by establshng a sequence of lemmas. The frst lemma shows that λ (f) can become at most 3 tmes as large as f, rrespectve of what the knot vector s. Lemma 9.4. Let P 2 (f) be as n (9.6). Then Proof. Fx an nteger. Then λ (f) 3 f,[t+1,t +2 ], for = 1,..., n. (9.7) λ (f) 1 = 2 f(t +1) + 2f(t +3/2 ) 1 2 f(t +2) from whch the result follows. ( ) f,[t+1,t +2 ] Snce the B-splne coeffcents of P 2 f are bounded t s easy to see that the splne P 2 f s also bounded by the same constant. Lemma 9.5. Select some nterval [t µ, t µ+1 ) of [t 3, t n+1 ). On ths nterval the splne P 2 f s bounded by P 2 f,[tµ,t µ+1 ] 3 f,[tµ 1,t µ+2 ]. (9.8) Proof. Fx x [t µ, t µ+1 ]. Snce the B-splnes are nonnegatve and form a partton of unty we have P 2 f(x) = µ =µ 2 λ (f)b,2,t (x) max λ (f) µ 2 µ 3 max f,[t µ 2 µ +1,t +2 ] = 3 f,[tµ 1,t µ+2 ], where we used Lemma 9.4. Ths completes the proof. The followng lemma shows that on one knot nterval the splne P 2 f approxmates f almost as well as the best quadratc polynomal over a slghtly larger nterval. The proof depends on a standard trck that we wll also use n the general case. Lemma 9.6. Let [t µ, t µ+1 ) be a subnterval of [t 3, t n+1 ). On ths nterval the error f P 2 f s bounded by f P 2 f,[tµ,t µ+1 ] 4 dst,[tµ 1,t µ+2 ](f, π 2 ). (9.9)

6 186 CHAPTER 9. APPROXIMATION THEORY AND STABILITY Proof. Let p π 2 be any quadratc polynomal. Snce P 2 p = p and P 2 s a lnear operator, applcaton of (9.8) to f p yelds f(x) (P2 f)(x) = f(x) p(x) ( (P2 f)(x) p(x) ) Snce p s arbtrary we obtan (9.9). f(x) p(x) + P2 (f p)(x) (1 + 3) f p,[tµ 1,t µ+2 ]. Proof of Theorem 9.2 for d = 2. Theorem 9.1 wth d = 2 states that dst,[a,b] (f, π 2 ) C 2 h 3 D 3 f,[a,b], (9.10) where h = b a and C 2 = 1/(2 3 3!). Specalsng ths estmate to the nterval [a, b] = [t µ 1, t µ+2 ] and combnng wth (9.9) we obtan (9.3) and hence (9.2) wth D 2 = 1/ The general case The general case s analogous to the quadratc case, but the detals are more nvolved. The crucal part s to fnd a suffcently good local approxmaton operator. The operator P 2 s a quas nterpolant that s based on local nterpolaton wth quadratc polynomals at the three ponts x,k = t +1 + k(t +2 t +1 )/2 for k = 0, 1, 2. Those ponts are located symmetrcally n the mddle subnterval of the support of the B-splne B,2. We wll follow the same strategy for general degree. The resultng quas-nterpolant wll be a specal case of the one gven n Theorem 8.7. The challenge s to choose the local nterpolaton ponts n such a way that the B-splne coeffcents of the approxmaton can be bounded ndependently of the knots, as n Lemma 9.4. The key s to let all the d + 1 ponts be unformly dstrbuted n the largest subnterval [a, b ] = [t µ, t µ+1 ] of [t +1, t +d ], Gven f C [a, b] we defne P d f S d,t by P d f(x) = x,k = a + k d (b a ), for k = 0, 1,..., d. (9.11) λ (f)b,d (x), where λ (f) = In ths stuaton Theorem 8.7 specalses to the followng. d w,k f(x,k ). (9.12) Lemma 9.7. Suppose that the functonals λ n (9.12) are gven by λ (f) = f(t +1 ) f t +d = t +1, whle f t +d > t +1 the coeffcents of λ (f) are gven by k=0 w,k = γ (p,k ), for k = 0, 1,..., d, (9.13) where γ (p,k ) s the th B-splne coeffcent of the polynomal p,k (x) = d j=0 j k Then the operator P d n (9.12) satsfes P d f = f for all f S d,t. x x,j x,k x,j. (9.14)

7 9.2. THE DISTANCE TO SPLINES 187 We really only need reproducton of polynomals of degree d, but snce all the nterpolaton ponts le n one knot nterval we automatcally get reproducton of all of S d,t. The frst challenge s to fnd a formula for the B-splne coeffcents of p,k. Blossomng makes ths easy. Lemma 9.8. Suppose the splne space S d,t s gven together wth the numbers v 1,..., v d. The th B-splne coeffcent of the polynomal p(x) = (x v 1 )... (x v d ) can be wrtten γ (p) = 1 (t +j1 v 1 ) (t +jd v d ), (9.15) d! (j 1,...,j d ) Π d where Π d s the set of all permutatons of the ntegers {1, 2,..., d}. Proof. By Theorem 4.14 we have γ (p) = B[p](t +1,..., t +d ), where B[p] s the blossom of p. It therefore suffces to verfy that the expresson (9.15) satsfes the three propertes of the blossom. Ths s smple and s left to the reader. Let us consder the specal case d = 2 as an example. The set of all permutatons of {1, 2} are Π 2 = {(1, 2), (2, 1)} and therefore ( γ (x v1 )(x v 2 ) ) = 1 ( ) (t +1 v 1 )(t +2 v 2 ) + (t +2 v 1 )(t +1 v 2 ). 2 The next and most dffcult step s to obtan a bound for λ (f). Theorem 9.9. Let P d (f) = n λ (f)b,d be the operator n Lemma 9.7. Then λ (f) K d f,[t+1,t +d ], = 1,..., n, (9.16) where depends only on d. K d = 2d ( ) d d(d 1) (9.17) d! Proof. Fx an nteger. We may as well assume that t +1 < t +d snce otherwse the result s obvous. From Lemma 9.8 we have w,k = (j 1,...,j d ) Π d r=1 d ( ) t+jr v r /d!, (9.18) x,k v r where (v r ) d r=1 = (x,0,..., x,k 1, x,k+1,..., x,d ). and Π d denotes the set of all permutatons of the ntegers {1, 2,..., d}. Snce the numbers t +jr and v r belongs to the nterval [t +1, t +d ] for all r we have the nequalty d (t +jr v r ) (t +d t +1 ) d. (9.19) r=1

8 188 CHAPTER 9. APPROXIMATION THEORY AND STABILITY We also note that x,k v r = (k q)(b a )/d for some q n the range 1 q d but wth q k. Takng the product over all r we therefore obtan d x,k v r = r=1 d q=0 q k k q (b a ) d ( ) b a d = k!(d k)! k!(d k)! d (9.20) ( ) t+d t d +1 d(d 1) for all values of k and r snce [a, b ] s the largest subnterval of [t +1, t +d ]. The sum n (9.18) contans d! terms whch means that and therefore d w,k k=0 [d(d 1)]d d! λ (f) f,[t+1,t +d ] whch s the requred nequalty. d k=0 ( ) d = 2d k d! [d(d 1)]d = K d d w,k K d f,[t+1,t +d ] (9.21) k=0 Theorem 9.9 s the central ngredent n the proof of Theorem 9.2, but t has many other consequences as well, some of whch we wll consder later n ths chapter. In fact Theorem 9.9 gves one of the key propertes of B-splnes. If f = n c B,d,t s a splne n S d,t we know that λ (f) = c. The nequalty (9.16) therefore states that a B-splne coeffcent s at most K d tmes larger than the splne t represents, where the constant K d s ndependent of the knots. A smlar concluson holds for d 2, see Lemma 9.4 and the defnton of P 0 and P 1 n (9.4) and (9.5). For later reference we record ths n a corollary. Corollary For any splne f = n c B,d n S d,t the sze of the B-splne coeffcents s bounded by c K d f,[t+1,t +d ], where the the constant K d depends only on d. From the bound on λ (f) we easly obtan a smlar bound for the norm of P d f. Theorem Let f be a functon n the space C [a, b]. On any subnterval [t µ, t µ+1 ) of [t d+1, t n+1 ) the approxmaton P d f s bounded by where K d s the constant n Theorem 9.9. P d f,[tµ,t µ+1 ] K d f,[tµ d+1,t µ+d ], (9.22) Proof. Fx an x n some nterval [t µ, t µ+1 ). Snce the B-splnes are nonnegatve and form a partton of unty we have by Theorem 9.9 P d f(x) µ = Ths completes the proof. =µ d λ (f)b,d,t (x) max µ d µ λ (f) K d max f,[t µ d µ +1,t +d ] = K d f,[tµ d+1,t µ+d ]

9 9.2. THE DISTANCE TO SPLINES 189 The followng corollary shows that P d f locally approxmates f essentally as well as the best polynomal approxmaton of f of degree d. Corollary On any subnterval [t µ, t µ+1 ) the error f P d f s bounded by f P d f,[tµ,t µ+1 ] (1 + K d ) dst,[tµ d+1,t µ+d ](f, π d ), (9.23) where K d s the constant n Theorem 9.9 Proof. We argue exactly as n the quadratc case. Let p π d be any polynomal n π d. Snce P d p = p and P d s a lnear operator we have f(x) (Pd f)(x) = f(x) p(x) ( (Pd f)(x) p(x) ) Snce p s arbtrary we obtan (9.23). f(x) p(x) + Pd (f p)(x) (1 + K d ) f p,[tµ d+1,t µ+d ]. Proof of Theorem 9.2 for general d. By Theorem 9.1 we have for any nterval [a, b] dst,[a,b] (f, π d ) C d h d+1 D d+1 f,[a,b], where h = b a and K d only depends on d. Combnng ths estmate on [a, b] = [t µ d+1, t µ+d ] wth (9.23) we obtan (9.3) and hence (9.2) wth D d = (K d + 1)D d. We have accomplshed our task of estmatng the dstance from a functon n C d+1 [a, b] to an arbtrary splne space S d,t. However, there are several unanswered questons. Perhaps the most obvous s whether the constant K d s the best possble. A moment s thought wll make you realse that t certanly s not. One reason s that we made use of some rather coarse estmates n the proof of Theorem 9.9. Another reason s that we may obtan better estmates by usng a dfferent approxmaton operator. In fact, t s qute easy to fnd a better operator whch s also a quas-nterpolant based on local nterpolaton. Instead of choosng the local nterpolaton ponts unformly n the largest subnterval of [t +1, t +d ], we smply choose the ponts unformly n [t +1, t +d ], x,k = t +1 + k d (t +d t +1 ), for k = 0, 1,..., d. It s easy to check that the bound (9.19) on the numerator stll holds whle the last estmate n the bound on the denomnator (9.20) s now unnecessary so we have d x,k v r = r=1 d q=0 q k k q k!(d k)! (t +d t +1 ) = d d d (t +d t +1 ) d. Ths gves a new constant K d = 2d d d. d! Note that the new approxmaton operator wll not reproduce the whole splne space for d > 2. Ths mproved constant can therefore not be used n Corollary The constant can be mproved further by choosng the nterpolaton ponts to be the extrema of the Chebyshev polynomal, adjusted to the nterval [t +1, t +d ].

10 190 CHAPTER 9. APPROXIMATION THEORY AND STABILITY 9.3 Stablty of the B-splne bass In order to compute wth polynomals or splnes we need to choose a bass to represent the functons. If a bass s to be sutable for computer manpulatons t should be reasonably nsenstve to round-off errors. In partcular, functons wth small functon values should have small coeffcents and vce versa. A bass wth ths property s sad to be well condtoned or stable and the stablty s measured by the condton number of the bass. In ths secton we wll study the condton number of the B-splne bass A general defnton of stablty The stablty of a bass can be defned qute generally. Instead of consderng polynomals we can consder a general lnear vector space where we can measure the sze of the elements through a norm; ths s called a normed lnear space. Defnton Let V be a normed lnear space. A bass (φ j ) for V s sad to be stable wth respect to a vector norm f there are small postve constants C 1 and C 2 such that C1 1 (c j ) c j φ j C2 (cj ), (9.24) j for all sets of coeffcents c = (c j ). Let C1 and C 2 denote the smallest possble values of C 1 and C 2 such that (9.24) holds. The condton number of the bass s then defned to be κ = κ((φ ) ) = C1 C 2. At the rsk of confuson we have used the same symbol both for the norm n V and the vector norm of the coeffcents. In our case V wll be some splne space S d,t and the bass (φ j ) wll be the B-splne bass. The norms we wll consder are the p-norms whch are defned by ( b 1/p ( ) 1/p f p = f p,[a,b] = f(x) dx) p and c p = c j p a where p s a real number n the range 1 p <. Here f s a functon on the nterval [a, b] and c = (c j ) s a real vector. For p = the norms are defned by f = f,[a,b] = max f(x) and c = (c j ) a x b = max c j, j In practce, the most mportant norms are the 1-, 2- and -norms. In Defnton 9.13 we requre the constants C 1 and C 2 to be small, but how small s small? There s no unque answer to ths queston, but t s typcally requred that C 1 and C 2 should be ndependent of the dmenson n of V, or at least grow very slowly wth n. Note that we always have κ 1, and κ = 1 f and only f we have equalty n both nequaltes n (9.24). A stable bass s desrable for many reasons, and the constant κ = C 1 C 2 crops up n many dfferent contexts. The condton number κ does n fact act as a sort of dervatve of the bass and gves a measure of how much an error n the coeffcents s magnfed n a functon value. j

11 9.3. STABILITY OF THE B-SPLINE BASIS 191 Proposton Suppose (φ j ) s a stable bass for V. If f = j c jφ j and g = j b jφ j are two elements n V wth f 0, then f g f c b κ c, (9.25) where κ s the condton number of the bass as n Defnton Proof. From (9.24), we have the two nequaltes f g C 2 (c j b j ) and 1/ f C 1 / (c j ). Multplyng these together gves the result. If we thnk of g as an approxmaton to f then (9.25) says that the relatve error n f g s bounded by at most κ tmes the relatve error n the coeffcents. If κ s small a small relatve error n the coeffcents gves a small relatve error n the functon values. Ths s mportant n floatng pont calculatons on a computer. A functon s usually represented by ts coeffcents relatve to some bass. Normally, the coeffcents are real numbers that must be represented nexactly as floatng pont numbers n the computer. Ths round-off error means that the computed splne, here g, wll dffer from the exact f. Proposton 9.14 shows that ths s not so serous f the perturbed coeffcents of g are close to those of f and the bass s stable. Proposton 9.14 also provdes some nformaton as to what are acceptable values of C1 and C 2. If for example κ = C 1 C 2 = 100 we rsk losng 2 decmal places n evaluaton of a functon; exactly how much accuracy one can afford to lose wll of course vary. One may wonder whether there are any unstable polynomal bases. It turns out that the power bass 1, x, x 2,..., on the nterval [0, 1] s unstable even for qute low degrees. Already for degree 10, one rsks losng as much as 4 or 5 decmal dgts n the process of computng the value of a polynomal on the nterval [0, 1] relatve to ths bass, and other operatons such as numercal root fndng s even more senstve Stablty of the B-splne bass, p = Snce splnes and B-splnes are defned va the knot vector, t s qute concevable that the condton number of the B-splne bass could become arbtrarly large for certan knot confguratons, for example n the lmt when two knots merge nto one. One of the key features of splnes s that ths cannot happen. Theorem There s a constant K d whch depends only on the polynomal degree d, such that for all splne spaces S d,t and all splnes f = n c B,d S d,t wth B-splne coeffcents c = (c ) n, the two nequaltes hold. K 1 d c f,[t1,t n+d ] c (9.26) Proof. We have already proved varants of the second nequalty several tmes; t follows snce B-splnes are nonnegatve and sum to (at most) 1. The frst nequalty s a consequence of Corollary The value of the constant K d s K 0 = K 1 = 1, K 2 = 3 whle t s gven by (9.17) for d > 2.

12 192 CHAPTER 9. APPROXIMATION THEORY AND STABILITY The condton number of the B-splne bass on the knot vector t wth respect to the - norm s usually denoted κ d,,t. By takng the supremum over all knot vectors we obtan the knot ndependent condton number κ d,, κ d, = sup κ d,,t. t Theorem 9.15 shows that κ d, s bounded above by K d. Although K d s ndependent of the knots, t grows qute quckly wth d and seems to ndcate that the B-splne bass may well be unstable for all but small values of d. However, by usng dfferent technques t s possble to fnd better estmates for the condton number, and t s ndeed known that the B-splne bass s very stable, at least for moderate values of d. It s smple to determne the condton number for d 2; we have κ 0, = κ 1, = 1 and κ 2, = 3. For d 3 t has recently been shown that κ d, = O(2 d ). The frst few values are known to be approxmately κ 3, and κ 4, Stablty of the B-splne bass, p < In ths secton we are gong to generalse Theorem 9.15 to any p-norm. Ths s useful n some contexts, especally the case p = 2 whch s closely related to least squares approxmaton. The proof uses standard tools from analyss, but may seem techncal for the reader who s not famlar wth the technques. Throughout ths secton p s a fxed real number n the nterval [1, ) and q s a related number defned by the dentty 1/p + 1/q = 1. A classcal nequalty for functons that wll be useful s the Hölder nequalty b a f(x)g(x) dx f p g q. We wll also need the Hölder nequalty for vectors whch s gven by b c (b ) n p (c ) n q. In addton to the Hölder nequaltes we need a fundamental nequalty for polynomals. Ths states that for any polynomal g π d and any nterval [a, b] we have g(x) C b a b a g(z) dz, for any x [a, b], (9.27) where the constant C only depends on the degree d. Ths s a consequence of the fact that all norms on a fnte dmensonal vector space are equvalent. In order to generalse the stablty result (9.26) to arbtrary p-norms we need to ntroduce a dfferent scalng of the B-splnes. We defne the p-norm B-splnes to be dentcally zero f t +d+1 = t and ( ) B p d + 1 1/p,d,t = B,d,t, (9.28) t +d+1 t otherwse. We can then state the p-norm stablty result for B-splnes.

13 9.3. STABILITY OF THE B-SPLINE BASIS 193 Theorem There s a constant K that depends only on the polynomal degree d, such that for all 1 p, all splne spaces S d,t and all splnes f = n c B p,d S d,t wth p-norm B-splne coeffcents c = (c ) n the nequaltes hold. K 1 c p f p,[t1,t m+d ] c p (9.29) Proof. We frst prove the upper nequalty. Let γ = (d + 1)/(t +d+1 t ) denote the pth power of the scalng factor n (9.28) for = 1,..., n and set [a, b] = [t 1, t n+d+1 ]. Rememberng the defnton of B p,d,t and the dentty 1/p + 1/q = 1 and applyng the Hölder nequalty for sums we obtan c B p,d = c γ 1/p B 1/p,d ( 1/q B,d c p γ B,d ) 1/p ( B,d ) 1/q. Rasng both sdes of ths nequalty to the pth power and recallng that B-splnes sum to (at most) 1 we obtan the nequalty c B p,d (x) p c p γ B,d (x) for any x R. (9.30) It can be shown that the ntegral of a B-splne s gven by t+d+1 Makng use of ths and (9.30) we fnd t B,d (x)dx = t +d+1 t d + 1 = 1 γ. b f p p,[a,b] = c B p,d (x) p b dx c p γ B,d (x) dx = a a c p. Takng pth roots on both sdes proves the upper nequalty. Consder now the lower nequalty. The splne f s gven as a lnear combnaton of p-norm B-splnes, but can very smply be wrtten as a lnear combnaton of the usual B-splnes, f = c B p,d = c γ 1/p B,d. From the frst nequalty n (9.26) we then obtan for each ( d + 1 t +d+1 t ) 1/p c K d max t +1 x t +d f(x), where the constant K d only depends on d. Extendng the maxmum to a larger subnterval and applyng the nequalty (9.27) we fnd c K d (d + 1) 1/p( t +d+1 t ) 1/p max t x t +d+1 f(x) CK d (d + 1) 1/p( ) t+d+1 1+1/p t +d+1 t f(y) dy. t

14 194 CHAPTER 9. APPROXIMATION THEORY AND STABILITY Next, we apply the Hölder nequalty for ntegrals to the product t +d+1 f(y) t 1 dy and obtan ( t+d+1 ) 1/p c CK d (d + 1) 1/p f(y) p dy. Rasng both sdes to the pth power and summng over we obtan c p C p K p d (d + t+d+1 1) 1 f(y) p dy C p K p d f p p,[a,b]. t Takng pth roots we obtan the lower nequalty n (9.29) wth K = CK d. 9.4 Convergence of the control polygon for splne functons Recall that for a splne functon f(x) = c B,d,t the control polygon s the pecewse lnear nterpolant to the ponts (t, c ), where t = (t t +d )/d s the th knot average. In ths secton we are gong to prove that the control polygon converges to the splne t represents when the knot spacng approaches zero. The man work s done n Lemma 9.17 whch shows that a corner of the control polygon s close to the splne snce c s close to f(t ), at least when the spacng n the knot vector s small. The proof of the lemma makes use of the fact that the sze of a B-splne coeffcent c can be bounded n terms of the sze of the splne on the nterval [t +1, t +d+1 ], whch we proved n Theorem 9.9 and Lemma 9.4 (and Secton 9.2.1), t c K d f [t+1,t +d ]. (9.31) The norm used here and throughout ths secton s the -norm. Lemma Let f be a splne n S d,t wth coeffcents (c ). Then c f(t ) K(t +d t +1 ) 2 D 2 f [t+1,t +d ], (9.32) where t = (t t +d )/d, the operator D 2 denotes (one-sded) dfferentaton (from the rght), and the constant K only depends on d. Proof. Let be fxed. If t +1 = t +d then we know from property 5 n Lemma 2.6 that B,d (t ) = 1 so c = f(t ) and there s nothng to prove. Assume for the rest of the proof that the nterval J = (t +1, t +d ) s nonempty. Snce J contans at most d 2 knots, t follows from the contnuty property of B-splnes that f has at least two contnuous dervatves n J. Let x 0 be a number n the nterval J and consder the splne g(x) = f(x) f(x 0 ) (x x 0 )Df(x 0 ) whch s the error n a frst order Taylor expanson of f at x 0. Ths splne les n S d,t and can therefore be wrtten as g = b B,d,t for sutable coeffcents (b ). More specfcally we have b = c f(x 0 ) (t x 0 )Df(x 0 ). Choosng x 0 = t we have b = c f(t ) and accordng to the nequalty (9.31) and the error term n frst order Taylor expanson we fnd c f(t ) = b K d g J K d(t +d t +1 ) 2 D 2 f J. 2 The nequalty (9.32) therefore holds wth K = K d /2 and the proof s complete.

15 9.4. CONVERGENCE OF THE CONTROL POLYGON FOR SPLINE FUNCTIONS195 Lemma 9.17 shows that the corners of the control polygon converge to the splne as the knot spacng goes to zero. Ths partly explans why the control polygon approaches the splne when we nsert knots. What remans s to show that the control polygon as a whole also converges to the splne. Theorem Let f = n c B,d be a splne n S d,t, and let Γ d,t (f) be ts control polygon. Then Γd,t (f) f [t 1,t n] Kh2 D 2 f [t1,t n+d+1 ], (9.33) where h = max {t +1 t } and the constant K only depends on d. Proof. As usual, we assume that t s d + 1-regular (f not we extend t wth d + 1-tuple knots at ether ends and add zero coeffcents). Suppose that x s n [t 1, t m] and let j be such that t j x < t j+1. Observe that snce the nterval J = (t j, t j+1 ) s nonempty we have t j+1 < t j+d+1 and J contans at most d 1 knots. From the contnuty property of B-splnes we conclude that f has a contnuous dervatve and the second dervatve of f s at least pecewse contnuous n J. Let g(x) = (t j+1 x)f(t j ) + (x t j )f(t j+1 ) t j+1 t j be the lnear nterpolant to f on ths nterval. We wll show that both Γ = Γ d,t (f) and f are close to g on J and then deduce that Γ s close to f because of the trangle nequalty Γ(x) f(x) Γ(x) g(x) + g(x) f(x). (9.34) Let us frst consder the dfference Γ g. Note that Γ(x) g(x) = (t j+1 x)(b j f(t j )) + (x t j )(b j+1 f(t j+1 )) t j+1 t j for any x n J. We therefore have Γ(x) g(x) max { bj f(t j), bj+1 f(t j+1) }, for x J. From Lemma 9.17 we then conclude that Γ(x) g(x) K 1 h 2 D 2 f J, x J, (9.35) where J = [t 1, t m+d+1 ] and K 1 s a constant that only depends on d. The second dfference f(x) g(x) n (9.34) s the error n lnear nterpolaton to f at the endponts of J. For ths process we have the standard error estmate f(x) g(x) 1 8 (t j+1 t j) 2 D 2 f J 1 8 h2 D 2 f J, x J. (9.36) If we now combne (9.35) and (9.36) as ndcated n (9.34), we obtan the Theorem wth constant K = K 1 + 1/8. Because of the factor h 2 n Theorem 9.18 we say (somewhat loosely) that the control polygon converges quadratcally to the splne.

16 196 CHAPTER 9. APPROXIMATION THEORY AND STABILITY Exercses for Chapter In ths exercse we wll study the order of approxmaton by the Schoenberg Varaton Dmnshng Splne Approxmaton of degree d 2. Ths approxmaton s gven by V d f = f(t )B,d, wth t = t +1 + t +d. d Here B,d s the th B-splne of degree d on a d + 1-regular knot vector t = (t ) n+d+1. We assume that t +d > t for = 2,..., n. Moreover we defne the quanttes a = t 1, b = t n+d+1, h = max 1 n t +1 t. We want to show that V d f s an O(h 2 ) approxmaton to a suffcently smooth f. We frst consder the more general splne approxmaton Ṽ d f = λ (f)b,d, wth λ (f) = w,0 f(x,0 ) + w,1 f(x,1 ). Here x,0 and x,1 are two dstnct ponts n [t, t +d ] and w,0, w,1 are constants, = 1,..., n. Before attemptng to solve ths exercse the reader mght fnd t helpful to revew Secton a) Suppose for = 1,..., n that w,0 and w,1 are such that w,0 + w,1 = 1 x,0 w,0 + x,1 w,1 = t Show that then Ṽdp = p for all p π 1. p(x) = 1 and p(x) = x.) (Hnt: Consder the polynomals b) Show that f we set x,0 = t for all then Ṽdf = V d f for all f, regardless of how we choose the value of x,1. In the rest of ths exercse we set λ (f) = f(t ) for = 1,..., n,.e. we consder V d f. We defne the usual unform norm on an nterval [c, d] by f [c,d] = sup f(x), c x d f C [c, d]. c) Show that for d + 1 l n V d f [tl,t l+1 ] f [t l d,t l ], f C [a, b]. d) Show that for f C [t l d, t l ] and d + 1 l n f V d f [tl,t l+1 ] 2 dst [t l d,t l ] (f, π 1 ).

17 9.4. CONVERGENCE OF THE CONTROL POLYGON FOR SPLINE FUNCTIONS197 e) Explan why the followng holds for d + 1 l n dst [t l d,t l ] (f, π 1 ) (t l t l d )2 D 2 f 8 [t l d,t l ]. f) Show that the followng O(h 2 ) estmate holds (Hnt: Verfy that t l t l d hd. ) f V d f [a,b] d2 4 h2 D 2 f [a,b]. 9.2 In ths exercse we want to perform a numercal smulaton experment to determne the order of approxmaton by the quadratc splne approxmatons V 2 f = P 2 f = f(t )B,2, wth t = t +1 + t +2, 2 ( 1 2 f(t +1) + 2f(t ) 1 2 f(t +2) ) B,2. We want to test the hypotheses f V 2 f = O(h 2 ) and f P 2 f = O(h 3 ) where h = max t +1 t. We test these on the functon f(x) = sn x on [0, π] for varous values of h. Consder for m 0 and n m = m the 3-regular knot vector t m = (t m )nm+3 on the nterval [0, π] wth unform spacng h m = π2 m. We defne V m 2 f = P m 2 f = f(t m +3/2 )Bm,2, wth t m = tm +1 + tm +2, 2 ( 1 2 f(tm +1) + 2f(t m +3/2 ) 1 2 f(tm +2) ) B m,2, and B,2 m s the th quadratc B-splne on tm. f V2 mf [0,π] and f P2 mf [0,π] we use E m V = max 0 j 100 f(jπ/100) V m 2 f(jπ/100), E m P = max 0 j 100 f(jπ/100) P m 2 f(jπ/100). As approxmatons to the norms Wrte a computer program to compute numercally the values of EV m and Em P for m = 0, 1, 2, 3, 4, 5, and the ratos EV m/em 1 V and EP m/em 1 P for 1 m 5. What can you deduce about the approxmaton order of the two methods? Make plots of V m 2 f, P m 2 f, f V m 2 f, and f P m 2 f for some values of m. 9.3 Suppose we have m 3 data ponts ( x, f(x ) ) m sampled from a functon f, where the abscssas x = (x ) m satsfy x 1 < < x m. In ths exercse we want to derve a local quas-nterpolaton scheme whch only uses the data values at the x s and whch has O(h 3 ) order of accuracy f the y-values are sampled from a smooth functon f. The method requres m to be odd.

18 198 CHAPTER 9. APPROXIMATION THEORY AND STABILITY From x we form a 3-regular knot vector by usng every second data pont as a knot t = (t j ) n+3 j=1 = (x 1, x 1, x 1, x 3, x 5,..., x m 2, x m, x m, x m ), (9.37) where n = (m + 3)/2. In the quadratc splne space S 2,t we can then construct the splne Q 2 f = λ j (f)b j,2, (9.38) j=1 where the B-splne coeffcents λ j (f) n j=1 are defned by the rule λ j (f) = 1 ( ) θj 1 f(x 2j 3 ) + θj 1 (1 + θ j ) 2 f(x 2j 2 ) θ j f(x 2j 1 ), (9.39) 2 for j = 1,..., n. Here θ 1 = θ n = 1 and for j = 2,..., n 1. θ j = x 2j 2 x 2j 3 x 2j 1 x 2j 2 a) Show that Q 2 smplfes to P 2 gven by (9.6) when the data abscssas are unformly spaced. b) Show that Q 2 p = p for all p π 2 and that because of the multple abscssas at the ends we have λ 1 (f) = f(x 1 ), λ n (f) = f(x m ), so only the orgnal data are used to defne Q 2 f. (Hnt: Use the formula n Exercse 1. c) Show that for j = 1,..., n and f C [x 1, x m ] where λ j (f) (2θ + 1) f,[tj+1,t j+2 ], θ = max 1 j n {θ 1 j, θ j }. d) Show that for l = 3,..., n, f C [x 1, x m ], and x [t l, t l+1 ] Q 2 (f)(x) (2θ + 1) f,[tl 1,t l+2 ]. e) Show that for l = 3,..., n and f C [x 1, x m ] f Q 2 f,[tl,t l+1 ] (2θ + 2) dst [tl 1,t l+2 ](f, π 2 ). f) Show that for f C 3 [x 1, x m ] we have the O(h 3 ) estmate where f Q 2 f,[x1,x m] K(θ) x 3 D 3 f,[x1,x m], x = max x j+1 x j j and the constant K(θ) only depends on θ.