CHAPTER 5 Spline Approximation of Functions and Data

Transcription

1 CHAPTER 5 Spline Approximtion of Functions nd Dt This chpter introduces number of methods for obtining spline pproximtions to given functions, or more precisely, to dt obtined by smpling function. In Section 5.1, we focus on locl methods where the pproximtion t point x only depends on dt vlues ner x. Connecting neighbouring dt points with stright lines is one such method where the vlue of the pproximtion t point only depends on the two nerest dt points. In order to get smoother pproximtions, we must use splines of higher degree. With cubic polynomils we cn prescribe, or interpolte, position nd first derivtives t two points. Therefore, given set of points with ssocited function vlues nd first derivtives, we cn determine sequence of cubic polynomils tht interpolte the dt, joined together with continuous first derivtives. This is the cubic Hermite interpolnt of Section In Section 5.2 we study globl cubic pproximtion methods where we hve to solve system of equtions involving ll the dt points in order to obtin the pproximtion. Like the locl methods in Section 5.1, these methods interpolte the dt, which now only re positions. The gin in turning to globl methods is tht the pproximtion my hve more continuous derivtives nd still be s ccurte s the locl methods. The cubic spline interpolnt with so clled nturl end conditions solves n interesting extreml problem. Among ll functions with continuous second derivtive tht interpolte set of dt, the nturl cubic spline interpolnt is the one whose integrl of the squre of the second derivtive is the smllest. This is the foundtion for vrious interprettions of splines, nd is ll discussed in Section 5.2. Two pproximtion methods for splines of rbitrry degree re described in Section 5.3. The first method is spline interpoltion with B-splines defined on some rther rbitrry knot vector. The disdvntge of using interpoltion methods is tht the pproximtions hve tendency to oscillte. If we reduce the dimension of the pproximting spline spce, nd insted minimize the error t the dt points this problem cn be gretly reduced. Such lest squres methods re studied in Section We end the chpter by discussing very simple pproximtion method, the Vrition Diminishing Spline Approximtion. This pproximtion scheme hs the desirble bility to trnsfer the sign of some of the derivtives of function to the pproximtion. This is 97

2 98 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA importnt since mny importnt chrcteristics of the shpe of function is closely relted to the sign of the derivtives. 5.1 Locl Approximtion Methods When we construct n pproximtion to dt, it is usully n dvntge if the pproximtion t point x only depends on the dt ner x. If this is the cse, chnging the dt in some smll re will only ffect the pproximtion in the sme re. The vrition diminishing pproximtion method nd in prticulr piecewise liner interpoltion hs this property, it is locl method. In this section we consider nother locl pproximtion method Piecewise liner interpoltion The simplest wy to obtin continuous pproximtion to set of ordered dt points is to connect neighbouring dt points with stright lines. This pproximtion is nturlly enough clled the piecewise liner interpolnt to the dt. It is clerly liner spline nd cn therefore be written s liner combintion of B-splines on suitble knot vector. The knots must be t the dt points, nd since the interpolnt is continuous, ech interior knot only needs to occur once in the knot vector. The construction is given in the following proposition. Proposition 5.1. Let (x i, y i ) m be set of dt points with x i < x i+1 for i = 1,..., m 1, nd construct the 2-regulr knot vector t s Then the liner spline g given by t = (t i ) m+2 = (x 1, x 1, x 2, x 3,..., x m 1, x m, x m ). g(x) = m y i B i,1 (x) stisfies the interpoltion conditions g(x i ) = y i, for i = 1,..., m 1, nd lim g(x) = y m. (5.1) x x m The lst condition sttes tht the limit of g from the left t x m is y m. If the dt re tken from function f so tht y i = f(x i ) for i = 1,..., m, the interpolnt g is often denoted by I 1 f. Proof. From Exmple 2.2 in Chpter 2, we see tht the B-spline B i,1 for 1 i m is given by (x x i 1 )/(x i x i 1 ), if x i 1 x < x i, B i,1 (x) = (x i+1 x)/(x i+1 x i ), if x i x < x i+1, 0, otherwise, where we hve set x 0 = x 1 nd x m+1 = x m. This mens tht B i,1 (x i ) = 1 for i < m nd lim x x m B m,1 (x) = 1, while B i,1 (x j ) = 0 for ll j i, so the interpoltion conditions (5.1) re stisfied.

3 5.1. LOCAL APPROXIMATION METHODS 99 The piecewise liner interpolnt preserves the shpe of the dt extremely well. The obvious disdvntge of this pproximtion is its lck of smoothness. Intuitively, it seems resonble tht if f is continuous, it should be possible to pproximte it to within ny ccurcy by piecewise liner interpolnts, if we let the distnce between the dt points become smll enough. This is indeed the cse. Note tht the symbol C j [, b] denotes the set of ll functions defined on [, b] with vlues in R whose first j derivtives re continuous. Proposition 5.2. Suppose tht = x 1 < x 2 < < x m = b re given points, nd set x = mx 1 i m 1 {x i+1 x i }. 1. If f C[, b], then for every ɛ > 0 there is δ > 0 such tht if x < δ, then f(x) I 1 f(x) < ɛ for ll x [, b]. 2. If f C 2 [, b] then for ll x [, b], f(x) (I 1 f)(x) 1 8 ( x)2 mx z b f (z), (5.2) f (x) (I 1 f) (x) 1 x mx 2 f (z). (5.3) z b Prt (i) of Proposition 5.2 sttes tht piecewise liner interpoltion to continuous function converges to the function when the distnce between the dt points goes to zero. More specificlly, given tolernce ɛ, we cn mke the error less thn the tolernce by choosing x sufficiently smll. Prt (ii) of Proposition 5.2 gives n upper bound for the error in cse the function f is smooth, which in this cse mens tht f nd its first two derivtives re continuous. The inequlity in (5.2) is often stted s piecewise liner pproximtion hs pproximtion order two, mening tht x is rised to the power of two in (5.2). The bounds in Proposition 5.2 depend both on x nd the size of the second derivtive of f. Therefore, if the error is not smll, it must be becuse one of these quntities re lrge. If in some wy we cn find n upper bound M for f, i.e., f (x) M, for x [, b], (5.4) we cn determine vlue of x such tht the error, mesured s in (5.2), is smller thn some given tolernce ɛ. We must clerly require ( x) 2 M/8 < ɛ. This inequlity holds provided x < 8ɛ/M. We conclude tht for ny ɛ > 0, we hve the impliction 8ɛ x < M = f(x) I 1f(x) < ɛ, for x [x 1, x m ]. (5.5) This estimte tells us how densely we must smple f in order to hve error smller thn ɛ everywhere. We will on occsions wnt to compute the piecewise liner interpolnt to given higher degree spline f. A spline does not necessrily hve continuous derivtives, but t lest we know where the discontinuities re. The following proposition is therefore meningful. Proposition 5.3. Suppose tht f S d,t for some d nd t with interior knots of multiplicity t most d (so f is continuous). If the brek points (x i ) m re chosen so s to include ll the knots in t where f is discontinuous, the bounds in (5.2) nd (5.3) continue to hold.

4 100 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA Cubic Hermite interpoltion The piecewise liner interpolnt hs the nice property of being locl construction: The interpolnt on n intervl [x i, x i+1 ] is completely defined by the vlue of f t x i nd x i+1. The other dvntge of f is tht it does not oscillte between dt points nd therefore preserves the shpe of f if x is smll enough. In this section we construct n interpolnt which, unlike the piecewise liner interpolnt, hs continuous first derivtive, nd which, like the piecewise liner interpolnt, only depends on dt vlues loclly. The price of the smoothness is tht this interpolnt requires informtion bout derivtives, nd shpe preservtion in the strong sense of the piecewise liner interpolnt cnnot be gurnteed. The interpolnt we seek is the solution of the following problem. Problem 5.4 (Piecewise Cubic Hermite Interpoltion). Let the discrete dt (x i, f(x i ), f (x i )) m with = x 1 < x 2 < < x m = b be given. Find function g = Hf tht stisfies the following conditions: 1. On ech subintervl (x i, x i+1 ) the function g is cubic polynomil. 2. The given function f is interpolted by g in the sense tht g(x i ) = f(x i ), nd g (x i ) = f (x i ), for i = 1,..., m. (5.6) A spline g tht solves Problem 5.4 must be continuous nd hve continuous first derivtive since two neighbouring pieces meet with the sme vlue f(x i ) nd first derivtive f (x i ) t join x i. Since Hf should be piecewise cubic polynomil, it is nturl to try nd define knot vector so tht Hf cn be represented s liner combintion of B-splines on this knot vector. To get the correct smoothness, we need t lest double knot t ech dt point. Since d = 3 nd we hve 2m interpoltion conditions, the length of the knot vector should be 2m + 4, nd we might s well choose to use 4-regulr knot vector. We chieve this by mking ech interior dt point knot of multiplicity two nd plce four knots t the two ends. This leds to the knot vector t = (t i ) 2m+4 = (x 1, x 1, x 1, x 1, x 2, x 2,..., x m 1, x m 1, x m, x m, x m, x m ), (5.7) which we cll the Cubic Hermite knot vector on x = (x 1,..., x m ). construct the solution to Problem 5.4. This llows us to Proposition 5.5. Problem 5.4 hs unique solution Hf in the spline spce S 3,t, where t is given in eqution (5.7). More specificlly, the solution is given by Hf = 2m c i B i,3, (5.8) where c 2i 1 = f(x i ) 1 3 x i 1f (x i ), c 2i = f(x i ) x if (x i ), for i = 1,..., m, (5.9) where x j = x j+1 x j, nd the points x 0 nd x m+1 re defined by x 0 = x 1 nd x m+1 = x m.

5 5.1. LOCAL APPROXIMATION METHODS () (b) Figure 5.1. Figure () shows the cubic Hermite interpolnt (solid) to f(x) = x 4 (dshed), see Exmple 5.6, while the error in this pproximtion is shown in (b). Proof. We leve the proof tht the spline defined by (5.9) stisfies the interpoltion conditions in Problem 5.4 to the reder. By construction, the solution is clerly cubic polynomil. Tht there is only one solution follows if we cn show tht the only solution tht solves the problem with f(x i ) = f (x i ) = 0 for ll i is the function tht is zero everywhere. For if the generl problem hs two solutions, the difference between these must solve the problem with ll the dt equl to zero. If this difference is zero, the two solutions must be equl. To show tht the solution to the problem where ll the dt re zero is the zero function, it is clerly enough to show tht the solution is zero in one subintervl. On ech subintervl the function Hf is cubic polynomil with vlue nd derivtive zero t both ends, nd it therefore hs four zeros (counting multiplicity) in the subintervl. But the only cubic polynomil with four zeros is the polynomil tht is identiclly zero. From this we conclude tht Hf must be zero in ech subintervl nd therefore identiclly zero. Let us see how this method of pproximtion behves in prticulr sitution. Exmple 5.6. We try to pproximte the function f(x) = x 4 on the intervl [0, 1] with only one polynomil piece so tht m = 2 nd [, b] = [x 1, x m] = [0, 1]. Then the cubic Hermite knots re just the Bernstein knots. From (5.9) we find (c 1, c 2, c 3, c 4) = (0, 0, 1/3, 1), nd (Hf)(x) = 1 3 3x2 (1 x) + x 3 = 2x 3 x 2. The two functions f nd Hf re shown in Figure 5.1. Exmple 5.7. Let us gin pproximte f(x) = x 4 on [0, 1], but this time we use two polynomil pieces so tht m = 3 nd x = (0, 1/2, 1). In this cse the cubic Hermite knots re t = (0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1), nd we find the coefficients c = (0, 0, 1/48, 7/48, 1/3, 1). The two functions f nd Hf re shown in Figure 5.1 (). With the extr knots t 1/2 (cf. Exmple 5.6), we get much more ccurte pproximtion to x 4. In fct, we see from the error plots in Figures 5.1 (b) nd 5.1 (b) tht the mximum error hs been reduced from 0.06 to bout 0.004, fctor of bout 15. Note tht in Exmple 5.6 the pproximtion becomes negtive even though f is nonnegtive in ll of [0, 1]. This shows tht in contrst to the piecewise liner interpolnt, the cubic Hermite interpolnt Hf does not preserve the sign of f. However, it is simple to give conditions tht gurntee Hf to be nonnegtive.

6 102 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA () (b) Figure 5.2. Figure () shows the cubic Hermite interpolnt (solid) to f(x) = x 4 (dshed) with two polynomil pieces, see Exmple 5.7, while the error in the pproximtion is shown in (b). Proposition 5.8. Suppose tht the function f to be pproximted by cubic Hermite interpoltion stisfies the conditions f(x i ) 1 3 x i 1f (x i ) 0, f(x i ) + 1 for i = 1,..., m. 3 x if (x i ) 0, Then the cubic Hermite interpolnt Hf is nonnegtive on [, b]. Proof. In this cse, the spline pproximtion Hf given by Proposition 5.5 hs nonnegtive B-spline coefficients, so tht (Hf)(x) for ech x is sum of nonnegtive quntities nd therefore nonnegtive. As for the piecewise liner interpolnt, it is possible to relte the error to the spcing in x nd the size of some derivtive of f. Proposition 5.9. Suppose tht f hs continuous derivtives up to order four on the intervl [x 1, x m ]. Then f(x) (Hf)(x) ( x)4 mx z b f (iv) (z), for x [, b]. (5.10) This estimte lso holds whenever f is in some spline spce S d,t provided f hs continuous derivtive t ll the x i. Proof. See text on numericl nlysis. The error estimte in (5.10) sys tht if we hlve the distnce between the interpoltion points, then we cn expect the error to decrese by fctor of 2 4 = 16. This is usully referred to s fourth order convergence. This behviour is confirmed by Exmples 5.6 nd 5.7 where the error ws reduced by fctor of bout 15 when x ws hlved. From Proposition 5.9, we cn determine spcing between dt points tht gurntees tht the error is smller thn some given tolernce. Suppose tht f (iv) (x) M, for x [, b].

7 5.2. CUBIC SPLINE INTERPOLATION 103 For ny ɛ > 0 we then hve ( ) 384ɛ 1/4 x = f(x) (Hf)(x) ɛ, for x [, b]. M When ɛ 0, the number ɛ 1/4 goes to zero more slowly thn the term ɛ 1/2 in the corresponding estimte for piecewise liner interpoltion. This mens tht when ɛ becomes smll, we cn usully use lrger x in cubic Hermite interpoltion thn in piecewise liner interpoltion, or equivlently, we generlly need fewer dt points in cubic Hermite interpoltion thn in piecewise liner interpoltion to obtin the sme ccurcy Estimting the derivtives Sometimes we hve function vlues vilble, but no derivtives, nd we still wnt smooth interpolnt. In such cses we cn still use cubic Hermite interpoltion if we cn somehow estimte the derivtives. This cn be done in mny wys, but one common choice is to use the slope of the prbol interpolting the dt t three consecutive dt-points. To find this slope we observe tht the prbol p i such tht p i (x j ) = f(x j ), for j = i 1, i nd i + 1, is given by where We then find tht δ i δ i 1 p i (x) = f(x i 1 ) + (x x i 1 )δ i 1 + (x x i 1 )(x x i ), x i 1 + x i After simplifiction, we obtin δ j = ( f(x j+1 ) f(x j ) ) / x j. p δ i δ i 1 i(x i ) = δ i 1 + x i 1. x i 1 + x i p i(x i ) = x i 1δ i + x i δ i 1 x i 1 + x i, for i = 2,..., m 1, (5.11) nd this we use s n estimte for f (x i ). Using cubic Hermite interpoltion with the choice (5.11) for derivtives is known s cubic Bessel interpoltion. It is equivlent to process known s prbolic blending. The end derivtives f (x 1 ) nd f (x m ) must be estimted seprtely. One possibility is to use the vlue in (5.11) with x 0 = x 3 nd x m+1 = x m Cubic Spline Interpoltion Cubic Hermite interpoltion works well in mny cses, but it is inconvenient tht the derivtives hve to be specified. In Section we sw one wy in which the derivtives cn be estimted from the function vlues. There re mny other wys to estimte the derivtives t the dt points; one possibility is to demnd tht the interpolnt should hve continuous second derivtive t ech interpoltion point. As we shll see in this section, this leds to system of liner equtions for the unknown derivtives so the loclity of the construction is lost, but we gin one more continuous derivtive which is importnt in some pplictions. A surprising property of this interpolnt is tht it hs the smllest second derivtive of ll C 2 -functions tht stisfy the interpoltion conditions. The cubic

8 104 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA spline interpolnt therefore hs number of geometric nd physicl interprettions tht we discuss briefly in Section Our strting point is m points = x 1 < x 2 < < x m = b with corresponding vlues y i = f(x i ). We re looking for piecewise cubic polynomil tht interpoltes the given vlues nd belongs to C 2 [, b]. In this construction, it turns out tht we need two extr conditions to specify the interpolnt uniquely. One of the following boundry conditions is often used. (i) g () = f () nd g (b) = f (b); H(ermite) (ii) g () = g (b) = 0; N(turl) (iii) g is continuous t x 2 nd x m 1. F(ree) (iv) D j g() = D j g(b) for j = 1, 2. P(eriodic) (5.12) The periodic boundry conditions re suitble for closed prmetric curves where f(x 1 ) = f(x m ). In order to formulte the interpoltion problems more precisely, we will define the pproprite spline spces. Since we wnt the splines to hve continuous derivtives up to order two, we know tht ll interior knots must be simple. For the boundry conditions H, N, nd F, we therefore define the 3-regulr knot vectors t H = t N = (t i ) m+6 = (x 1, x 1, x 1, x 1, x 2, x 3,..., x m 1, x m, x m, x m, x m ), t F = (t i ) m+2 = (x 1, x 1, x 1, x 1, x 3, x 4,..., x m 2, x m, x m, x m, x m ). (5.13) This leds to three cubic spline spces S 3,tH, S 3,tN nd S 3,tF, ll of which will hve two continuous derivtives t ech interior knot. Note tht x 2 nd x m 1 re missing in t F. This mens tht ny h S 3,tF will utomticlly stisfy the free boundry conditions. We consider the following interpoltion problems. Problem Let the dt (x i, f(x i )) m with = x 1 < x 2 < < x m = b be given, together with f (x 1 ) nd f (x m ) if they re needed. For Z denoting one of H, N, or F, we seek spline g = g Z = I Z f in the spline spce S 3,tZ, such tht g(x i ) = f(x i ) for i = 1, 2,..., m, nd such tht boundry condition Z holds. We consider first Problem 5.10 in the cse of Hermite boundry conditions. Our im is to show tht the problem hs unique solution, nd this requires tht we study it in some detil. It turns out tht ny solution of Problem 5.10 H hs remrkble property. It is the interpolnt which, in some sense, hs the smllest second derivtive. To formulte this, we need to work with integrls of the splines. An interprettion of these integrls is tht they re generliztions of the dot product or inner product for vectors. Recll tht if u nd v re two vectors in R n, then their inner product is defined by u, v = u v = n u i v i, nd the length or norm of u cn be defined in terms of the inner product s ( n ) 1/2. u = u, u 1/2 = u 2 i

9 5.2. CUBIC SPLINE INTERPOLATION 105 nd The corresponding inner product nd norm for functions re u, v = b u(x)v(x)dx = b uv ( b ) 1/2 ( u = u(t) 2 b dt = u 2) 1/2. It therefore mkes sense to sy tht two functions u nd v re orthogonl if u, v = uv = 0. The first result tht we prove sys tht the error f I H f is orthogonl to fmily of liner splines. Lemm Denote the error in cubic spline interpoltion with Hermite end conditions by e = f I H f, nd let t be the 1-regulr knot vector t = (t i ) m+2 = (x 1, x 1, x 2, x 3,..., x m 1, x m, x m ). Then the second derivtive of e is orthogonl to the spline spce S 1,t. In other words b e (x)h(x) dx = 0, for ll h S 1,t. Proof. Dividing [, b] into the subintervls [x i, x i+1 ] for i = 1,..., m 1, nd using integrtion by prts, we find b e h = m 1 xi+1 m 1 ( e h = e h x i Since e () = e (b) = 0, the first term is zero, m 1 x i+1 x i xi+1 ) e h. x i e h x i+1 = e (b)h(b) e ()h() = 0. (5.14) x i For the second term, we observe tht since h is liner spline, its derivtive is equl to some constnt h i in the subintervl (x i, x i+1 ), nd therefore cn be moved outside the integrl. Becuse of the interpoltion conditions we hve e(x i+1 ) = e(x i ) = 0, so tht m 1 This completes the proof. xi+1 x i e h = m 1 h i xi+1 x i e (x) dx = 0. We cn now show tht the cubic spline interpolnt solves minimiztion problem. In ny minimiztion problem, we must specify the spce over which we minimize. The spce in this cse is E H (f), which is defined in terms of the relted spce E(f) E(f) = { g C 2 [, b] g(x i ) = f(x i ) for i = 1,..., m }, E H (f) = { g E(f) g () = f () nd g (b) = f (b) }. (5.15)

10 106 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA The spce E(f) is the set of ll functions with continuous derivtives up to the second order tht interpolte f t the dt points. If we restrict the derivtives t the ends to coincide with the derivtives of f we obtin E H (f). The following theorem shows tht the second derivtive of cubic interpolting spline hs the smllest second derivtive of ll functions in E H (f). Theorem Suppose tht g = I H f is the solution of Problem 5.10 H. Then b ( g (x) ) b 2 ( dx h (x) ) 2 dx for ll h in EH (f), (5.16) with equlity if nd only if h = g. Proof. Select some h E H (f) nd set e = h g. Then we hve b b h 2 ( = e + g ) b b 2 = e e g + b g 2. (5.17) Since g S 3,tH we hve g S 1,t, where t is the knot vector given in Lemm Since g = I H h = I H f, we hve e = h I H h so we cn pply Lemm 5.11 nd obtin b e g = 0. We conclude tht b h 2 b g 2. To show tht we cn only hve equlity in (5.16) when h = g, suppose tht b h 2 = b g 2. Using (5.17), we observe tht we must hve b e 2 = 0. But since e is continuous, this mens tht we must hve e = 0. Since we lso hve e() = e () = 0, we conclude tht e = 0. This cn be shown by using Tylor s formul Since e = 0, we end up with g = h. e(x) = e() + (x )e () + x e (t)(x t) dt. Lemm 5.11 nd Theorem 5.12 llow us to show tht the Hermite problem hs unique solution. Theorem Problem 5.10 H hs unique solution. Proof. We seek function in S 3,tH such tht g = I H f = m+2 c i B i,3 m+2 j=1 m+2 j=1 c j B j,3 (x i ) = f(x i ), for i = 1,..., m, c j B j,3(x i ) = f (x i ), for i = 1 nd m. (5.18) This is liner system of m + 2 equtions in the m + 2 unknown B-spline coefficients. From liner lgebr we know tht such system hs unique solution if nd only if the

11 5.2. CUBIC SPLINE INTERPOLATION 107 corresponding system with zero right-hnd side only hs the zero solution. This mens tht existence nd uniqueness of the solution will follow if we cn show tht Problem 5.10 H with zero dt only hs the zero solution. Suppose tht g S 3,tH solves Problem 5.10 H with zero dt. Clerly g = 0 is solution. According to Theorem 5.12, ny other solution must lso minimize the integrl of the second derivtive. By the uniqueness ssertion in Theorem 5.12, we conclude tht g = 0 is the only solution. We hve similr results for the nturl cse. Lemm If e = f I N f nd t the knot vector t = (t i ) m = (x 1, x 2, x 3,..., x m 1, x m ), the second derivtive of e is orthogonl to S 1,t, b e (x)h(x) dx = 0, for ll h in S 1,t. Proof. The proof is similr to Lemm h S 1,t now stisfies h() = h(b) = 0. The reltion in (5.14) holds since every Lemm 5.14 llows us to prove tht the cubic spline interpoltion problem with nturl boundry conditions hs unique solution. Theorem Problem 5.10 N hs unique solution g = I N f. The solution is the unique function in C 2 [, b] with the smllest possible second derivtive in the sense tht b ( g (x) ) b 2 ( dx h (x) ) 2 dx, for ll h E(f), with equlity if nd only if h = g. Proof. The proof of Theorem 5.12 crries over to this cse. We only need to observe tht the nturl boundry conditions imply tht g S 1,t. From this it should be cler tht the cubic spline interpolnts with Hermite nd nturl end conditions re extrordinry functions. If we consider ll continuous functions with two continuous derivtives tht interpolte f t the x i, the cubic spline interpolnt with nturl end conditions is the one with the smllest second derivtive in the sense tht the integrl of the squre of the second derivtive is minimized. This explins why the N boundry conditions in (5.12) re clled nturl. If we restrict the interpolnt to hve the sme derivtive s f t the ends, the solution is still cubic spline. For the free end interpolnt we will show existence nd uniqueness in the next section. No minimiztion property is known for this spline Interprettions of cubic spline interpoltion Tody engineers use computers to fit curves through their dt points; this is one of the min pplictions of splines. But splines hve been used for this purpose long before computers were vilble, except tht t tht time the word spline hd different mening. In industries like for exmple ship building, thin flexible ruler ws used to drw curves.

12 108 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA The ruler could be clmped down t fixed dt points nd would then tke on nice smooth shpe tht interpolted the dt nd minimized the bending energy in ccordnce with the physicl lws. This llowed the user to interpolte the dt in visully plesing wy. This flexible ruler ws known s drftmns spline. The physicl lws governing the clssicl spline used by ship designers tell us tht the ruler will tke on shpe tht minimizes the totl bending energy. The linerized bending energy is given by g 2, where g(x) is the position of the centerline of the ruler. Outside the first nd lst fixing points the ruler is unconstrined nd will tke the shpe of stright line. From this we see tht the nturl cubic spline models such linerized ruler. The word spline ws therefore nturl choice for the cubic interpolnts we hve considered here when they were first studied systemticlly in 1940 s. The cubic spline interpolnt lso hs relted, geometric interprettion. From differentil geometry we know tht the curvture of function g(x) is given by κ(x) = g (x) (1 + (g (x)) 2) 3/2. The curvture κ(x) mesures how much the function curves t x nd is importnt in the study of prmetric curves. If we ssume tht 1 + g 2 1 on [, b], then κ(x) g (x). The cubic spline interpolnts I H f nd I N f cn therefore be interpreted s the interpolnts with the smllest linerized curvture Numericl solution nd exmples If we were just presented with the problem of finding the C 2 function tht interpolte given function t some points nd hve the smllest second derivtive, without the knowledge tht we obtined in Section 5.2, we would hve to work very hrd to write relible computer progrm tht could solve the problem. With Theorem 5.15, the most difficult prt of the work hs been done, so tht in order to compute the solution to sy Problem 5.10 H, we only hve to solve the liner system of equtions (5.18). Let us tke closer look t this system. We order the equtions so tht the boundry conditions correspond to the first nd lst eqution, respectively. Becuse of the locl support property of the B-splines, only few unknowns pper in ech eqution, in other words we hve bnded liner system. Indeed, since t i+3 = x i, we see tht only {B j,3 } i+3 j=i cn be nonzero t x i. But we note lso tht x i is locted t the first knot of B i+3,3, which mens tht B i+3,3 (x i ) = 0. Since we lso hvet B j,3 (x 1) = 0 for j 3 nd B j,3 (x m) = 0 for j m, we conclude tht the system cn be written in the tridigonl form α 1 γ 1 β 2 α 2 γ 2 Ac = β m+1 α m+1 γ m+1 β m+2 α m+2 where the elements of A re given by c 1 c 2. c m+1 c m+2 α 1 = B 1,3(x 1 ), α m+2 = B m+2,3(x m ), γ 1 = B 2,3(x 1 ), β m+2 = B m+1,3(x m ), = f (x 1 ) f(x 1 ). f(x m ) f (x m ) β i+1 = B i,3 (x i ), α i+1 = B i+1,3 (x i ), γ i+1 = B i+2,3 (x i ). = f, (5.19) (5.20)

13 5.2. CUBIC SPLINE INTERPOLATION () (b) Figure 5.3. Cubic spline interpoltion to smoothly vrying dt () nd dt with shrp corners (b). The elements of A cn be computed by one of the tringulr lgorithms for B-bses. For H 3 f we hd explicit formuls for the B-spline coefficients tht only involved few function vlues nd derivtives, in other words the pproximtion ws locl. In cubic spline interpoltion the sitution is quite different. All the equtions in (5.19) re coupled nd we hve to solve liner system of equtions. Ech coefficient will therefore in generl depend on ll the given function vlues which mens tht the vlue of the interpolnt t point lso depends on ll the given function vlues. This mens tht cubic spline interpoltion is not locl process. Numericlly it is quite simple to solve (5.19). It follows from the proof of Theorem 5.13 tht the mtrix A is nonsingulr, since otherwise the solution could not be unique. Since it hs tridigonl form it is recommended to use Gussin elimintion. It cn be shown tht the elimintion cn be crried out without chnging the order of the equtions (pivoting), nd detiled error nlysis shows tht this process is numericlly stble. In most cses, the underlying function f is only known through the dt y i = f(x i ), for i = 1,..., m. We cn still use Hermite end conditions if we estimte the end slopes f (x 1 ) nd f (x m ). A simple estimte is f () = d 1 nd f (b) = d 2, where d 1 = f(x 2) f(x 1 ) x 2 x 1 nd d 2 = f(x m) f(x m 1 ) x m x m 1. (5.21) More elborte estimtes like those in Section re of course lso possible. Another possibility is to turn to nturl nd free boundry conditions which lso led to liner systems similr to the one in eqution (5.19), except tht the first nd lst equtions which correspond to the boundry conditions must be chnged ppropritely. For nturl end conditions we know from Theorem 5.15 tht there is unique solution. Existence nd uniqueness of the solution with free end conditions is estblished in Corollry The free end condition is prticulrly ttrctive in B-spline formultion, since by not giving ny knot t x 2 nd x m 1 these conditions tke cre of themselves. The free end conditions work well in mny cses, but extr wiggles cn sometimes occur ner the ends of the rnge. The Hermite conditions give us better control in this respect. Exmple In Figure 5.3 () nd 5.3 (b) we show two exmples of cubic spline interpoltion. In both cses we used the Hermite boundry conditions with the estimte in (5.21) for the slopes. The dt to be interpolted is shown s bullets. Note tht in Figure 5.3 () the interpolnt behves very nicely nd predictbly between the dt points.

14 110 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA In comprison, the interpolnt in Figure 5.3 (b) hs some unexpected wiggles. This is chrcteristic feture of spline interpoltion when the dt hve sudden chnges or shrp corners. For such dt, lest squres pproximtion by splines usully gives better results, see Section Generl Spline Approximtion So fr, we hve minly considered spline pproximtion methods tilored to specific degrees. In prctice, cubic splines re undoubtedly the most common, but there is n obvious dvntge to hve methods vilble for splines of ll degrees. In this section we first consider spline interpoltion for splines of rbitrry degree. The optiml properties of the cubic spline interpolnt cn be generlized to spline interpolnts of ny odd degree, but here we only focus on the prcticl construction of the interpolnt. Lest squres pproximtion, which we study in Section 5.3.2, is completely different pproximtion procedure tht often give better results thn interpoltion, especilly when the dt chnges bruptly like in Figure 1.6 (b) Spline interpoltion Given points (x i, y i ) m, we gin consider the problem of finding spline g such tht g(x i ) = y i, i = 1,..., m. In the previous section we used cubic splines where the knots of the spline were locted t the dt points. This works well if the dt points re firly evenly spced, but cn otherwise give undesirble effects. In such cses the knots should not be chosen t the dt points. However, how to choose good knots in generl is difficult. In some cses we might lso be interested in doing interpoltion with splines of degree higher thn three. We could for exmple be interested in smooth representtion of the second derivtive of f. However, if we wnt f to be continuous, sy, then the degree d must be higher thn three. We therefore consider the following interpoltion problem. Problem Let there be given dt ( ) m x i, y i nd spline spce S d,t whose knot vector t = (t i ) m+d+1 stisfies t i+d+1 > t i, for i = 1,..., m. Find spline g in S d,t such tht m g(x i ) = c j B j,d (x i ) = y i, for i = 1,..., m. (5.22) j=1 The equtions in (5.22) form system of m equtions in m unknowns. In mtrix form these equtions cn be written B 1,d (x 1 )... B m,d (x 1 ) c 1 y 1 Ac = =. = y. (5.23) B 1,d (x m )... B m,d (x m ) c m y m Theorem 5.18 gives necessry nd sufficient conditions for this system to hve unique solution, in other words for A to be nonsingulr. Theorem The mtrix A in (5.23) is nonsingulr if nd only if the digonl elements i,i = B i,d (x i ) re positive for i = 1,... m. Proof. See Theorem 9.6 in Chpter 9.

15 5.3. GENERAL SPLINE APPROXIMATION 111 The condition tht the digonl elements of A should be nonzero cn be written t i < x i+1 < t i+d+1, i = 1, 2,..., m, (5.24) provided we llow x i = t i if t i = = t i+d. Conditions (5.24) re known s the Schoenberg- Whitney nesting conditions. As n ppliction of Theorem 5.18, let us verify tht the coefficient mtrix for cubic spline interpoltion with free end conditions is nonsingulr. Corollry Cubic spline interpoltion with free end conditions (Problem 5.10 F) hs unique solution. Proof. The coefficients of the interpolnt re found by solving liner system of equtions of the form (5.22). Recll tht the knot vector t = t F is given by t = (t i ) m+4 = (x 1, x 1, x 1, x 1, x 3, x 4,..., x m 2, x m, x m, x m, x m ). From this we note tht B 1 (x 1 ) nd B 2 (x 2 ) re both positive. Since t i+2 = x i for i = 3,..., m 2, we lso hve t i < x i 1 < t i+4 for 3 i m 2. The lst two conditions follow similrly, so the coefficient mtrix is nonsingulr. For implementtion of generl spline interpoltion, it is importnt to mke use of the fct tht t most d + 1 B-splines re nonzero for given x, just like we did for cubic spline interpoltion. This mens tht in ny row of the mtrix A in (5.22), t most d + 1 entries re nonzero, nd those entries re consecutive. This gives A bnd structure tht cn be exploited in Gussin elimintion. It cn lso be shown tht nothing is gined by rerrnging the equtions or unknowns in Gussin elimintion, so the equtions cn be solved without pivoting Lest squres pproximtion In this chpter we hve described number of spline pproximtion techniques bsed on interpoltion. If it is n bsolute requirement tht the spline should pss exctly through the dt points, there is no lterntive to interpoltion. But such perfect interpoltion is only possible if ll computtions cn be performed without ny round-off error. In prctice, ll computtions re done with floting point numbers, nd round-off errors re inevitble. A smll error is therefore lwys present nd must be tolerble whenever computers re used for pproximtion. The question is wht is tolerble error? Often the dt re results of mesurements with certin known resolution. To interpolte such dt is not recommended since it mens tht the error is lso pproximted. If it is known tht the underlying function is smooth, it is usully better to use method tht will only pproximte the dt, but pproximte in such wy tht the error t the dt points is minimized. Lest squres pproximtion is generl nd simple pproximtion method for ccomplishing this. The problem cn be formulted s follows. Problem Given dt (x i, y i ) m with x 1 < < x m, positive rel numbers w i for i = 1,..., m, nd n n-dimensionl spline spce S d,t, find spline g in S d,t which solves the minimiztion problem min h S d,t m w i (y i h(x i )) 2. (5.25)

16 112 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA The expression (5.25) tht is minimized is sum of the squres of the errors t ech dt point, weighted by the numbers w i which re clled weights. This explins the nme lest squres pproximtion, or more precisely weighted lest squres pproximtion. If w i is lrge in comprison to the other weights, the error y i h(x i ) will count more in the minimiztion. As the the weight grows, the error t this dt point will go to zero. On the other hnd, if the weight is smll in comprison to the other weights, the error t tht dt point gives little contribution to the totl lest squres devition. If the weight is zero, the pproximtion is completely independent of the dt point. Note tht the ctul vlue of the weights is irrelevnt, it is the reltive size tht mtters. The weights therefore provides us with the opportunity to ttch mesure of confidence to ech dt point. If we know tht y i is very ccurte dt vlue we cn give it lrge weight, while if y i is very inccurte we cn give it smll weight. Note tht it is the reltive size of the weights tht mtters, nturl neutrl vlue is therefore w i = 1. From our experience with interpoltion, we see tht if we choose the spline spce S d,t so tht the number of B-splines equls the number of dt points nd such tht B i (x i ) > 0 for ll i, then the lest squres pproximtion will gree with the interpolnt nd give zero error, t lest in the bsence of round off errors. Since the whole point of introducing the lest squres pproximtion is to void interpoltion of the dt, we must mke sure tht n is smller thn m nd tht the knot vector is pproprite. This ll mens tht the spline spce S d,t must be chosen ppropritely, but this is not esy. Of course we would like the spline spce to be such tht good pproximtion g cn be found. Good, will hve different interprettions for different pplictions. A sttisticin would like g to hve certin sttisticl properties. A designer would like n estheticlly plesing curve, nd mybe some other shpe nd tolernce requirements to be stisfied. In prctice, one often strts with smll spline spce, nd then dds knots in problemtic res until hopefully stisfctory pproximtion is obtined. Different points of view re possible in order to nlyze Problem 5.20 mthemticlly. Our pproch is bsed on liner lgebr. Our tsk is to find the vector c = (c 1,..., c n ) of B-spline coefficients of the spline g solving Problem The following mtrix-vector formultion is convenient. Lemm Problem 5.20 is equivlent to the liner lest squres problem min c R n Ac b 2, where A R m,n nd b R m hve components i,j = w i B j (x i ) nd b i = w i y i, (5.26) nd for ny u = (u 1,..., u m ), u = u u2 m, is the usul Eucliden length of vector in R m.

17 5.3. GENERAL SPLINE APPROXIMATION 113 Proof. Suppose c = (c 1,..., c n ) re the B-spline coefficients of some h S d,t. Then Ac b 2 2 = = = m ( n ) 2 i,j c j b i j=1 m ( n wi B j (x i )c j ) 2 w i y i j=1 m ) 2. w i (h(x i ) y i This shows tht the two minimiztion problems re equivlent. In the next lemm, we collect some fcts bout the generl liner lest squres problem. Recll tht symmetric mtrix N is positive semidefinite if c T Nc 0 for ll c R n, nd positive definite if in ddition c T Nc > 0 for ll nonzero c R n. Lemm Suppose m nd n re positive integers with m n, nd let the mtrix A in R m,n nd the vector b in R m be given. The liner lest squres problem min Ac c R b 2 (5.27) n lwys hs solution c which cn be found by solving the liner set of equtions A T Ac = A T b. (5.28) The coefficient mtrix N = A T A is symmetric nd positive semidefinite. It is positive definite, nd therefore nonsingulr, nd the solution of (5.27) is unique if nd only if A hs linerly independent columns. Proof. Let spn(a) denote the n-dimensionl liner subspce of R m spnned by the columns of A, spn(a) = {Ac c R n }. From bsic liner lgebr we know tht vector b R m cn be written uniquely s sum b = b 1 + b 2, where b 1 is liner combintion of the columns of A so tht b 1 spn(a), nd b 2 is orthogonl to spn(a), i.e., we hve b T 2 d = 0 for ll d in spn(a). Using this decomposition of b, nd the Pythgoren theorem, we hve for ny c R n, Ac b 2 = Ac b 1 b 2 2 = Ac b b 2 2. It follows tht Ac b 2 2 b for ny c Rn, with equlity if Ac = b 1. A c = c such tht Ac = b 1 clerly exists since b 1 is in spn(a), nd c is unique if nd only if A hs linerly independent columns. To derive the liner system for c, we note tht ny c tht is minimizing stisfies Ac b = b 2. Since we lso know tht b 2 is orthogonl to spn(a), we must hve d T (Ac b) = c T 1 A T (Ac b) = 0 for ll d = Ac 1 in spn(a), i.e., for ll c 1 in R n. But this is only possible if A T (Ac b) = 0. This proves (5.28).

18 114 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA () (b) Figure 5.4. Figure () shows the cubic spline interpoltion to the noisy dt of Exmple 5.24, while lest squres pproximtion to the sme dt is shown in (b). The n n-mtrix N = A T A is clerly symmetric nd c T Nc = Ac 2 2 0, (5.29) for ll c R n, so tht N is positive semi-definite. From (5.29) we see tht we cn find nonzero c such tht c T Nc = 0 if nd only if Ac = 0, i.e., if nd only if A hs linerly dependent columns. We conclude tht N is positive definite if nd only if A hs linerly independent columns. Applying these results to Problem 5.20 we obtin. Theorem Problem 5.20 lwys hs solution. The solution is unique if nd only if we cn find sub-sequence (x il ) n l=1 of the dt bsciss such tht B l (x il ) 0 for l = 1,..., n. Proof. By Lemm 5.21 nd Lemm 5.22 we conclude tht Problem 5.20 lwys hs solution, nd the solution is unique if nd only if the mtrix A in Lemm 5.21 hs linerly independent columns. Now A hs linerly independent columns if nd only if we cn find subset of n rows of A such tht the squre submtrix consisting of these rows nd ll columns of A is nonsingulr. But such mtrix is of the form treted in Theorem Therefore, the submtrix is nonsingulr if nd only if the digonl elements re nonzero. But the digonl elements re given by B l (x il ). Theorem 5.23 provides nice condition for checking tht we hve unique lest squres spline pproximtion to given dt set; we just hve to check tht ech B-spline hs its own x il in its support. To find the B-spline coefficients of the pproximtion, we must solve the liner system of equtions (5.28). These equtions re clled the norml equtions of the lest squres system nd cn be solved by Cholesky fctoriztion of bnded mtrix followed by bck substitution. The lest squres problem cn lso be solved by computing QR-fctoriztion of the mtrix A; for both methods we refer to stndrd text on numericl liner lgebr for detils.

19 5.4. THE VARIATION DIMINISHING SPLINE APPROXIMATION 115 Exmple Lest squres pproximtion is especilly pproprite when the dt is known to be noisy. Consider the dt represented s bullets in Figure 5.4 (). These dt were obtined by dding rndom perturbtions in the intervl [ 0.1, 0.1] to the function f(x) = 1. In Figure 5.4 () we show the cubic spline interpolnt (with free end conditions) to the dt, while Figure 5.4 (b) shows the cubic lest squres pproximtion to the sme dt, using no interior knots. We see tht the lest squres pproximtion smoothes out the dt nicely. We lso see tht the cubic spline interpolnt gives nice pproximtion to the given dt, but it lso reproduces the noise tht ws dded rtificilly. Once we hve mde the choice of pproximting the dt in Exmple 5.24 using cubic splines with no interior knots, we hve no chnce of representing the noise in the dt. The flexibility of cubic polynomils is nowhere ner rich enough to represent ll the oscilltions tht we see in Figure 5.4 (), nd this gives us the desired smoothing effect in Figure 5.4 (b). The dvntge of the method of lest squres is tht it gives resonbly simple method for computing resonbly good pproximtion to quite rbitrry dt on quite rbitrry knot vectors. But it is lrgely the knot vector tht decides how much the pproximtion is llowed to oscillte, nd good methods for choosing the knot vector is therefore of fundmentl importnce. Once the knot vector is given there re in fct mny pproximtion methods tht will provide good pproximtions. 5.4 The Vrition Diminishing Spline Approximtion In this section we describe simple, but very useful method for obtining spline pproximtions to function f defined on n intervl [, b]. This method is generliztion of piecewise liner interpoltion nd hs nice shpe preserving behviour. For exmple, if the function f is positive, then the spline pproximtion will lso be positive. Definition Let f be given continuous function on the intervl [, b], let d be given positive integer, nd let t = (t 1,..., t n+d+1 ) be d + 1-regulr knot vector with boundry knots t d+1 = nd t n+1 = b. The spline given by (V f)(x) = n f(t j)b j,d (x) (5.30) j=1 where t j = (t j t j+d )/d re the knot verges, is clled the Vrition Diminishing Spline Approximtion of degree d to f on the knot vector t. The pproximtion method tht ssigns to f the spline pproximtion V f is bout the simplest method of pproximtion tht one cn imgine. Unlike some of the other methods discussed in this chpter there is no need to solve liner sustem. To obtin V f, we simply evlute f t certin points nd use these function vlues s B-spline coefficients directly. Note tht if ll interior knots occur less thn d + 1 times in t, then = t 1 < t 2 <... < t n 1 < t n = b. (5.31) This is becuse t 1 nd t n+d+1 do not occur in the definition of t 1 nd t n so tht ll selections of d consecutive knots must be different. Exmple Suppose tht d = 3 nd tht the interior knots of t re uniform in the intervl [0, 1], sy t = (0, 0, 0, 0, 1/m, 2/m,..., 1 1/m, 1, 1, 1, 1). (5.32) For m 2 we then hve t = (0, 1/(3m), 1/m, 2/m,..., 1 1/m, 1 1/(3m), 1). (5.33)

20 116 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA () (b) Figure 5.5. The exponentil function together with the cubic vrition diminishing pproximtion of Exmple 5.26 in the specil cse m = 5 is shown in (). The error in the pproximtion is shown in (b). Figure 5.5 () shows the cubic vrition diminishing pproximtion to the exponentil function on the knot vector in (5.32) with m = 5, nd the error is shown in Figure 5.5 (b). The error is so smll tht it is difficult to distinguish between the two functions in Figure 5.5 (). The vrition diminishing spline cn lso be used to pproximte functions with singulrities, tht is, functions with discontinuities in derivtive of first or higher orders. Exmple Suppose we wnt to pproximte the function f(x) = 1 e 50 x, x [ 1, 1], (5.34) by cubic spline V f. In order to construct suitble knot vector, we tke closer look t the function, see Figure 5.6 (). The grph of f hs cusp t the origin, so f is discontinuous nd chnges sign there. Our spline pproximtion should therefore lso hve some kind of singulrity t the origin. Recll from Theorem 3.19 tht B-spline cn hve discontinuous first derivtive t knot provided the knot hs multiplicity t lest d. Since we re using cubic splines, we therefore plce triple knot t the origin. The rest of the interior knots re plced uniformly in [ 1, 1]. A suitble knot vector is therefore t = ( 1, 1, 1, 1, 1 + 1/m,..., 1/m, 0, 0, 0, 1/m,..., 1 1/m, 1, 1, 1, 1). (5.35) The integer m is prmeter which is used to control the number of knots nd thereby the ccurcy of the pproximtion. The spline V f is shown in Figure 5.6 () for m = 4 together with the function f itself. The error is shown in Figure 5.6 (b), nd we note tht the error is zero t x = 0, but quite lrge just outside the origin. Figures 5.6 (c) nd 5.6 (d) show the first nd second derivtives of the two functions, respectively. Note tht the sign of f nd its derivtives seem to be preserved by the vrition diminishing spline pproximtion. The vrition diminishing spline pproximtion is very simple procedure for obtining spline pproximtions. In Exmple 5.27 we observed tht the pproximtion hs the sme sign s f everywhere, nd more thn this, even the sign of the first two derivtives is preserved in pssing from f to the pproximtion V f. This is importnt since the sign of the derivtive gives importnt informtion bout the shpe of the grph of the function. A nonnegtive derivtive for exmple, mens tht the function is nondecresing, while nonnegtive second derivtive roughly mens tht the function is convex, in other words it curves in the sme direction everywhere. Approximtion methods tht preserve the sign of the derivtive re therefore importnt in prcticl modelling of curves. We will now study such shpe preservtion in more detil.

21 5.4. THE VARIATION DIMINISHING SPLINE APPROXIMATION () (b) (c) (d) Figure 5.6. Figure () shows the function f(x) = 1 e 50 x (dshed) nd its cubic vrition diminishing spline pproximtion (solid) on the knot vector described in Exmple 5.27, nd the error in the pproximtion is shown in Figure (b). The first derivtive of the two functions is shown in (c), nd the second derivtives in (d).

22 118 CHAPTER 5. SPLINE APPROXIMATION OF FUNCTIONS AND DATA Preservtion of bounds on function Sometimes it is importnt tht the mximum nd minimum vlues of function re preserved under pproximtion. Splines hve some very useful properties in this respect. Lemm Let g be spline in some spline spce S d,t. smllest nd lrgest B-spline coefficients, Then g is bounded by its min{c i } i i c i B i (x) mx{c i }, for ll x R. (5.36) i Proof. Let c mx be the lrgest coefficient. Then we hve c i B i (x) c mx B i (x) = c mx B i (x) c mx, i i i since i B i(x) 1. This proves the second inequlity in (5.36). The proof of the first inequlity is similr. Any plot of spline with its control polygon will confirm the inequlities (5.36), see for exmple Figure Figure 5.7. A cubic spline with its control polygon. extrem of the spline. Note how the extrem of the control polygon bound the With Lemm 5.28 we cn show tht bounds on function re preserved by its vrition diminishing pproximtion. Proposition Let f be function tht stisfies f min f(x) f mx for ll x R. Then the vrition diminishing spline pproximtion to f from some spline spce S d,t hs the sme bounds, f min (V f)(x) f mx for ll x R. (5.37) Proof. Recll tht the B-spline coefficients c i of V f re given by c i = f(t i ) where t i = (t i t i+d )/d.