CHAPTER 8 Quasi-interpolation methods

Transcription

1 CHAPTER 8 Qusi-interpoltion methods In Chpter 5 we considered number of methods for computing spline pproximtions. The strting point for the pproximtion methods is dt set tht is usully discrete nd in the form of function vlues given t set of bscisss. The methods in Chpter 5 roughly fll into two ctegories: globl methods nd locl methods. A globl method is one where ny B-spline coefficient depends on ll initil dt points, wheres locl method is one where B-spline coefficient only depends on dt points tken from the neighbourhood of the support of the corresponding B-spline. Typicl globl methods re cubic spline interpoltion nd lest squres pproximtion, while cubic Hermite interpoltion nd the Schoenberg vrition diminishing spline pproximtion re populr locl methods. In this chpter we re going to describe generl recipe for developing locl spline pproximtion methods. This will enble us to produce n infinite number of pproximtion schemes tht cn be tilored to ny specil needs tht we my hve or tht our given dt set dicttes. In principle, the methods re locl, but by llowing the re of influence for given B-spline coefficient to grow, our generl recipe my even encompss the globl methods in Chpter 5. The recipe we describe produces pproximtion methods known under the collective term qusi-interpoltion methods. Their dvntge is their flexibility nd their simplicity. There is considerble freedom in the recipe to produce tilor-mde pproximtion schemes for initil dt sets with specil structure. Qusi-interpolnts lso llow us to estblish importnt properties of B-splines. In the next chpter we will employ them to study how well given function cn be pproximted by splines, nd to show tht B-splines form stble bsis for splines. 8.1 A generl recipe A spline pproximtion method consists of two min steps: First the degree nd knot vector re determined, nd then the B-spline coefficients of the pproximtion re computed from given dt ccording to some formul. For some methods like spline interpoltion nd lest squres pproximtion, this formul corresponds to the solution of liner system of equtions. In other cses, like cubic Hermite interpoltion nd Schoenberg s Vrition Diminishing spline pproximtion, the formul for the coefficients is given directly in terms of given vlues of the function to be interpolted. 161

2 162 CHAPTER 8. QUASI-INTERPOLATION METHODS The bsic ide The bsic ide behind the construction of qusi-interpolnts is very simple. We focus on how to compute the B-spline coefficients of the pproximtion nd ssume tht the degree nd knot vector re known. The procedure depends on two versions of the locl support property of B-splines tht we know well from erlier chpters: (i) The B-spline B j is nonzero only within the intervl [t j, t j+d+1 ], nd (ii) on the intervl [t µ, t µ+1 ) there re only d + 1 B-splines in S d,t tht re nonzero so spline g in S d,t cn be written s g(x) = µ i=µ d b ib i (x) when x is restricted to this intervl. Suppose we re to compute n pproximtion g = i c ib i in S d,t to given function f. To compute c j we cn select one knot intervl I = [t µ, t µ+1 ] which is subintervl of [t j, t j+d+1 ]. We denote the restriction of f to this intervl by f I nd determine n pproximtion g I = µ i=µ d b ib i to f I. One of the coefficients of g I will be b j nd we fix c j by setting c j = b j. The whole procedure is then repeted until ll the c i hve been determined. It is importnt to note the flexibility of this procedure. In choosing the intervl I we will in generl hve the d + 1 choices µ = j, j,..., j +d (fewer if there re multiple knots). As we shll see below we do not necessrily hve to restrict I to be one knot intervl; ll tht is required is tht I [t µ, t µ+d+1 ] is nonempty. When pproximting f I by g I we hve vst number of possibilities. We my use interpoltion or lest squres pproximtion, or ny other pproximtion method. Suppose we settle for interpoltion, then we hve complete freedom in choosing the interpoltion points within the intervl I. In fct, there is so much freedom tht we cn hve no hope of exploring ll the possibilities. It turns out tht some of this freedom is only pprent to produce useful qusiinterpolnts we hve to enforce certin conditions. With the generl setup described bove, useful restriction is tht if f I should hppen to be polynomil of degree d then g I should reproduce f I, i.e., in this cse we should hve g I = f I. This hs the importnt consequence tht if f is spline in S d,t then the pproximtion g will reproduce f exctly (prt from rounding errors in the numericl computtions). To see why this is the cse, suppose tht f = ˆb i i B i is spline in S d,t. Then f I will be polynomil tht cn be written s f I = µ ˆb i=µ d i B i. Since we hve ssumed tht polynomils will be reproduced we know tht g I = f I so µ i=µ d b ib i = µ i=µ d ˆb i B i, nd by the liner independence of the B-splines involved we conclude tht b i = ˆb i for i = µ d,..., µ. But then we see tht c j = b j = ˆb j so g will gree with f. An pproximtion scheme with the property tht P f = f for ll f in spce S is to reproduce the spce A more detiled description Hopefully, the bsic ide behind the construction of qusi-interpolnts becme cler bove. In this section we describe the construction in some more detil with the generlistions mentioned before. We first write down the generl procedure for determining qusiinterpolnts nd then comment on the different steps fterwrds. Algorithm 8.1 (Construction of qusi-interpolnts). Let the spline spce S d,t of dimension n nd the rel function f defined on the intervl [t d+1, t n+1 ] be given, nd suppose tht t is d + 1-regulr knot vector. To pproximte f from the spce S d,t perform the following steps for j = 1, 2,..., n:

3 8.1. A GENERAL RECIPE Choose subintervl I = [t µ, t ν ] of [t d+1, t n+1 ] with the property tht I (t j, t j+d+1 ) is nonempty, nd let f I denote the restriction of f to this intervl. 2. Choose locl pproximtion method P I nd determine n pproximtion g I to f I, µ g I = P I f I = b i B i, (8.1) on the intervl I. i=ν d 3. Set coefficient j of the globl pproximtion P f to b j, i.e., c j = b j. The spline P f = n c jb j will then be n pproximtion to f. The coefficient c j obviously depends on f nd this dependence on f is often indicted by using the nottion λ j f for c j. This will be our norml nottion in the rest of the chpter. An importnt point to note is tht the restriction S d,t,i of the spline spce S d,t to the intervl I cn be written s liner combintion of the B-splines {B i } µ i=ν d. These re exctly the B-splines whose support intersect the interior of the intervl I, nd by construction, one of them must clerly be B j. This ensures tht the coefficient b j tht is needed in step 3 is computed in step 2. Algorithm 8.1 generlises the simplified procedure in Section in tht I is no longer required to be single knot intervl in [t j, t j+d+1 ]. This gives us considerbly more flexibility in the choice of locl pproximtion methods. Note in prticulr tht the clssicl globl methods re included s specil cses since we my choose I = [t d+1, t n+1 ]. As we mentioned in Section 8.1.1, we do not get good pproximtion methods for free. If P f is going to be decent pproximtion to f we must mke sure tht the locl methods used in step 2 reproduce polynomils or splines. Lemm 8.2. Suppose tht ll the locl methods used in step 2 of Algorithm 8.1 reproduce ll polynomils of some degree d 1 d. Then the globl pproximtion method P will lso reproduce polynomils of degree d 1. If ll the locl methods reproduce ll the splines in S d,t,i then P will reproduce the whole spline spce S d,t. Proof. The proof of both clims follow just s in the specil cse in Section 8.1.1, but let us even so go through the proof of the second clim. We wnt to prove tht if ll the locl methods P I reproduce the locl spline spces S d,t,i nd f is spline in S d,t, then P f = f. If f is in S d,t we clerly hve f = n ˆb i=1 i B i for pproprite coefficients (ˆb i ) n i=1, nd the restriction of f to I cn be represented s f I = µ ˆb i=ν d i B i. Since P I reproduces S d,t,i we will hve P I f I = f I or µ µ b i B i = ˆbi B i. i=ν d i=ν d The liner independence of the B-splines involved over the intervl I then llows us to conclude tht b i = ˆb i for ll indices i involved in this sum. Since j is one the indices we therefore hve c j = b j = ˆb j. When this holds for ll vlues of j we obviously hve P f = f.

4 164 CHAPTER 8. QUASI-INTERPOLATION METHODS The reder should note tht if I is single knot intervl, the locl spline spce S d,t,i reduces to the spce of polynomils of degree d. Therefore, when I is single knot intervl, locl reproduction of polynomils of degree d leds to globl reproduction of the whole spline spce. Why does reproduction of splines or polynomils ensure tht P will be good pproximtion method? We will study this in some detil in Chpter 9, but s is often the cse the bsic ide is simple: The functions we wnt to pproximte re usully nice nd smooth, like the exponentil functions or the trigonometric functions. An importnt property of polynomils is tht they pproximte such smooth functions well, lthough if the intervl becomes wide we my need to use polynomils of high degree. A quntittive mnifesttion of this phenomenon is tht if we perform Tylor expnsion of smooth function, then the error term will be smll, t lest if the degree is high enough. If our pproximtion method reproduces polynomils it will pick up the essentil behviour of the Tylor polynomil, while the pproximtion error will pick up the essence of the error in the Tylor expnsion. The pproximtion method will therefore perform well whenever the error in the Tylor expnsion is smll. If we reproduce spline functions we cn essentilly reproduce Tylor expnsions on ech knot intervl s long s the function we pproximte hs t lest the sme smoothness s the splines in the spline spce we re using. So insted of incresing the polynomil degree becuse we re pproximting over wide intervl, we cn keep the spcing in the knot vector smll nd thereby keep the polynomil degree of the spline low. Another wy to view this is tht by using splines we cn split our function into suitble pieces tht ech cn be pproximted well by polynomils of reltively low degree, even though this is not possible for the complete function. By constructing qusi-interpolnts s outlined bove we obtin pproximtion methods tht ctully utilise this pproximtion power of polynomils on ech subintervl. In this wy we cn produce good pproximtions even to functions tht re only piecewise smooth. 8.2 Some qusi-interpolnts It is high time to try out our new tool for constructing pproximtion methods. Let us see how some simple methods cn be obtined from Algorithm Piecewise liner interpoltion Perhps the simplest, locl pproximtion method is piecewise liner interpoltion. We ssume tht our n-dimensionl spline spce S 1,t is given nd tht t is 2-regulr knot vector. For simplicity we lso ssume tht ll the interior knots re simple. The function f is given on the intervl [t 2, t n+1 ]. To determine c j we choose the locl intervl to be I = [t j, t j+1 ]. In this cse, we hve no interior knots in I so S 1,t,I is the two dimensionl spce of liner polynomils. A bsis for this spce is given by the two liner B-splines B j 1 nd B j, restricted to the intervl I. A nturl cndidte for our locl pproximtion method is interpoltion t t j ndt j+1. On the intervl I, the B-spline B j 1 is stright line with vlue 1 t t j nd vlue 0 t t j+1, while B j is stright line with vlue 0 t t j nd vlue 1 t t j+1. The locl interpolnt cn therefore be written P I 1 f(x) = f(t j )B j 1 (x) + f(t j+1 )B j (x). From Algorithm 8.1 we know tht the coefficient multiplying B j is the one tht should multiply B j lso in our globl pproximtion, in other words c j = λ j f = f(t j+1 ). The

5 8.2. SOME QUASI-INTERPOLANTS 165 globl pproximtion is therefore P 1 f(x) = n f(t j+1 )B j (x). i=1 Since stright line is completely chrcterised by its vlue t two points, the locl pproximtion will lwys give zero error nd therefore reproduce ll liner polynomils. Then we know from Lemm 8.2 tht P 1 will reproduce ll splines S 1,t. This my seem like unnecessry formlism in this simple cse where the conclusions re lmost obvious, but it illustrtes how the construction works in very trnsprent sitution A 3-point qudrtic qusi-interpolnt In our repertoire of pproximtion methods, we only hve one locl, qudrtic method, Schoenberg s vrition diminishing spline. With the qusi-interpolnt construction it is esy to construct lterntive, locl methods. Our strting point is qudrtic spline spce S 2,t bsed on 3-regulr knot vector with distinct interior knots, nd function f to be pproximted by scheme which we denote P 2. The support of the B-spline B j is [t j, t j+3 ], nd we choose our locl intervl s I = [t j+1, t j+2 ]. Since I is one knot intervl, we need locl pproximtion method tht reproduces qudrtic polynomils. One such method is interpoltion t three distinct points. We therefore choose three distinct points x j,0, x j,1 nd x j,2 in I. Some degree of symmetry is lwys good guide so we choose x j,0 = t j+1, x j,1 = t j+1 + t j+2, x j,2 = t j+2. 2 To determine P2 I f we hve to solve the liner system of three equtions in the three unknowns b j 1, b j nd b j+1 given by P I 2 f(x j,k ) = j+1 i=j 1 b i B i (x j,k ) = f(x j,k ), for k = 0, 1, 2. With the id of tool like Mthemtic we cn solve these equtions symboliclly. The result is tht b j = 1 2 ( f(t j+1) + 4f(t j+3/2 ) f(t j+2 ) ), where t j+3/2 = (t j+1 + t j+2 )/2. The expressions for b j 1 nd b j+1 re much more complicted nd involve the knots t j nd t j+3 s well. The simplicity of the expression for b j stems from the fct tht x j,1 ws chosen s the midpoint between t j+1 nd t j+2. The expression for b j is vlid whenever t j+1 < t j+2 which is not the cse for j = 1 nd j = n since t 1 = t 2 = t 3 nd t n+1 = t n+2 = t n+3. But from Lemm 2.12 we know tht ny spline g in S 3,t will interpolte its first nd lst B-spline coefficient t these points so we simply set c 1 = f(t 1 ) nd c n = f(t n+1 ). Hving constructed the locl interpolnts, we hve ll the ingredients necessry to

6 166 CHAPTER 8. QUASI-INTERPOLATION METHODS construct the qusi-interpolnt P 2 f = n λ jfb j, nmely f(t 1 ), when j = 1; λ j f = 1 2 ( f(x j,0) + 4f(x j,1 ) f(x j,2 ), when 1 < j < n; f(t n+1 ), when j = n. Since the locl pproximtion reproduced the locl spline spce (the spce of qudrtic polynomils in this cse), the complete qusi-interpolnt will reproduce the whole spline spce S 2,t A 5-point cubic qusi-interpolnt The most commonly used splines re cubic, so let us construct cubic qusi-interpolnt. We ssume tht the knot vector is 4-regulr nd tht the interior knots re ll distinct. As usul we focus on the coefficient c j = λ j f. It turns out tht the choice I = [t j+1, t j+3 ] is convenient. The locl spline spce S 3,t,I hs dimension 5 nd is spnned by the (restriction of the) B-splines {B i } j+2 i=j 2. We wnt the qusi-interpolnt to reproduce the whole spline spce nd therefore need P I to reproduce S 3,t,I. We wnt to use interpoltion s our locl pproximtion method, nd we know from Chpter 5 tht spline interpoltion reproduces the spline spce s long s it hs unique solution. The solution is unique if the coefficient mtrix of the resulting liner system is nonsingulr, nd from Theorem 5.18 we know tht B-spline coefficient mtrix is nonsingulr if nd only if its digonl is positive. Since the dimension of S 3,t,I is 5 we need 5 interpoltion points. We use the three knots t j+1, t j+2 nd t j+3 nd one point from ech of the knot intervls in I, x j,0 = t j+1, x j,1 (t j+1, t j+2 ), x j,2 = t j+2, x j,3 (t j+2, t j+3 ), x j,4 = t j+3. Our locl interpoltion problem is j+2 i=j 2 b i B i (x j,k ) = f(x j,k ), for k = 0, 1,..., 4. In mtrix-vector form this becomes B j 2 (x j,0 ) B j 1 (x j,0 ) b j 2 f(x j,0 ) B j 2 (x j,1 ) B j 1 (x j,1 ) B j (x j,1 ) B j (x j,1 ) 0 b j 1 B j 2 (x j,2 ) B j 1 (x j,2 ) B j (x j,2 ) B j (x j,2 ) B j (x j,2 ) b j 0 B j 1 (x j,3 ) B j (x j,3 ) B j (x j,3 ) B j (x j,3 ) b j+1 = f(x j,1 ) f(x j,2 ) f(x j,3 ) B j (x j,4 ) B j (x j,4 ) b j+2 f(x j,4 ) when we insert the mtrix entries tht re zero. Becuse of the wy we hve chosen the interpoltion points we see tht ll the entries on the digonl of the coefficient mtrix will be positive so the mtrix is nonsingulr. The locl problem therefore hs unique solution nd will reproduce S 3,t,I. The expression for λ j f is in generl rther complicted, but in the specil cse where the width of the two knot intervls is equl nd x j,2 nd x j,4 re chosen s the midpoints of the two intervls we end up with λ j f = 1 ( f(tj+1 ) 8f(t 6 j+3/2 ) + 20f(t j+2 ) 8f(t j+5/2 ) + f(t j+3 ) )

7 8.2. SOME QUASI-INTERPOLANTS 167 where t j+3/2 = (t j+1 + t j+2 )/2 nd t j+5/2 = (t j+2 + t j+3 )/2. Unfortuntely, this formul is not vlid when j = 1, 2, n 1 or n since then one or both of the knot intervls in I collpse to one point. However, our procedure is sufficiently generl to derive lterntive formuls for computing the first two coefficients. The first vlue of j for which the generl procedure works is j = 3. In this cse I = [t 4, t 6 ] nd our interpoltion problem involves the B-splines {B i } 5 i=1. This mens tht when we solve the locl interpoltion problem we obtin B-spline coefficients multiplying ll of these B-splines, including B 1 nd B 2. There is nothing stopping us from using the sme intervl I for computtion of severl coefficients, so in ddition to obtining λ 3 f from this locl interpolnt, we lso use it s our source for the first two coefficients. In the specil cse when the interior knots re uniformly distributed nd x 3,1 = t 9/2 nd x 3,3 = t 11/2, we find λ 1 f = f(t 4 ), λ 2 f = 1 18( 5f(t4 ) + 40f(t 9/2 ) 36f(t 5 ) + 18f(t 11/2 ) f(t 6 ) ). In generl, the second coefficient will be much more complicted, but the first one will not chnge. This sme procedure cn obviously be used to determine vlues for the lst two coefficients, nd under the sme conditions of uniformly distributed knots nd interpoltion points we find λ n 1 f = 1 18( f(tn 1 ) + 18f(t n 1/2 ) 36f(t n ) + 40f(t n+1/2 ) 5f(t n+1 ) ), λ n f = f(t n+1 ) Some remrks on the constructions In ll our constructions, we hve derived specific formuls for the B-spline coefficients of the qusi-interpolnts in terms of the function f to be pproximted, which mkes it nturl to use the nottion c j = λ j f. To do this, we hd to solve the locl liner system of equtions symboliclly. When the systems re smll this cn be done quite esily with computer lgebr system like Mple or Mthemtic, but the solutions quickly become complicted nd useless unless the knots nd interpoltion points re nicely structured, preferbly with uniform spcing. The dvntge of solving the equtions symboliclly is of course tht we obtin explicit formuls for the coefficients once nd for ll nd cn void solving equtions when we pproximte prticulr function. For generl knots, the locl systems of equtions usully hve to be solved numericlly, but qusi-interpolnts cn nevertheless prove very useful. One sitution is rel-time processing of dt. Suppose we re in sitution where dt re mesured nd need to be fitted with spline in rel time. With globl pproximtion method we would hve to recompute the whole spline ech time we receive new dt. This would be cceptble t the beginning, but s the dt set grows, we would not be ble to compute the new pproximtion quickly enough. We could split the pproximtion into smller pieces t regulr intervls, but qusi-interpolnts seem to be perfect tool for this kind of ppliction. In rel-time ppliction the dt will often be mesured t fixed time intervls, nd s we hve seen it is then esy to construct qusi-interpolnts with explicit formuls for the coefficients. Even if this is not prcticble becuse the explicit expressions re not

8 168 CHAPTER 8. QUASI-INTERPOLATION METHODS vilble or become too complicted, we just hve to solve simple, liner set of equtions to determine ech new coefficient. The importnt fct is tht the size of the system is constnt so tht we cn hndle lmost rbitrrily lrge dt sets, the only limittion being vilble storge spce. Another importnt feture of qusi-interpolnts is their flexibility. In our constructions we hve ssumed tht the function we pproximte cn be evluted t ny point tht we need. This my sometimes be the cse, but often the function is only prtilly known by few discrete, mesured vlues t specific bscisss. The procedure for constructing qusi-interpolnts hs so much inherent freedom tht it cn be dpted in number of wys to virtully ny specific sitution, whether the whole dt set is vilble priori or the pproximtion hs to be produced in rel-time s the dt is generted. 8.3 Qusi-interpolnts re liner opertors Now tht we hve seen some exmples of qusi-interpolnts, let us exmine them from more generl point of view. The bsic ingredient of qusi-interpolnts is the computtion of ech B-spline coefficient, nd we hve hve used the nottion c j = λ j f = λ j (f) to indicte tht ech coefficient depends on f. It is useful to think of λ j s function tht tkes n ordinry function s input nd gives rel number s output; such functions re usully clled functionls. If we go bck nd look t our exmples, we notice tht in ech cse the dependency of our coefficient functionls on f is quite simple: The function vlues occur explicitly in the coefficient expressions nd re not multiplied or operted on in ny wy other thn being dded together nd multiplied by rel numbers. This is fmilir from liner lgebr. Definition 8.3. In the construction of qusi-interpolnts, ech B-spline coefficient is computed by evluting liner functionl. A liner functionl λ is mpping from suitble spce of functions S into the rel numbers R with the property tht if f nd g re two rbitrry functions in S nd α nd β re two rel numbers then λ(αf + βg) = αλf + βλg. Linerity is necessry property of functionl tht is being used to compute B-spline coefficients in the construction of qusi-interpolnts. If one of the coefficient functionls re not liner, then the resulting pproximtion method is not qusi-interpolnt. Linerity of the coefficient functionls leds to linerity of the pproximtion scheme. Lemm 8.4. Any qusi-interpolnt P is liner opertor, i.e., for ny two dmissible functions f nd g nd ny rel numbers α nd β, P (αf + βg) = αp f + βp g. Proof. Suppose tht the liner coefficient functionls re (λ j ) n. Then we hve P (αf + βg) = n n n λ j (αf + βg)b i = α λ j fb i + β λ j gb i = αp f + βp g i=1 i=1 i=1 which demonstrtes the linerity of P.

9 8.4. DIFFERENT KINDS OF LINEAR FUNCTIONALS AND THEIR USES 169 This lemm is simple, but very importnt since there re so mny powerful mthemticl tools vilble to nlyse liner opertors. In Chpter 9 we re going to see how well given function cn be pproximted by splines. We will do this by pplying bsic tools in the nlysis of liner opertors to some specific qusi-interpolnts. 8.4 Different kinds of liner functionls nd their uses In our exmples of qusi-interpolnts in Section 8.2 the coefficient functionls were ll liner combintions of function vlues, but there re other functionls tht cn be useful. In this section we will consider some of these nd how they turn up in pproximtion problems Point functionls Let us strt by recording the form of the functionls tht we hve lredy encountered. The coefficient functionls in Section 8.2 were ll in the form λf = l w i f(x i ) (8.2) i=0 for suitble numbers (w i ) l i=0 nd (x i) l i=0. Functionls of this kind cn be used if procedure is vilble to compute vlues of the function f or if mesured vlues of f t specific points re known. Most of our qusi-interpolnts will be of this kind. Point functionls of this type occur nturlly in t lest two situtions. The first is when the locl pproximtion method is interpoltion, s in our exmples bove. The second is when the locl pproximtion method is discrete lest squres pproximtion. As simple exmple, suppose our spline spce is S 2,t nd tht in determining c j we consider the single knot intervl I = [t j+1, t j+2 ]. Suppose lso tht we hve 10 function vlues t the points (x j,k ) 9 k=0 in this intervl. Since the dimension of S 2,t,I is 3, we cnnot interpolte ll 10 points. The solution is to perform locl lest squres pproximtion nd determine the locl pproximtion by minimising the sum of the squres of the errors, min g S 2,t,I 9 ( g(xj,k ) f(x j,k ) ) 2. k=0 The result is tht c j will be liner combintion of the 10 function vlues, c j = λ j f = 9 w j,k f(x j,k ). k= Derivtive functionls In ddition to function vlues, we cn lso compute derivtives of function t point. Since differentition is liner opertor it is esy to check tht functionl like λf = f (x i ) is liner. The most generl form of derivtive functionl bsed t point tht we will consider is r λf = w k f (k) (x) k=0

10 170 CHAPTER 8. QUASI-INTERPOLATION METHODS where x is suitble point in the domin of f. We will construct qusi-interpolnt bsed on this kind of coefficient functionls in Section By combining derivtive functionls bsed t different points we obtin λf = l r i w i,k f (k) (x i ) i=0 k=0 where ech r i is nonnegtive integer. A typicl functionl of this kind is the divided difference of function when some of the rguments re repeted. Such functionls re fundmentl in interpoltion with polynomils. Recll tht if the sme rgument occurs r + 1 times in divided difference, this signifies tht ll derivtives of order 0, 1,..., r re to be interpolted t the point. Note tht the point functionls bove re derivtive functionls with r i = 0 for ll i Integrl functionls The finl kind of liner functionls tht we will consider re bsed on integrtion. typicl functionl of this kind is A λf = f(x)φ(x) dx (8.3) where φ is some fixed function. Becuse of bsic properties of integrtion, it is esy to check tht this is liner functionl. Just s with point functionls, we cn combine severl functionls like the one in (8.3) together, λf = w 0 f(x)φ 0 (x) dx + w 1 f(x)φ 1 (x) dx + + w l f(x)φ l (x) dx, where (w i ) l i=0 re rel numbers nd {φ i} l i=0 re suitble functions. Note tht the righthnd side of this eqution cn be written in the form (8.3) if we define φ by φ(x) = w 0 φ 0 (x) + w 1 φ 1 (x) + + w l φ l (x). Point functionls cn be considered specil cse of integrl functionls. For if φ ɛ is function tht is positive on the intervl I ɛ = (x i ɛ, x i + ɛ) nd I ɛ φ ɛ = 1, then we know from the men vlue theorem tht I ɛ f(x)φ ɛ (x) dx = f(ξ) for some ξ in I ɛ, s long s f is nicely behved (for exmple continuous) function. If we let ɛ tend to 0 we clerly hve lim f(x)φ ɛ (x) dx = f(x i ), (8.4) ɛ 0 I ɛ so by letting φ in (8.3) be nonnegtive function with smll support round x nd unit integrl we cn come s close to point interpoltion s we wish. If we include the condition tht φ dx = 1, then the nturl interprettion of (8.3) is tht λf gives weighted verge of the function f, with φ(x) giving the weight of the

11 8.4. DIFFERENT KINDS OF LINEAR FUNCTIONALS AND THEIR USES 171 function vlue f(x). A specil cse of this is when φ is the constnt φ(x) = 1/(b ); then λf is the trditionl verge of f. From this point of view the limit (8.4) is quite obvious: if we tke the verge of f over ever smller intervls round x i, the limit must be f(x i ). The functionl f(x) dx is often referred to s the first moment of f. As the nme suggests there re more moments. The i + 1st moment of f is given by f(x)x i dx. Moments of function occur in mny pplictions of mthemtics like physics nd the theory of probbility Preservtion of moments nd interpoltion of liner functionls Interpoltion of function vlues is populr pproximtion method, nd we hve used it repetedly in this book. However, is it good wy to pproximte given function f? Is it not bit hphzrd to pick out few, rther rbitrry, points on the grph of f nd insist tht our pproximtion should reproduce these points exctly nd then ignore ll other informtion bout f? As n exmple of wht cn hppen, suppose tht we re given set of function vlues ( x i, f(x i ) ) m nd tht we use piecewise liner interpoltion i=1 to pproximte the underlying function. If f hs been smpled densely nd we interpolte ll the vlues, we would expect the pproximtion to be good, but consider wht hppens if we interpolte only two of the vlues. In this cse we cnnot expect the resulting stright line to be good pproximtion. If we re only llowed to reproduce two pieces of informtion bout f we would generlly do much better by reproducing its first two moments, i.e., the two integrls f(x) dx nd f(x)x dx, since this would ensure tht the pproximtion would reproduce some of the verge behviour of f. Reproduction of moments is quite esy to ccomplish. If our pproximtion is g, we just hve to ensure tht the conditions g(x)x i dx = f(x)x i dx, i = 0, 1,..., n 1 re enforced if we wnt to reproduce n moments. In fct, this cn be viewed s generlistion of interpoltion if we view interpoltion to be preservtion of the vlues of set of liner functionls (ρ i ) n i=1, ρ i g = ρ i f, for i = 1, 2,..., n. (8.5) When ρ i f = f(x)xi 1 dx for i = 1,..., n we preserve moments, while if ρ i f = f(x i ) for i = 1,..., n we preserve function vlues. Suppose for exmple tht g is required to lie in the liner spce spnned by the bsis {ψ j } n. Then we cn determine coefficients (c j) n so tht g(x) = n c jψ j (x) stisfies the interpoltion conditions (8.5) by inserting this expression for g into (8.5). By exploiting the linerity of the functionls, we end up with the n liner equtions c 1 ρ i (ψ 1 ) + c 2 ρ i (ψ 2 ) + + c n ρ i (ψ n ) = ρ i (f), i = 1,..., n

12 172 CHAPTER 8. QUASI-INTERPOLATION METHODS in the n unknown coefficients (c i ) n i=1. In mtrix-vector form this becomes ρ 1 (ψ 1 ) ρ 1 (ψ 2 ) ρ 1 (ψ n ) c 1 ρ 1 (f) ρ 2 (ψ 1 ) ρ 2 (ψ 2 ) ρ 1 (ψ n ) c = ρ 2 (f).. (8.6) ρ n (ψ 1 ) ρ n (ψ 2 ) ρ n (ψ n ) ρ n (f) A fundmentl property of interpoltion by point functionls is tht the only polynomil of degree d tht interpoltes the vlue 0 t d + 1 points is the zero polynomil. This corresponds to the fct tht when ρ i f = f(x i ) nd ψ i (x) = x i for i = 0,..., d, the mtrix in (8.6) is nonsingulr. Similrly, it turns out tht the only polynomil of degree d whose d + 1 first moments vnish is the zero polynomil, which corresponds to the fct tht the mtrix in (8.6) is nonsingulr when ρ i f = f(x)xi dx nd ψ i (x) = x i for i = 0,..., d. If the equtions (8.6) cn be solved, ech coefficient will be liner combintion of the entries on the right-hnd side, c j = λ j f = w j,1 ρ 1 (f) + w j,2 ρ 2 (f) + + w j,n ρ n (f). We recognise this s (8.2) when the ρ i correspond to point functionls, wheres we hve c j = λ j f = w j,1 f(x) dx + w j,2 f(x)x dx + + w j,n f(x)x n 1 dx = c n f(x) ( w j,1 + w j,2 x + + w j,n x n 1) dx when the ρ i correspond to preservtion of moments Lest squres pproximtion In the discussion of point functionls, we mentioned tht lest squres pproximtion leds to coefficients tht re liner combintions of point functionls when the error is mesured by summing up the squres of the errors t given set of dt points. This is nturlly termed discrete lest squres pproximtion. In continuous lest squres pproximtion we minimise the integrl of the squre of the error. If the function to be pproximted is f nd the pproximtion g is required to lie in liner spce S, we solve the minimistion problem min g S ( f(x) g(x) ) 2 dx. If S is spnned by (ψ i ) n i=1, we cn write g s g = n i=1 c iψ nd the minimistion problem becomes min (c 1,...,c n) R n ( f(x) n 2 c i ψ(x)) dx. To determine the minimum we differentite with respect to ech coefficient nd set the derivtives to zero which leds to the so-clled norml equtions n c i ψ i (x)ψ j (x) dx = i=1 i=1 ψ j (x)f(x) dx, for j = 1,..., n.

13 8.5. ALTERNATIVE WAYS TO CONSTRUCT COEFFICIENT FUNCTIONALS 173 If we use the nottion bove nd introduce the liner functionls ρ i f = ψ i(x)f(x) represented by the bsis functions, we recognise this liner system s n instnce of (8.6). In other words, lest squres pproximtion is nothing but interpoltion of the liner functionls represented by the bsis functions. In prticulr, preservtion of moments corresponds to lest squres pproximtion by polynomils Computtion of integrl functionls In our discussions involving integrl functionls we hve tcitly ssumed tht the vlues of integrls like f(x)ψ(x) dx re redily vilble. This is certinly true if both f nd ψ re polynomils, nd it turns out tht it is lso true if both f nd ψ re splines. However, if f is some generl function, then the integrl cnnot usully be determined exctly, even when ψ i is polynomil. In such situtions we hve to resort to numericl integrtion methods. Numericl integrtion mounts to computing n pproximtion to n integrl by evluting the function to be integrted t certin points, multiplying the function vlues by suitble weights, nd then dding up to obtin the pproximte vlue of the integrl, f(x) dx w 0 f(x 0 ) + w 1 f(x 1 ) + + w l f(x l ). In other words, when it comes to prcticl implementtion of integrl functionls we hve to resort to point functionls. In spite of this, integrl functionls nd continuous lest squres pproximtion re such importnt concepts tht it is well worth while to hve n exct mthemticl description. And it is importnt to remember tht we do hve exct formuls for the integrls of polynomils nd splines. 8.5 Alterntive wys to construct coefficient functionls In Section 8.2 we constructed three qusi-interpolnts by following the generl procedure in Section 8.1. In this section we will deduce two lterntive wys to construct qusiinterpolnts Computtion vi evlution of liner functionls Let us use the 3-point, qudrtic qusi-interpolnt in subsection s n exmple. In this cse we used I = [t j+1, t j+2 ] s the locl intervl for determining c j = λ j f. This ment tht the locl spline spce S 2,t,I become the spce of qudrtic polynomils on I which hs dimension three. This spce is spnned by the three B-splines {B i } j+1 i=j 1 nd interpoltion t the three points t j+1, t j+3/2 = t j+1 + t j+2, t j+2 2 llowed us to determine locl interpolnt g I = j+1 i=j 1 b ib i whose middle coefficient b j we used s λ j f. An lterntive wy to do this is s follows. Since g I is constructed by interpoltion t the three points t j+1, t j+3/2 nd t j+2, we know tht λ j f cn be written in the form λ j f = w 1 f(t j+1 ) + w 2 f(t j+3/2 ) + w 3 f(t j+2 ). (8.7)

14 174 CHAPTER 8. QUASI-INTERPOLATION METHODS We wnt to reproduce the locl spline spce which in this cse is just the spce of qudrtic polynomils. This mens tht (8.7) should be vlid for ll qudrtic polynomils. Reproduction of qudrtic polynomils cn be ccomplished by demnding tht (8.7) should be exct when f is replced by the three elements of bsis for S 2,t,I. The nturl bsis to use in our sitution is the B-spline bsis {B i } j+1 i=j 1. Inserting this, we obtin the system λ j B j 1 = w 1 B j 1 (t j+1 ) + w 2 B j 1 (t j+3/2 ) + w 3 B j 1 (t j+2 ), λ j B j = w 1 B j (t j+1 ) + w 2 B j (t j+3/2 ) + w 3 B j (t j+2 ), λ j B j+1 = w 1 B j+1 (t j+1 ) + w 2 B j+1 (t j+3/2 ) + w 3 B j+1 (t j+2 ). in the three unknowns w 1, w 2 nd w 3. The left-hnd sides of these equtions re esy to determine. Since λ j f denotes the jth B-spline coefficient, it is cler tht λ j B i = δ i,j, i.e., it is 1 when i = j nd 0 otherwise. To determine the right-hnd sides we hve to compute the vlues of the B-splines. For this it is useful to note tht the w j s in eqution (8.7) cnnot involve ny of the knots other thn t j+1 nd t j+2 since generl polynomil knows nothing bout these knots. This mens tht we cn choose the other knots so s to mke life simple for ourselves. The esiest option is to choose the first three knots equl to t j+1 nd the lst three equl to t j+2. But then we re in the Bézier setting, nd we know tht the B-splines in this cse will hve the sme vlues if we choose t j+1 = 0 nd t j+2 = 1. The knots re then (0, 0, 0, 1, 1, 1) which mens tht t j+3/2 = 1/2. If we denote the B-splines on these knots by { B i } 3 i=1, we cn replce B i in (8.5.1) by B i j+2 for i = 1, 2, 3. We cn now simplify (8.5.1) to 0 = w 1 B1 (0) + w 2 B1 (1/2) + w 3 B1 (1), 1 = w 1 B2 (0) + w 2 B2 (1/2) + w 3 B2 (1), 0 = w 1 B3 (0) + w 2 B3 (1/2) + w 3 B3 (1). If we insert the vlues of the B-splines we end up with the system w 1 + w 2 /4 = 0, w 2 /2 = 1, w 2 /4 + w 3 = 0, which hs the solution w 1 = 1/2, w 2 = 2 nd w 3 = 1/2. In conclusion we hve λ j f = f(t j+1) + 4f(t j+3/2 ) f(t j+2 ), 2 s we found in Section This pproch to determining the liner functionl works quite generlly nd is often the esiest wy to compute the weights (w i ) Computtion vi explicit representtion of the locl pproximtion There is third wy to determine the expression for λ j f. For this we write down n explicit expression for the pproximtion g I. Using the 3-point qudrtic qusi-interpolnt s our

15 8.6. TWO QUASI-INTERPOLANTS BASED ON POINT FUNCTIONALS 175 exmple gin, we introduce the bbrevitions = t j+1, b = t j+3/2 nd c = t j+2. We cn write the locl interpolnt g I s g I (x) = (x b)(x c) (x )(x c) (x )(x b) f() + f(b) + ( b)( c) (b )(b c) (c )(c b) f(c), s it is esily verified tht g I then stisfies the three interpoltion conditions g I () = f(), g I (b) = f(b) nd g I (c) = f(c). Wht remins is to write this in terms of the B-spline bsis {B i } j+1 i=j 1 nd pick out coefficient number j. Recll tht we hve the nottion γ j(f) for the jth B-spline coefficient of spline f. Coefficient number j on the left-hnd side is λ j f. On the right, we find the B-spline coefficients of ech of the three polynomils nd dd up. The numertor of the first polynomil is (x b)(x c) = x 2 (b + c)x + bc. To find the jth B-spline of this polynomil, we mke use of Corollry 3.5 which tells tht, when d = 2, we hve γ j (x 2 ) = t j+1 t j+2 = c nd γ j (x) = (t j+1 + t j+2 )/2 = ( + c)/2 = b, respectively. The jth B-spline coefficient of the first polynomil is therefore ( c (b + c)b + bc ) γ j = ( b)( c) c b 2 ( b)( c) (8.8) which simplifies to 1/2 since b = ( + c)/2. Similrly, we find tht the jth B-spline coefficient of the second nd third polynomils re 2 nd 1/2, respectively. The complete jth B-spline coefficient of the right-hnd side of (8.8) is therefore f()/2+2f(b) f(c)/2. In totl, we hve therefore obtined λ j f = γ j (g I ) = f(t j+1) 2 + 2f(t j+3/2 ) f(t j+2), 2 s required. This generl procedure lso works generlly, nd we will see nother exmple of it in Section Two qusi-interpolnts bsed on point functionls In this section we consider two prticulr qusi-interpolnts tht cn be constructed for ny polynomil degree. They my be useful for prcticl pproximtion problems, but we re going to use them to prove specil properties of spline functions in Chpters 9 nd 10. Both qusi-interpolnts re bsed on point functionls: In the first cse ll the points re identicl which leds to derivtive functionls, in the second cse ll the points re distinct A qusi-interpolnt bsed on the Tylor polynomil A very simple locl, polynomil pproximtion is the Tylor polynomil. This leds to qusi-interpolnt bsed on derivtive functionls. Even though we use splines of degree d, our locl pproximtion cn be of lower degree; in Theorem 8.5 this degree is given by r. Theorem 8.5 (de Boor-Fix). Let r be n integer with 0 r d nd let x j be number in [t j, t j+d+1 ] for j = 1,..., n. Consider the qusi-interpolnt Q d,r f = n λ j (f)b j,d, where λ j (f) = 1 d! r ( 1) k D d k ρ j,d (x j )D k f(x j ), (8.9) k=0

16 176 CHAPTER 8. QUASI-INTERPOLATION METHODS nd ρ j,d (y) = (y t j+1 ) (y t j+d ). Then Q d,r reproduces ll polynomils of degree r nd Q d,d reproduces ll splines in S d,t. Proof. To construct Q d,r we let I be the knot intervl tht contins x j nd let the locl pproximtion g I = P I r f be the Tylor polynomil of degree r t the point x j, g I (x) = P I r f(x) = r (x x j ) k D k f(x j ). k! k=0 To construct the liner functionl λ j f, we hve to find the B-spline coefficients of this polynomil. We use the sme pproch s in Section For this Mrsden s identity, (y x) d = n ρ j,d (y)b j,d (x), will be useful. Setting y = x j, we see tht the jth B-spline coefficient of (x j x) d is ρ j,d (x j ). Differentiting Mrsden s identity d k times with respect to y, setting y = x i nd rerrnging, we obtin the jth B-spline coefficient of (x x j ) k /k! s Summing up, we find tht γ j ( (x xj ) k /k! ) = ( 1) k D d k ρ j,d (x j )/d! for k = 0,..., r. λ j (f) = 1 d! r ( 1) k D d k ρ j,d (x j )D k f(x j ). k=0 Since the Tylor polynomil of degree r reproduces polynomils of degree r, we know tht the qusi-interpolnt will do the sme. If r = d, we reproduce polynomils of degree d which gree with the locl spline spce S d,t,i since I is single knot intervl. The qusi-interpolnt therefore reproduces the whole spline spce S d,t in this cse. Exmple 8.6. We find D d ρ j,d (y)/d! = 1, D d 1 ρ j,d (y)/d! = y t j, where t j = tj t j+d. (8.10) d For r = 1 nd x j = t j we therefore obtin Q d,r f = nx f(t j )B j,d which is the Vrition Diminishing spline pproximtion. For d = r = 2 we obtin Q 2,2f = nx f(xj) (x j t j+3/2 )Df(x j) (xj tj+1)(xj tj+2)d2 f(x j) B j,2. (8.11) while for d = r = 3 nd x j = t j+2 we obtin Q 3,3f = nx 1 f(tj+2)+ 3 (tj+3 2tj+2 +tj+1)df(tj+2) 1 6 (tj+3 tj+2)(tj+2 tj+1)d2 f(t j+2) B j,3. (8.12) We leve the detiled derivtion s problem for the reder.

17 8.6. TWO QUASI-INTERPOLANTS BASED ON POINT FUNCTIONALS 177 Since Q d,d f = f for ll f S d,t it follows tht the coefficients of spline f = n c jb j,d cn be written in the form c j = 1 d! d ( 1) k D d k ρ j,d (x j )D k f(x j ), for j = 1,..., n, (8.13) k=0 where x j is ny number in [t j, t j+d+1 ] Qusi-interpolnts bsed on evlution Another nturl clss of liner functionls is the one where ech λ j used to define Q is constructed by evluting the dt t r + 1 distinct points t j x j,0 < x j,1 < < x j,r t j+d+1 (8.14) locted in the support [t j, t j+d+1 ] of the B-spline B j,d for j = 1,..., n. We consider the qusi-interpolnt n P d,r f = λ j,r (f)b j,d, (8.15) where λ j,r (f) = r w j,k f(x j,k ). (8.16) k=0 From the preceding theory we know how to choose the constnts w j,k so tht P d,r f = f for ll f π r. Theorem 8.7. Let S d,t be spline spce with d + 1-regulr knot vector t = (t i ) n+d+1 i=1. Let (x j,k ) r k=0 be l + 1 distinct points in [t j, t j+d+1 ] for j = 1,..., n, nd let w j,k be the jth B-spline coefficient of the polynomil p j,k (x) = r r=0 r k x x j,r x j,k x j,r. Then P d,r f = f for ll f π r, nd if r = d nd ll the numbers (x j,k ) r k=0 lie in one subintervl t j t lj x j,0 < x j,1 < < x j,r t lj +1 t j+d+1 (8.17) then P d,d f = f for ll f S d,t. Proof. It is not hrd to see tht p j,k (x j,i ) = δ k,i, k, i = 0,..., r so tht the polynomil r Pd,r I f(x) = p j,k (x)f(x j,k ) stisfies the interpoltion conditions Pd,r I f(x j,r) = f(x j,r ) for ll j nd r. therefore follows from the generl recipe. k=0 The result

18 178 CHAPTER 8. QUASI-INTERPOLATION METHODS In order to give exmples of qusi-interpolnts bsed on evlution we need to know the B-spline coefficients of the polynomils p j,k. We will return to this in more detil in Chpter 9, see (9.14) in the cse r = d. A similr formul cn be given for r < d. Exmple 8.8. For r = 1 we hve nd (8.15) tkes the form P d,1 f = p j,0(x) = nx xj,1 x x j,1 x j,0, p j,1(x) = x xj,0 x j,1 x j,0 xj,1 t j x j,1 x j,0 f(x j,0) + t j x j,0 x j,1 x j,0 f(x j,1) B j,d. (8.18) This qusi-interpolnt reproduces stright lines for ny choice of t j x j,0 < x j,1 t j+d+1. If we choose x j,0 = t j the method simplifies to P d,1 f = nx This is gin the Vrition diminishing method of Schoenberg. f(t j )B j,d. (8.19) Exercises for Chpter In this exercise we ssume tht the points (x i,k ) nd the spline spce S d,t re s in Theorem 8.7. ) Show tht for r = d = 2 n [ (tj+1 x j,1 )(t j+2 x j,2 ) + (t j+2 x j,1 )(t j+1 x j,2 ) P 2,2 f = f(x j,0 ) 2(x j,0 x j,1 )(x j,0 x j,2 ) + (t j+1 x j,0 )(t j+2 x j,2 ) + (t j+2 x j,0 )(t j+1 x j,2 ) f(x j,1 ) 2(x j,1 x j,0 )(x j,1 x j,2 ) + (t ] j+1 x j,0 )(t j+2 x j,1 ) + (t j+2 x j,0 )(t j+1 x j,1 ) f(x j,2 ) 2(x j,2 x j,0 )(x j,2 x j,1 ) B j,2 (8.20) b) Show tht (8.20) reduces to the opertor (9.4) for suitble choice of (x j,k ) 2 k= Derive the following opertors Q d,l nd show tht they re exct for π r for the indicted r. Agin we the points (x j,k ) nd the spline spce S d,t re is in Theorem 8.7. Which of the opertors reproduce the whole spline spce? ) Q d,0 f = n f(x j)b j,d, (r = 0). b) Q d,1 f = n [ f(xj ) + (t j x j )Df(x j ) ] B j,d, (r = 1). c) Q d,1 f = n f(t j )B j,d, (r = 1). d) Q 2,2 f = n [ f(xj ) (x j t j+3/2 )Df(x j ) (x j t j+1 )(x j t j+2 )D 2 f(x j ) ] B j,2, (r=2).

19 8.6. TWO QUASI-INTERPOLANTS BASED ON POINT FUNCTIONALS 179 e) Q 2,2 f = n [ f(tj+3/2 ) 1 2 (t j+2 t j+1 ) 2 D 2 f(t j+3/2 ) ] B j,2, (r = 2). f) Q 3,3 f = n [ f(tj+2 ) (t j+3 2t j+2 + t j+1 )Df(t j+2 ) 1 6 (t j+3 t j+2 )(t j+2 t j+1 )D 2 f(t j+2 ) ] B j,3, (r = 3).