Construction of ECT-B-splines, a survey

Transcription

1 Annaes Mathematicae et Informaticae 32 (2005) pp Construction of ECT-B-spines, a survey Günter W. Mühbach a and Yuehong Tang b a Institut für Angewandte Mathematik, Universität Hannover, Germany e-mai: mb@ifam.uni-hannover.de b Department of Mathematics, Nanjing University of Aeronautics and Astronautics P. R. China Abstract s-dimensiona generaized poynomias are inear combinations of functions forming an ECT-system on a compact interva with coefficients from R s. ECT-spine curves in R s are constructed by gueing together at interva endpoints generaized poynomias generated from different oca ECT-systems via connection matrices. If they are nonsinguar, ower trianguar and totay positive there is a basis of the space of 1-dimensiona ECT-spines consisting of functions having minima compact supports normaized to form a nonnegative partition of unity. Its functions are caed ECT-B-spines. One way (which is semiconstructiona) to prove existence of such a basis is based upon zero bounds for ECT-spines. A constructiona proof is based upon a definition of ECT-B-spines by generaized divided differences extending Schoenberg s cassica construction of ordinary poynomia B-spines. This fact epains why ECT-B-spines share many properties with ordinary poynomia B-spines. In this paper we survey such constructiona aspects of ECT-spines which in particuar situations reduce to cassica resuts. Key Words: ECT-systems, ECT-B-spines, ECT-spine curves, de-boor agorithm AMS Cassification Number: 41A15, 41A05 1. ECT-systems and their duas, ret- and ET-systems Let J be a nontrivia compact subinterva of the rea ine R. A system of functions U = (u 1,..., u n ) in C n 1 (J; R) is caed an extended Tchebycheff system supported in part by INTAS

2 96 G. W. Mühbach and Y. Tang (ET-system, for short) of order n on J provided for a T = (t 1,..., t n ), t 1... t n, t j J, V u 1,..., u n t 1,..., t n := det (D ν j r u i (t j )) i,j=1,...,n > 0 r with ν j := max{ : t j = t j 1 =... = t j t 1 }, j = 1,..., n, (1.1) f(x+h) f(x) where Df(x) := im h 0 h denotes the operator of differentiation appropriatey one sided at an endpoint of J. Then span U wi be caed an ET-space of dimension n on J. An ET-system U = (u 1,..., u n ) is caed compete or an ECTsystem provided (u 1,..., u k ) is an ET-system of order k on J for k = 1,..., n. The foowing characterization of ECT-systems is we known [8] p. 376f, [30] p. 364: Theorem 1.1. Let u 1,..., u n be of cass C n 1 (J; R). Then the foowing assertions are equivaent: (i) (u 1,..., u n ) is an ECT-system of order n on J. (ii) A Wronskian determinants W (u 1,..., u k )(x) = det ( D j 1 u i (x) ) j=1,...,k i=1,...,k > 0 k = 1,..., n; x J are positive on J. (iii) There exist positive weight functions w j C n j (J; R), j = 1,..., n, and for every c J coefficients c j,i R such that u j (x) = w 1 (x) j 1 + x c t2 t3 tj 1 w 2 (t 2 ) w 3 (t 3 )... w j (t j ) dt j... dt 2 (1.2) c c c c j,i u i (x), j = 1,..., n; x J. i=1 Ceary, the functions s j (x, c) := u j (x) j 1 i=1 c j,i u i (x) j = 1,..., n satisfy s j (x, c) = w 1 (x) h j 1 (x, c; w 2,..., w j ) j = 1,..., n (1.3) where h 0 (x, c) := 1 and for 1 m n h m (x, c; w 1,..., w m ) := x c w 1 (t) h m 1 (t, c; w 2,..., w m )dt. The system (1.3) (s 1,..., s n ) forms a specia basis of span{u 1,..., u n ) which we ca an ECT-system in canonica form with respect to c.

3 Construction of ECT-B-spines, a survey 97 Exampe 1.2. If w j = 1 for j = 1,..., n where 1 denotes the constant function equa to one then (x c)m h m (x, c; 1,..., 1) = m! (x c)j 1 s j (x, c) = (j 1)! m = 0,..., n j = 1,..., n and and span{s 1,..., s n } = π n 1, the space of ordinary poynomias of degree n 1 or of order n at most. Exampe 1.3. (cf. aso [3], [31]) If n 3 and w j = 1 for j = 1,..., n 2, w n 1 (x) = (n 2)! (x a + ε) n 1, w (n 1)(b a + 2ε)(x a + ε)n 2 n(x) = (b + ε x) n with ε > 0 a parameter, then for any c [a, b] s j (x, c) = s n 1 (x, c) = s n (x, c) = (x c)j 1, j = 1,..., n 2 (1.4) (j 1)! (x c) n 2 (x a + ε)(c a + ε) n 2 (1.5) (x c) n 1 (b a + 2ε) (x a + ε)(b + ε x)(b + ε c) n 1 (1.6) is a Cauchy-Vandermonde-system in canonica form with respet to c whose first n 2 functions are poynomias and the ast two are proper rationa functions, s n 1 having a poe of order 1 at x = a ε and s n having poes of order 1 at x = a ε and at x = b + ε. Associated with an ECT-system (1.2) or (1.3) are the inear differentia operators D 0 u = u, D j u = D( u w j ) j = 1,..., n ˆL j u = D j D 0 u j = 0,..., n L j u = 1 ˆLj u w j+1 j = 0,..., n 1. For µ N by L[f](t) := (L 0 f(t),..., L µ 1 f(t)) T we denote the ECT-derivative vector of dimension µ of a sufficienty smooth function f. Aso, we wi use the imits L µ [f](t ) := im τ t 0 Lµ [f](τ), L µ [f](t+) := im τ t+0 Lµ [f](τ). Obviousy, ker ˆL j = span{u 1,..., u j }, j = 1,..., n, and L j s j+1 (x, c) = 1 j = 0,..., n 1,

4 98 G. W. Mühbach and Y. Tang L j s +1 (c, c) = δ j, j, = 0,..., n 1. There is a Tayor s Theorem with respect to ECT-systems. The initia vaue probem ˆL n u(x) = f(x), x J L j u(c) = c j, j = 0,..., n 1, with f C(J; R) and c j R given, has the soution n 1 u(x) = c j s j+1 (x, c) + j=0 x c f(t)s n (x, t)dt. (1.7) Associated with any ECT-system U = (s j ) n j=1 of order n on J in canonica form with respect to c J with weights w 1,..., w n its dua canonica system U = (s i )n i=1 with respect to c J is defined by s j,n(x, c) := h j 1 (x, c; w n,..., w n+2 j ) j = 1,..., n. (1.8) It is again an ECT-system of order n on J with weights (w 1,..., w n) = (1, w n,..., w 2 ) provided w j C max{n j,j 2} (J; R), j = 2,..., n. (1.9) Assuming this, with the dua canonica ECT-system with respect to c associated are the inear differentia operators D0f = f, D1f = Df, Dj f f = D( ), j = 2,..., n w n+2 j ˆL j f = Dj D0f, j = 0,..., n, L 0f = f, L 1 j f = ˆL w j f, j = 1,..., n. n+1 j The function g(x, y) := { w 1 (x)h n 1 (x, y; w 2,..., w n ) x y 0 otherwise has the characteristic behaviour of a Green s function for the differentia operator L n 1 acting on the varabe x, i.e. In particuar, for x, y, c J L n 1 g n (x, y) x=y = 0 L n 1 g n (x, y) x=y+ = L n 1 s n (x, y) x=y = 1. h(x, y) := s n (x, y) = w 1 (x)h n 1 (x, y; w 2,..., w n )

5 Construction of ECT-B-spines, a survey 99 n = ( 1) n k s k (x, c)s n+1 k,n(y, c) (1.10) k=1 = ( 1) n 1 w 1 (x)h n 1 (y, x; w n,..., w 2 ) = ( 1) n 1 w 1 (x)s n,n(y, x) where the right hand side of (1.10) is independent of c [10]. Exampe 1.1. (continued) If w 1 =... = w n = 1, then s (x c)j 1 j (x, c) = j = 1,..., n, and (1.10) reduces to the Binomia Theorem (j 1)!, s n (x, y) = h(x, y) = (x y)n 1 (n 1)! = n n k (x c)k 1 ( 1) (k 1)! k=1 (y c)n k. (n k)! Exampe 1.2. (continued, cf. [31]) If n 3 and the weight functions are taken as in Exampe 1.2 then (w 1,..., w n) = (1, w n, w n 1,..., w 2 ), and if for any c [a, b] then γ(k, n, c) := (n 2)! (c a + ε)k 1 n (k 3)! (n 1)! δ(k, n) := (b a + 2ε) (k 3)! ( λ(ν, n) := ( 1) n 1 ν n 1 ν µ(k, n, ν, c) := 1 ( k 3 ν n ν 1 k 3 κ=0 ( k 3 κ ) ( 1) k 3 κ n 2 κ ) (b a + 2ε) ν, 1 ν n 1 ) ν+k n 2 i=0 ( 1) k i ( ν + k n 2 i i (c a + ε)ν+k n 2 i (b a + 2ε), n k 2 ν n 1 ν 1 i 1 ψ ν := ψ ν (x, b, c, ε) := (b + ε x) ν 1 (b + ε c) ν, 1 ν n 1 and for 3 k n s 1,n(x, c) = 1 n 1 s 2,n(x, c) = ψ ν α(2, n, ν, c), α(2, n, ν, c) = λ(ν, n) (1.11) ν=1 ) n 1 s k,n(x, c) = ψ ν α(k, n, ν, c) (1.12) ν=1

6 100 G. W. Mühbach and Y. Tang where α(k, n, ν, c) = { γ(k, n, c)λ(ν, n) 1 ν n k + 1 γ(k, n, c)λ(ν, n) + δ(k, n)µ(k, n, ν, c) n k + 2 ν n 1. The representations (1.11) and (1.12) are proved by cacuating the integras according to the definition of the dua system in its canonica form with respect to c. In exampe 1.2 according to (1.10) h(x, y) = ( 1) n 1 s n,n(y, x). Let J be a subinterva of the rea ine R that is open to the right. For n N 0 et C n r (J; R) := {f C(J; R) : for every x J and for ν = 1,..., n there exists the right derivative of f of order ν at x and J x D ν r f(x) is right continuous}. A system of functions U = (u 1,..., u n ) in Cr n 1 (J; R) is caed a right-sided extended Tchebycheff system (ret-system, for short) of order n on J provided for a T = (t 1,..., t n ), t 1... t n, t j J, V u 1,..., u n t 1,..., t n := det (Dr νj u i (t j )) i,j=1,...,n > 0 r f(x+h) f(x) with ν j defined by (1.1) where D r f(x) := im h 0+ h denotes the operator of ordinary right differentiation. Then span U wi be caed an ret-space of dimension n on J. If q span U where U is an ret-system of order n on J, a point x 0 J is caed a zero of q of right mutipicity ν 0 iff q(x 0 ) = 0, Drq(x 1 0 ) = 0,..., D ν 0 1 r q(x 0 ) = 0, Dr ν0 q(x 0 ) 0. The foowing characterization of ret-spaces is an immediate consequence of the Aternative Theorem of Linear Agebra, as is the corresponding we known characterization for ET-spaces (cf. [8], p. 376). Theorem 1.4. (i) (u 1,..., u n 1, u n ) or (u 1,..., u n 1, u n ) is an ret-sytem of order n on J. (ii) Every nontrivia eement of span(u 1,..., u n ) has at most n 1 zeros in J counting right mutipicities. (iii) Every probem of right sided Hermite interpoation given points t 1... t n in J, given f Cr n 1 (J; R), H(U, T +, f) : find q span U such that Dr νj q(t j ) = Dr νj f(t j ) j = 1,..., n has a unique soution. (1.13)

7 Construction of ECT-B-spines, a survey 101 Anaogousy, eft sided ET-systems and ET-spaces and reated concepts as the probem of eft sided Hermite interpoation H(U, T, f) are defined. In the anaysis of dua functionas to ECT-B-spines naturay certain ret-and ET-spaces arise (see (5.1) and (5.2) beow) that are no ET-spaces. If U = (u 1,..., u n ) is an ret-system on J then the eading coefficient (that before u n ) of the unique q span U that soves H(U, T +, f) is caed the right sided generaized divided difference of f with respect to u 1,..., u n and with nodes t 1,..., t n. By Cramer s rue it is [ ] u1,..., u V u 1,...,u n 1,f r n t 1,...,t n 1,t n f = t 1,..., t n r V u 1,...,u n 1,u n r. t 1,...,t n 1,t n Deveoping the numerator determinant aong its ast coumn one sees V u 1,...,u n 1 r t 1,...,t n 1 [ ] u1,..., u n n f = c j D ν j r f(t j ), c n = t 1,..., t n r j=1 V u1,...,un 1,un t 1,...,t n 1,t n r (1.14) with coefficients c j that do not depend on f. For ET- or ET-systems we use simiar notations with the suffix r repaced by or omitted, respectivey. It is known [17] that if (u 1,..., u n+1 ), (u 1,..., u n ) are ECT-systems, and, if n 2, aso (u 1,..., u n 1 ) is an ECT-system, then if t 1 t n+1 [ u1,..., u n+1 t 1,..., t n+1 ] f = [ u1,...,u n ] t 2,...,t n+1 f [ ] u1,...,u n t 2,...,t n+1 u n+1 [ u1,...,u n ] t 1,...,t n f [ ]. u1,...,u n t 1,...,t n u n+1 This formua hods for the right or eft sided generaized divided differences as we [25]. 2. rect-spines; the spaces S n (U, A +, M, X) and S n (U, A +, ξ ext ) Assume that x is a rea number and that in nontrivia cosed intervas J 0 = [a, x] and J 1 = [x, b] eft and right to x there are given two ECT-systems of order n with weights w [i] j L [i] j U [0] := U n [0] := (u [0] 1,..., u[0] n ), U [1] := U n [1] := (u [1] 1,..., u[1] n ), (j = 1,..., n; i = 0, 1) and associated inear differentia operators (j = 0,..., n 1; i = 0, 1), correspondingy. Suppose that µ is an integer, 0 µ n, and that A is a square (n µ) dimensiona rea matrix which is nonsinguar. A function s : [a, b] R such that s [a,x) span U [0] and s [x,b] span U [1] and L [1]n µ [s](x+) = A L [0]n µ [s](x ) (2.1)

8 102 G. W. Mühbach and Y. Tang where L [i]n µ [s](t) (i = 0, 1) denote the ECT-derivative vectors of s at t of dimension n µ is caed (U [0], U [1], A) smooth of order n µ at x. The equations (2.1) are caed the connection equations of s at the knot x and A is caed a connection matrix at x. We aow 0 µ n where in case µ = n there is no condition on s at x. In case µ = 0 the knot x is a knot with no freedom. If 1 µ n at x, given s on [a, x), there are µ degrees of freedom in extending s to [x, b] as a function beonging to span U [1] such that s Cr n 1 ([a, b]; R). Symmetricay, if 1 µ n at x, given s on (x, b], there are µ degrees of freedom in extending s to [a, x] as a function beonging to span U [0] such that s C n 1 ([a, b]; R). It shoud be observed that (U [0], U [1], A) smoothness in genera does not impy smoothness in the ordinary sense. But it is not hard to give conditions that a function being (U [0], U [1], A) smooth at x of order n µ is smooth at x of order m in the usua sense [31]. Let [a, b] R be either a nontrivia compact interva or the rea ine. By X we denote a finite or a bi-infinite partition of [a, b] respectivey, i.e. X = {x 0,..., x k+1 } with a = x 0 < x 1 <... < x k+1 = b or X = (x i ) i Z with... < x 1 < x 0 < x 1 <... and im i ± x i = ±. The points of X which are not endpoints are caed inner knots and endpoints are caed auxiiary knots. The index sets for inner knots are { {1,..., k} if X = {x 0,..., x k+1 } K X := Z if X = (x i ) i Z. In any case by = (J i ), J i := [x i, x i+1 ) and ˇ = ( ˇJ i ), ˇJi := (x i, x i+1 ] for a i except the ast resp. first we denote the corresponding partition of [a, b] into subintervas caed r- resp. -knot intervas where in case of a finite partition of a compact interva the ast r resp. first knot interva is J k := [x k, x k+1 ] resp. ˇJ 0 = [x 0, x 1 ]. Assume that on each cosed interva Ji = [x i, x i+1 ] the system U [i] n = ( u [i] 1,..., u[i] n is an ECT-system of order n with associated weight functions ) (2.2) w [i] j C n j ( J i ; (0, )), j = 1,..., n (2.3) and associated inear differentia operators L [i] j and ECT-derivative vectors ( ) T L [i]µ [f](t) = L [i] 0 f(t),..., L[i] µ 1 f(t) of dimension µ. By U = U n = ( U [i]) we denote the sequence of ECT-systems. Assume that corresponding to the inner knots we are given a sequence of integers M = (µ i ), 0 i µ i n, and a sequence of nonsinguar matrices A = A n = (A [i] ), A [i] R (n µ i) (n µ i ).

9 Construction of ECT-B-spines, a survey 103 A function s : [a, b] R is caed an rect- resp. ECT-spine function on [a, b] with respect to the generating sequences U, A, M, X provided s Ji span U [i] resp. s ˇJi span U [i] for a i and s is (U [i 1], U [i], A [i] ) smooth at x i for a inner knots. (2.4) The sets of a such functions wi be denoted by S n (U, A, M, X) and Š n (U, A, M, X), respectivey. Ceary, every rect-spine function is right continous everywhere and jumps may occur ony at the knots. If a ECT-systems U [i] have the first weight function and a connection matrices A [i] have the form w [i] 1 (x) = 1 x J i, for a i (2.5) A [i] = diag (1, Ā[i] ) (2.6) where µ i n 1 and Ā[i] R (n 1 µi) (n 1 µi) is nonsinguar for a i then S n (U, A, M, X) and Šn(U, A, M, X) C([a, b]; R) and both spaces contain the constant functions. In the seque we sha treat rect-spaces ony. Ceary, every resut for rectspines has an anaogue for ECT-spines. Under the assumptions (2.5), (2.6) and that A = A + := (A [i] ) i where for every i the connection matrix A [i] is nonsinguar, ower trianguar, totay positive (2.7) it is possibe to construct for the space S n (U, A +, M, X) a oca support basis (N j ) that is normaized to form a nonnegative partition of unity. In order to give the definitions the foowing notation is usefu. For any partition X = (x i ) of [a, b], finite or biinfinite, with corresponding sequence of mutipicities of inner knots M = (µ i ) such that 1 µ i n for a i, we denote by ξ resp. by ξ ext the weaky increasing sequence of inner resp. of a knots where auxiiary knots by definition have mutipicity n, each repeated according to its mutipicity, the enumeration being fixed by the convention ξ 1 = ξ 2 =... = ξ µ1 = x 1. In this case we wi aso use the notation S n (U, A +, M, X) = S n (U, A +, ξ ext ). By ϕ : j i j we denote the mapping which assigns to each ξ j the unique knot such that ξ j = x ij. Then x ij X = ϕ(ξ ext ), M ext := (µ i ) with µ i = card ϕ 1 ({x i }). It wi be convenient to use the index set { J ϕ = Jϕ n { n + 1,..., µ} if [a, b] is compact := Z if [a, b] = R.

10 104 G. W. Mühbach and Y. Tang Observe that the sequences ξ or ξ ext are we defined as nonvoid sequences of µ := µ i terms aso in case 0 µ i n for a i provided 1 µ 1 n. Ony in case that a inner knots have mutipicities zero, M = (0) i, we have ξ = ( ), a void sequence. Remark 2.1. The space S n (U, A +, M, X) was introduced by Barry [1], p Barry has constructed de Boor-Fix functionas first and used them to derive existence of a oca support basis for this space. ECT-spines are studied from a bossom point of view by Mazure [13],[14],[16] and Pottmann [15],[27] and more recenty by Prautzsch [28], and from a constructive point of view by Mühbach [23],[24]. Cardina ECT-spines with simpe knots are discussed in [31]. Remark 2.2. If U [i] = U n Ji where U n is a fixed goba ECT-system of order n on [a, b] and A [i] is the (n µ i ) dimensiona identity matrix then S n (U, A, M, X) is the space of Tchebycheff spines of order n on [a, b] with knots x 1,..., x k of mutipicities µ 1,..., µ k, respectivey. Remark 2.3. If U [i] = (1, x,..., x n 1 ) Ji for i = 0,..., k then S n = S n (U, A, M, X) = S n (x 1,..., x k A [1],..., A [k] ) is the space of piecewise ordinary poynomias of order n generated by connection matrices A [i] considered by Dyn and Micchei [4], p. 321, and by Barry et a [2]. If moreover each A [i] is an identity matrix then S n is the we known Schoenberg space of ordinary poynomia spine functions of order n with knots x i of mutiicity µ i, i = 1,..., k. According to the definitions given an rect-spine s S n (U, A, M, X) may be represented by s = i n j=1 c [i] j u [i] j meaning that s Ji = n j=1 c [i] j u [i] j, a i with coefficients c [i] j (j = 1,..., n µ i ) that are reated by the connection equations (2.4). There remain µ i degrees of freedom for s right to x i. From this it is easiy seen that with the usua pointwise defined agebraic operations S n (U, A, M, X) is a inear space over the reas whose dimension is d = dim S n (U, A, M, X) = n + µ, µ = µ i (2.8) where the sum is extended over a inner knots. A basis generaizing the truncated powers is constructed as foows. It wi be suficient to consider the case of a compact interva [a, b]. For i = 0 et b j (x) J0 := s [0] j (x, c), j = 1,..., n, where

11 Construction of ECT-B-spines, a survey 105 s [0] 1,..., s[0] n is the prescribed ECT-system on J 0 in its canonica form with respect to a fixed point c with x 0 c x 1. Then extend b j to J 1 such that the extension satisfies the connection equations (2.4) at x 1. Since A [1] is nonsinguar there is a µ 1 parameter famiy of such extensions. Actuay, in extending the basic functions b j to the right for every knot x i we choose in the connection equations (2.4) the connection matrix of the form C [i] := diag (A [i], I µi ) R n n (2.9) where I ν denotes the identity matrix of dimension ν requiring L [i] b j (x i +) = L [i 1] b j (x i ), = n µ i,..., n 1, i = 1,..., k. If 1 i k and j = n + i 1 =1 µ + m, m = 1,..., µ i, take b j (x) Ji = s [i] n µ i +m(x, x i ) where s [i] 1,..., s[i] n is the ECT-system on J i in its canonica form with respect to c = x i, extend b j to the eft by zero and to the right across each knot x p, i + 1 p k, via the connection equations (2.4) with the connection matrices (2.9). By construction, the functions b 1,..., b d beong to S n (U, A, M, X) and they are ineary independent on [a, b]. Since their cardinaity equas the dimension of S n (U, A +, M, X) we have constructed a basis of this space. 3. A zero bound for spines in S n (U, A +, M, X) We use the zero counting convention due to Goodman [6]). In this section and in the rest of the paper we make the basic assumptions (2.5),(2.6) and (2.7) which ensure, in particuar, 1 S n (U, A +, M, X) C([a, b]; R). Definition 3.1. Let f S n (U, A +, M, X) and t (a, b). We set 1 there exists ε > 0 such that f is positive on (t, t + ε) f(t) + := 0 there exists ε > 0 such that f vanishes identicay on (t, t + ε) 1 there exists ε > 0 such that f is negative on (t, t + ε) 1 there exists ε > 0 such that f is positive on (t ε, t) f(t) := 0 there exists ε > 0 such that f vanishes identicay on (t ε, t) 1 there exists ε > 0 such that f is negative on (t ε, t). If f is not identicay zero in some neighborhood of t then f(t) + f(t) 0. In this case there exist nonnegative integers, r n 1 such that f(t ) = f (t ) =... = f ( 1) (t ) = f(t+) = f (t+) =... = f (r 1) (t+) = 0 and f () (t )f (r) (t+) 0. Let q := max (, r). We say that f has a point zero of mutipicity m at the point t where { q if f(t) f(t) + ( 1) q > 0 m = q + 1 if f(t) f(t) + ( 1) q < 0

12 106 G. W. Mühbach and Y. Tang As a consequence, f(t) + f(t) = ( 1) m. If x 0 α < β x k+1 we set k(α, β) := α<x <β µ. Definition 3.2. Let f S n (U, A +, M, X) and a α < β b. (i) If f(α) f(β) + 0 and f(x) = 0 for α < x < β, then α and β are knots, α = x p and β = x q with 0 < p < q < k + 1, and we say that f has an interva zero [α, β] of mutipicity Z(f [α, β]) = n k(α, β). (ii) If f(x) = 0 for a a x < β whie f(β) + 0, then β is a knot, β = x q with 0 < q < k + 1, and we say that f has an interva zero [a, β] of mutipicity Z(f [a, β]) = n + k(a, β). (iii) If f(x) = 0 for a α < x b whie f(α) 0, then α is a knot, α = x p with 0 < p < k + 1, and we say that f has an interva zero [α, b] of mutipicity Z(f [α, b]) = n + k(α, b). The tota number of zeros of f in an interva J wi be denoted by Z(f J). Dyn and Micchei [4], p have estabished a zero bound for poynomia spines via connection matrices as in remark 2.3 under the basic assumptions (2.6) and (2.7). A carefu examination of their proof shows that it can be adapted to the space S n (U, A +, M, X). The reason is that aso for ECT-spaces there hods a Budan-Fourier-Theorem [30], p From this as in [4] a Boudan-Fourier- Theorem for S n (U, A +, M, X) can be derived (see theorem 3.3 of [25]), and this in turn yieds the foowing Theorem 3.3. Let f S n (U, A +, M, X) with X = (x 0,..., x k+1 ) being a partition of a compact interva [a, b]. Under the basic assumptions (2.5),(2.6) and (2.7) if f is not identicay zero then Z(f [x 0, x k+1 ]) n 1 + µ, µ = k µ i. For the particuar case that a mutipicities are zero aso Barry [1] has given this bound. i=1 Coroary 3.4. If f S n (U, A +, M, X) is not the zero function and vanishes identicay on ([a, x 1 ) and on (x k, b]) then Z(f (x 1, x k )) max{µ n 1, 0}.

13 Construction of ECT-B-spines, a survey 107 It is the situation of coroary 3.4 that is needed for constructing a B-spine basis for the space S n (U, A +, M, X) or rect-spines. For the space of piecewise ordinary poynomias of order n via totay positive connection matrices, Dyn and Micchei [4] have constructed such a basis. Again, a carefu inspection of their proof shows that it carries over to rect-spines yieding the foowing theorem. Theorem 3.5. Suppose that n 2 and [a, b] R is compact. Under the basic assumptions (2.5),(2.6) and (2.7) with 1 µ i n 1 for i = 1,..., k, then there is a basis (N n j )µ j= n+1 of the space S n(u, A +, ξ ext ) having the properties N j (x) := N n j (x) := N j (x ξ j,..., ξ j+n ) S n (U, A +, ξ ext ) N j (x) > 0 x (ξ j, ξ j+n ) N j (x) = 0 x / [ξ j, ξ j+n ] N () j (ξ j +) = 0 for = 0,..., n 1 µ + j, Dn µ+ j + N j (ξ j ) > 0, N () j (ξ j+n ) = 0 for = 0,..., n 1 µ j+n, Dn µ j+n N j (ξ j+n ) < 0, µ N j (x) = 1 x [a, b]. j= n+1 Here µ ± j := #{ 0 : ξ j = ξ j± } denote the right and eft mutipicities of a knot ξ j in the sequence (ξ ) µ+n = n+1. Another proof of theorem 3.5 based upon right sided generaized divided differences can be found in [24]. It shoud be remarked that for arbitrary knot sequences tota positivity of the connection matrices is a sufficient condition to ensure existence of a oca support basis forming a nonnegative partition of unity. As shown by Mazure [14] it is not necessary. It is an open probem to give conditions which are necessary and sufficient for existence of such a basis. Given arbitrary nonsinguar connection matrices, for Chebycheff spines oca support bases forming a partition of unity exist for knot sequences which are dense in the set of a possibe knot sequences, as is shown recenty by Prautzsch [28]. 4. Interpoation properties of the spine spaces S n (U, A +, M, X) Consider the spine space S n = S n (U, A +, M, X) with X = (x i ) k+1 i=0 a partition of a compact interva [a, b]. Assume that there are given d nodes or interpoation points y j, Y = (y 1,..., y d ) where x 0 y 1 y 2... y d x k+1. (4.1) Here d denotes the dimension (2.8) of the space S n. Since its eements are continuous functions that are piecewise generaized poynomias of order n we assume

14 108 G. W. Mühbach and Y. Tang that ν j := max { 0 : y j = y j 1 =... = y j } n 1, j = 1,..., d, (4.2) i.e. each node has mutipicity not greater than n. For every node y j there is a unique integer h such that y j = x h, h {0,..., k + 1} or y j intj h, h {0,..., k}. When y j = x h with h {1,..., k} we suppose that ν j + µ h n 1, j = 1,..., d. (4.3) This condition is caed the accumuation condition. It aows that nodes are knots. Ony finite endpoints or knots of mutipicity 0 may be nodes of mutipicity n. If a node y j equas an inner knot x h whose mutipicity µ h is not zero then the accumuation condition guarantees that for every f S n the rect-derivative of highest order L [h] ν j f(y j +) does exist. Then aso D νj + f(y j +) exists. We consider the probem H(S n, Y +, f) of right sided Hermite interpoation (cf. (1.13)) given y 1... y d, y j [a, b], given f C+ H(S n, Y +, f) : N ([a, b]; R) with N = max j=1,...,d ν j, find s S n (U, A +, M, X) such that D ν j + s(y j ) = D ν j + f(y j ), j = 1,..., d. The foowing theorem gives conditions which are necessary and sufficient for this probem to have a unique soution. Theorem 4.1. We make the assumptions (2.5),(2.6),(2.7),(4.1),(4.2) and (4.3). Then the foowing assertions are equivaent (i) H(S n, Y +, f) has a unique soution for every admissibe function f. (ii) (iii) y i < ξ i < y i+n i = 1,..., µ, µ := y i M i i = 1,..., d, k µ. =1 where [x 0, ξ i ) i = 1,..., n M i := (ξ i n, ξ i ) i = n + 1,..., d n (ξ i n, x k+1 ] i = d n + 1,..., d.

15 Construction of ECT-B-spines, a survey 109 The conditions (ii) and (iii) are caed the mixing conditions of the first resp. second kind. Theorem 4.1 generaizes in part the interpoation theorems of Schoenberg and Whitney [29] for ordinary poynomia spines with simpe knots, of Karin and Zieger [9] for Chebycheffian spines with mutipe knots and an interpoation theorem of Dyn and Micchei [4] for poynomia spines via totay positive connection matrices. It is consistent with theorems 4.67 and 9.33 of Schumaker [30] on right sided Hermite interpoation by ordinary poynomia spines or by Tchebycheffian spines, respectivey, since a our interpoation functions are continuous and the nodes satisfy the conditions (4.2) and (4.3). For the same reasons it is aso consistent with the particuar case q = d of the more genera resut of Lyche and Schumaker [11] on modified Hermite interpoation by LB-spines. In case M = (0) when a inner knots have mutipicity zero the mixing conditions of both kinds are void. We then have Coroary 4.2. Under the assumptions of theorem 4.1 the space S n (U, A +, (0), X) is an ret-space of order n consisting of continuous functions. It has a basis p 1,..., p n such that V p 1,..., p n y 1,..., y n := det ( D ν j + p i (y j ) ) > 0 + for a y 1... y n in [a, b]. Coroary 4.3. Under the assumptions of theorem 4.1 for a systems of nodes y 1... y d in [a, b] V := V b 1,..., b d y 1,..., y d 0 + with strict inequaity iff the mixing conditions hod. Here b 1,..., b d is the basis of generaized truncated powers constructed in section 2. Coroary 4.4. Under the assumptions of theorem 4.1 and of theorem 3.2 we have V N n+1,..., N d n y 1,..., y d 0 (4.4) + with strict inequaity iff the mixing conditions hod. Here (N j ) d n j= n+1 is the basis of theorem 3.5. It is an open probem if the generaized Vandermonde matrix of (4.4) for simpe nodes is totay positive as in the particuar case of ordinary poynomia B-spines. Coroary 4.5. Under the assumptions of theorem 4.1 and assuming that each connection matrix can be partitioned according to A [i] = diag(1, A [i] ) where A[i] 1, A[i] 2 1

16 110 G. W. Mühbach and Y. Tang and A [i] 2 are square matrices of dimensions n m 1 and m, respectivey, that both satisfy (2.7), then the space S n (U, A +, (0), X) is an ret-space of order n that has an ret-subspace of order n m. There is a basis p 1,..., p n of S n (U, A +, (0), X) such that V p 1,..., p n m y 1,..., y n m := det (D ν j r p (y j )) > 0 r for a y 1... y n m in [a, b]. Coroary 4.6. Under the assumptions of coroary 4.5 every nontrivia f span {p 1,..., p n m } S n (U, A +, (0), X) has at most n m 1 zeros in [a, b]. Coroary 4.7. Assuming (2.5) and that each connection matrix A [i] is a nonsinguar positive diagona matrix with a [i] 11 = 1, then the space S n(u, A +, (0), X) is an rect-space of order n, i.e. this space has a basis p 1,..., p n such that for m = 1,..., n V p 1,..., p m y 1,..., y m := det (D ν j r p (y j )) > 0 r for a y 1... y m in [a, b]. In the situation of coroary 4.7 every interpoation probem H(S n, Y +, f) can be soved recursivey either using Newton s method via generaized divided differences [17], [18], [21], or using the generaized Nevie-Aitken agorithm [19], [20]. This proves particuar usefu in computing the spine weights recursivey that occur in the recurrence reation for rect-b-spines (see (7.5) beow). 5. Póya-poynomias and Marsden s identity generaized to ECT-spines As in the preceding sections we adopt the genera assumptions (2.5), (2.6) and (2.7). Let S n (U, A +, ξ ext ) = S n (U, A +, M, X) be an rect-spine space as in section 2. Assuming for the weights of every oca ECT-system (1.9) we set C A + := (C [i] ) i KX with C [i] = diag(a [i], I µi ) E A + := (E [i] ) i KX with E [i]t := R 1 ( C [i]) 1 R where R = R n is the n dimensiona orthogona matrix defined by ( 1) 0 R T := ( 1) ( 1) n

17 Construction of ECT-B-spines, a survey 111 We use the spaces P n := P n (U, C A +, X) := S n (U, C A +, (0), X) := {f : f C n 1 r ([a, b]; R), (5.1) f Ji span U [i], f is (U [i 1], U [i], C [i] ) smooth at x i, i K X } and P n := P n (U, E A +, X) := Šn(U, E A +, (0), X) := {f : f C n 1 ([a, b]; R), (5.2) f ˇJi span U [i], f is (U [i 1], U [i], E [i] ) smooth at x i, i K X }. Ceary, P n S n (U, A +, M, X). From the assumption (2.6) it foows that E [i] = diag (I µi, Ē[i] n 1 µ i, 1), Ē [i] n 1 µ i = Rn 1 µ (Ā[i] ) T Rn 1 µi i. (5.3) is a n-dimensiona square matrix which satisfies (2.7). Later we wi aso use spaces having P n as a subspace and P n+1 := S n+1 (Û, ĈA +, (0), X) Pn+1 := Šn+1(Û, ÊA +, (0), X) having Pn as a subspace. They are defined by the extensions [i] Û = (Û ) (Ĉ[i] ), Ĉ A + =, Û = (Û ) [i], Ê A + = i KX i KX i KX where Û [i] (w [i] 1,..., w[i] n+1 (Ê[i] ) i KX = (u [i] 1,..., u[i] n, u [i] n+1 ) is an ECT-system generated by the weights 1,..., u [i] n, u [i] n+1 ) is an ) = (1, w[i] 2,..., w[i] n, ŵ [i] n+1 ) and Û [i] = (u [i] ECT-system generated by the weights (w [i] 1,......, w [i] n+1, w [i] n+1 arbitrariy, where now we assume that ) = (1, w[i] n,..., w [i] 2 ). Here 0 < ŵ[i] n+1 C0 ( J i ; R) and 0 < w [i] n+1 C0 ( J i ; R) may be chosen w [i] j C max{n+1 j,j 1} (J i ; R), j = 2,..., n. The connection matrices for P n resp. for Pn+1 for every i K X are defined by Ĉ [i] := diag (C [i], 1) resp. Ê [i] := diag(e [i], 1). Here Ê[i] = diag(i µi, E [i] n 1 µ i, 1, 1) if A [i] may be partitioned as in (2.6). According to coroary 4.2 P n resp. Pn is an ret- resp. ET-space of order n each and Pn+1 is an ET-space of order n + 1. By coroary 4.5 the space Pn+1 has a basis q 1,..., q n+1 (5.4) such that for ν = 0, 1, 2 the system q 1,..., q n 1+ν is an ET-sytem of order n 1+ν. Such a basis is obtained by fixing in any knot interva J i the oca ECT-system (1.8) in canonica form with respect to any c [x i, x i+1 ] (s [i] 1,n+1 (x, c),..., s[i] n+1,n+1 (x, c)),

18 112 G. W. Mühbach and Y. Tang x (x i, x i+1 ], and extending these functions by the connection equations of Pn+1 to the eft and right of J i. Since s [i] j,n = s[i] j,n+1, j = 1,..., n, the basis q 1,..., q n+1 of Pn+1 constructed this way under the hypothesis (2.6) indeed yieds in the sections q 1,..., q n 1 and q 1,..., q n ET-systems of orders n 1 and n, respectivey. Generaized Póya poynomias are defined for j J ϕ by denoting by rf [ u1,...,u n t 1,...,t n (y) the interpoation remain- ] der where pf M j (y) = Mj n (x) = [ ] = M j (y ξ j+1,..., ξ j+n 1 ) = ( 1) n 1 q 1,..., q n 1 rq n ξ j+1,..., ξ j+n 1 [ u1,...,u n t 1,...,t n ] (y), (y) := f(y) pf [ u1,...,u n t 1,...,t n (y) is the soution of the Hermite interpoation probem ] H(U, T, f). M j Pn has exacty n 1 zeros ξ j+1,..., ξ j+n 1, counting eft mutipicities, and no other zeros, and M j has eading coefficient ( 1) n 1 in every interva ˇJ i. Therefore M j is positive for x < ξ j+1. It is not hard to show that every n consecutive generaized Póya poynomias (M j ) +n 1, J ϕ, form a basis of j= span{q 1,..., q n }. Barry [1] has constructed de Boor-Fix functionas n 1 Λ j (x)[f] := ( 1) n 1 p L [r] p f(x) L [r] n 1 p M j(x), x J r, ξ j < x < ξ j+n. p=0 Actuay, it is easiy derived from (1.10) that under our genera assumptions (2.5), (2.6) and (2.7) for every j J ϕ and f S n (U, A +, ξ ext ) the function x Λ j (x)[f] is a constant function of x (ξ j, ξ j+n ). As a consequence, S n (U, A +, M, X) f Λ j (f) := Λ j (x)[f], x (ξ j, ξ j+n ) is a we defined inear functiona for every j J ϕ. It foows that they are dua functionas for the rect-b-spine basis (N j ) of order n. This is due to Barry [1], cf. aso [24]. Theorem 5.1. Λ j (N i ) = δ i,j i, j J ϕ. The foowing theorem wi be basic both for the definition of the rect-b-spines in terms of generaized divided differences and for their genera recursion reation. It s proof is a straightforward extension of the proof due to Dyn and Micchei [4] of the simiar resut for poynomia spines via connection matrices.

19 Construction of ECT-B-spines, a survey 113 Theorem 5.2. Under the assumptions (2.5), (2.6), (2.7) there exists a unique function [a, b] [a, b] (x, y) h(x, y) such that (i) for each y [a, b] h(, y) P n, (ii) for each x [a, b] h(x, ) P n. (iii) Whenever for some K X x J and y ˇJ then h(x, y) = w [] 1 (x)h n 1(x, y; w [] 2,..., w[] n ) = s [] n (x, y) n = ( 1) n k s [] k (x, c) s[] n+1 k,n (y, c) k=1 with c J arbitrary. n = w [] 1 (x) h k 1 (x, c; w [] 2,..., w[] k )h n k(y, c; w [] k+1,..., w[] n ) k=1 (iv) For i K X fixed, c J i and j = 1,..., n et p j (, c) P n be defined by p j (x, c) = w [i] 1 (x)h j 1(x, c; w [i] 2,..., w[i] j ) = s[i] j (x, c), x J i, and for i K X fixed, c J i and j = 1,..., n et q j (, c) P n be defined by q j (y, c) = h j 1 (y, c; w [i] n,..., w [i] n+2 j ) = s[i] j,n (y, c), y ˇJ i. (5.5) Then the function h has the representation h(x, y) = n ( 1) n k p k (x, c)q n+1 k (y, c), (x, y) [a, b] [a, b], c J i (5.6) k=1 where the right hand side is independent of i and of c J i. The function h wi be caed generating function of rect-b-spines for reasons that wi become cear soon. Remark 5.3. If U [i] = (1, x,..., x n 1 ) Ji for a i then h(x, y) = (x y)n 1 (n 1)! whenever x J and y ˇJ for some and (5.6) reduces to formua (3.67) of [4]. If moreover A [i] = I n µi for a i then (5.6) reduces to the binomia theorem (x y) n 1 (n 1)! = n n k (x c)k 1 ( 1) (k 1)! k=1 (y c) n k. (n k)! When U [i] = U Ji where U = (u 1,..., u n ) is an ECT-system on [a, b] and for a i Z A [i] = I n µi is an identity matrix then (5.6) reduces to Marsden s identity for Tchebycheff spines [30] p. 382.

20 114 G. W. Mühbach and Y. Tang The next theorem is a generaization of Marsden s identity to rect-b-spines. Theorem 5.4. Under the assumptions (2.5), (2.6) and (2.7) for the function h of theorem 5.2 there hods h(x, y) = i J ϕ N i (x)m i (y) for a (x, y) [a, b] [a, b]. Remark 5.5. Theorems 5.1 and 5.4 are equivaent in the sense that each is a consequence of the other [24]. 6. rect-b-spines defined by generaized divided differences Let g(x, y) := { h(x, y) x y 0 otherwise (6.1) where h is the function of theorem 3.5 By the properties of h, with y fixed, as a function of x, g(x, y) beongs piecewise, for x y and for x < y, to P n S n (U, A +, M, X). If x is fixed, as a function of y, g(x, y) beongs piecewise, for x y and for x < y, to Pn. The function g is separatey, with respect to x, in Cr n 1 ([a, b]; R), and with respect to y, in C n 1 ([a, b]; R), since h has this property, and the (n 1)st ECT-derivative of g with respect to x at x = y has the characteristic jump discontinuity of a Green s function: for every i im x y L[i] ν g(x, y) = 0 ν = 0,..., n 1, if x i < y x i+1 im x y+ L[i] ν g(x, y) = { 0 ν = 0,..., n 2 1 ν = n 1 if x i y < x i+1. We ca (6.1) the Green s function with respect to the spaces P n (U, C A +, X) and P n (U, E A +, X). Definition 6.1. For j J ϕ and x [a, b] Ñj n (x) := Ñ(x ξ j,..., ξ j+n ) ([ ] [ ] ) := ( 1) n q 1,..., q n q 1,..., q n g(x, ) g(x, ). (6.2) ξ j+1,..., ξ j+n ξ j,..., ξ j+n 1 Here we have made use of the notation (1.14) for the eft sided generaized divided differences of a function, and the functions q 1,..., q n are those defined in Theorem 5.2.

21 Construction of ECT-B-spines, a survey 115 Theorem 6.2. For j J ϕ and x [a, b] [ ] Ñj n (x) = ( 1) n q1,..., q n+1 f n,j ξ j,..., ξ j+n [ ] q 1,..., q n f n,j = ξ j+1,..., ξ j+n q n+1 g(x, ), (6.3) [ ] q 1,..., q n ξ j,..., ξ j+n 1 q n+1. Here q 1,..., q n+1 P n+1 are the functions defined by (5.4) and q 1,..., q n are those of Theorem 5.2. Remark 6.3. In case of ordinary poynomia spines of order n where a connection matrices are identity matrices (6.3) simpifies to Ñ n j (x) = ( 1) n (ξ j+n ξ j )[ξ j,..., ξ j+n ] (x ) n 1 + where [ξ j,..., ξ j+n ] f denotes the ordinary eft sided divided difference of the function f C n 1 (J; R) with respect to the poynomias of degree n at most. In case of Tchebycheffian spines of order n where a connection matrices are identity matrices (6.3) extends Lyche s definition (6.2) of Tchebycheffian B-spines [10]. In [24] it is proved Theorem 6.4. For j J ϕ Ñ n j = N n j. Here N n j are the rect-b-spines of theorem 3.5. Remark 6.5. It is definition 6.1 which, under suitabe assumptions, eads to a recursive method for computing ECT-B-spines and ECT-spine curves deveoped in section 7 [25]. In [31] cardina ECT-B-spines with simpe knots defined by connection matrices are computed directy according to theorem 6.2 Actuay, there the eft sided generaized divided differences are computed directy via a certain characteristic poynomia wherein aso the Tayor s expansion (1.7) with respect to an ECT-system and that with respect to its dua are invoved. 7. Computing rect-b-spines recursivey Recursive methods for computing B-spines (normaized to form a nonnegative partition of unity) for spines of particuar casses are we known. For ordinary poynomia spines (with connection matrices that a are identity matrices) best known is the deboor-mansion-cox recurrence reation (7.3),(7.4) which is a twoterm recursion. Using a contour integra approach Waz [32] has proved more genera more-term recursions. For Tchebycheff spines there is a two-term recursion due to Lyche [10] where the spine weights are expressed as quotients of determinants. For a wide cass of Tchebycheff spines, LB-spines and compex spines Dyn and Ron [5] have given four-term recursions. It shoud be noticed that our constructive approach does not cover trigonometric B-spines. Stabe two-term recursions for ordinary trigonometric B-spines (with

22 116 G. W. Mühbach and Y. Tang a connection matrices equa to identity matrices) are due to Lyche and Winther [12] and more genera ones due to Waz [33]. It is Lyche s approach to Tchebycheff B-spines [10] that can be extended to rect-b-spines (and simiary to ECT-B-spines). Two ideas are basic in estabishing the recurrence reation. One is due to Lyche defining auxiiary B-spines of ower orders which are used as intermediate resuts in computing the B-spines N n of order n. As before we adopt the assumptions (2.5),(2.6) and (2.7). Definition 7.1. For n N, k = 1,..., n and x R and j Z et N k,n j (x) := ( 1) n q 1(,c),...,q n(,c) ξ j+1,...,x,..., x,...,ξ j+k g(x, ) }{{} n k q 1 (,c),...,q n (,c) ξ j,...,x,..., x,...,ξ j+k 1 g(x, ) if ξ j < ξ j+k and ξ j x < ξ j+k }{{} n k 0 otherwise (7.1) Here g is defined by (6.1) and the q (y, c) are defined by (5.5). It can be shown [25] that N k,n j (x) is independent of c J i and of i Z. We ca these functions auxiiary B-spines of suborder k of the B-spine N n of order n such that ξ ξ j < ξ j+k ξ +n. The B-spines of owest orders are N 1,1 j (x) = N 1 j (x) = N(x ξ j, ξ j+1 ) = { 1 if ξ j x < ξ j+1 0 otherwise. It is not hard to show that the auxiiary B-spines have simiar properties as the B-spines themseves (see theorem 3.2) though, at east in genera, they do not beong to S n (U, A +, ξ ext ). The second basic idea dates back to Popoviciu [26]. He has modified generaized divided differences by introducing further nodes as is done in definition 7.1. This idea was eaborated for ECT-systems by Lyche [10] and for ET-systems by Mühbach and Tang in emmata 4.2 and 4.3 of [25]. Remark 7.2. In case of ordinary poynomia spines of order n the N k,n j poynomia B-spines of order k: are the N k,n j (x) = ( 1) k (ξ j+k ξ j )[ξ j,..., ξ j+k ] (x ) k 1 + = N k,k j (x), (7.2) k = 1,..., n. It is we known that the poynomia B-spines (7.2) can be computed by the de Boor-Mansion-Cox recursion N k+1,n j (x) = λ k,n j (x)n k,n j (x) + µ k,n j+1 k,n (x)nj+1(x), k = 1,..., n 1 (7.3)

23 Construction of ECT-B-spines, a survey 117 starting with N 1,n j (x) = N 1,1 j (x) = { 1 if ξ j x < ξ j+1 0 otherwise where the coefficients are the Nevie-Aitken weights of poynomia interpoation that are independent of n λ k,n j (x) = x ξ j ξ j+k ξ j, µ k,n j+1 (x) = 1 λk,n j+1 (x) = ξ j+k+1 x. (7.4) ξ j+k+1 ξ j+1 They occur in the Nevie-Aitken interpoation formua p k+1 f[ξ j,..., ξ j+k ](x) = x ξ j ξ j+k ξ j p k f[ξ j+1,..., ξ j+k ](x) + ξ j+k x ξ j+k ξ j p k f[ξ j,..., ξ j+k 1 ](x) where p f[z 1,..., z ] is that poynomia of order interpoating the function f at the nodes z 1,..., z in the sense of Hermite. The equaity (7.2) reies on the factorization of agebraic poynomias. For arbitrary ECT-systems s [i] there is no simiar interpretation of the auxiiary ECT-B-spines of suborder k as in (7.2). Ony for particuar ECT-systems, for instance of rationa functions with prescribed poes there is a simiar interpretation (see section 8). However, under suitabe assumptions aso in the genera case the auxiiary ECT-B-spines of suborder k can be computed recursivey by a de Boor-ike recursion. It turns out that the spine weight factors aso in the genera case can be interpreted in terms of interpoation theory with respect to an ET-system q 1,..., q n. They are again certain generaized Nevie-Aitken weights. Theorem 7.3. Suppose that N k,n j (x) for k = 1,..., n and a j Z is defined by (7.1). Assume that the basis (5.5) q 1 (, c),..., q n (, c) of P n (U, E A +, X) is an ET-system on [a, b] of order n that has the property that aso q 1 (, c),..., q n 1 (, c) is an ET-systems on [a, b]. Here c J i and i Z are arbitrary. In view of (5.3) according to coroary 4.5 this hods true due to the basic assumption (2.6). Then for k = 1,..., n 1, x R and j Z N k+1,n j with the initiaization (x) = λ k,n j N 1,n j (x) = (x) N k,n j (x) + µ k,n k,n j+1 (x) Nj+1 (x) (7.5) { 1 if ξ j x < ξ j+1 0 for a other x, and N 1,n j (x) 0 iff ξ j = ξ j+1, where the spine weights can be computed by the foowing formuas: λ k,n j (x) 0 if ξ j =... = ξ j+k

24 118 G. W. Mühbach and Y. Tang 0 if x = ξ j λ k,n D µ+ (ξ j ) 1 rq n[ξ j+1,...,x,..., x,...,ξ j+k 1 ] (ξ j) }{{} j (x) := n k rq n[ξ j+1,...,x,..., x,...,ξ j+k ] (ξ j) D µ+ (ξ j ) 1 } {{ } n k 1 if ξ j < x ξ j+k where µ + (ξ j ) is the mutipicity of ξ j in (ξ j, ξ j+1,..., ξ j+k ) and the interpoation remainders are with respect to the ET-system q 1 (, c),..., q n 1 (, c) with c R arbitrary. Moreover µ k,n j+1 (x) = µ k,n j+1 (x) 0 if ξ j+1 =... = ξ j+k+1 1 if x = ξ j+1 D µ (ξ j+k+1 ) 1 rq n [ξ j+2,...,x,..., x,...,ξ j+k ] (ξ j+k+1 ) }{{} n k rq n [ξ j+1,...,x,..., x,...,ξ j+k ] (ξ j+k+1 ) D µ (ξ j+k+1 ) 1 } {{ } n k 1 if ξ j+1 < x ξ j+k+1 where µ (ξ j+k+1 ) is the mutipicity of ξ j+k+1 in (ξ j+1,..., ξ j+k+1 ). Moreover, we have 0 < λ k,n j (x) < 1 if ξ j < x < ξ j+k, and x λ k,n j (x) is eft continuous everywhere and stricty increasing from 0 to 1 for ξ j < x < ξ j+k. Simiary, we have 0 < µ k,n j+1 (x) < 1 if ξ j+1 < x < ξ j+k+1, and x µ k,n j+1 (x) is eft continuous everywhere and stricty decreasing from 1 to 0 for ξ j+1 < x < ξ j+k+1. Remark 7.4. The spine weight factors can be interpreted in terms of interpoation theory, cf. [25] remark 4.2. Remark 7.5. If for every i the connection matrix A [i] is a diagona matrix with ony positive diagona eements then the basis q 1 (, c),..., q n (, c) is an ECTsystem, and a weights λ k,n j (x), µ k,n j+1 (x) can be computed recursivey since the remainders rq n [y 1,..., y n ] (y) can be computed recursivey (cf. [19]). Remark 7.6. It shoud be observed that theorem 7.3 aso covers the case of Bézier- ECT-spines. This type of spines arises if we consider a compact interva [a, b], a finite partition X = (x i ) k+1 i=0 of [a, b], a = x 0 < x 1 <... < x k < x k+1 = b, into knot intervas J i = [x i, x i+1 ), i = 0,..., k 1, with the ast knot interva

25 Construction of ECT-B-spines, a survey 119 J k = [x k, x k+1 ] resp. ˇJi = (x i, x i+1 ], i = 1,..., k, with the first knot interva ˇJ 0 = [x 0, x 1 ], with mutipicities µ 0 = µ k+1 = n, µ i = 0 for i = 1,..., k, with oca ECT-systems (2.2) on J i generated by weights (2.3) satisfying (2.5), with fu connection matrices A [i] R n n that satisfy (2.6) and (2.7). Notice that the ECT-system U [0] on J 0 = [x 0, x 1 ] may be extended as an ECT-system to a arger interva Ĵ 0 = [x 0 δ, x 1 ] for δ > 0 simpy by extending the weights w [0] j to Ĵ0 maintaining the smoothness properties (2.3). According to theorem 7.3 for ξ j x < ξ j+1 the B-spine curve of order n s(x) = j =j n+1 c N n (x) where the contro points c j n+1,..., c j R s (s N) are given can be computed recursivey by the foowing de Boor-ike agorithm. Agorithm 7.7 initiaisation: agorithm: output: c k+1,n i (x) = λ n k,n i c 1,n (x) := c = j n + 1,..., j (x) c k,n i (x) + (1 λ n k,n i (x)) c k,n i 1 (x), ξ j x < ξ j+1, i = j n + k + 1,..., j, k = 1,..., n 1. c n,n j (x) = s(x), ξ j x < ξ j+1. Moreover, at eve k (k = 1,..., n) there hods s(x) = j =j n+k c k,n (x) N n+1 k,n (x), ξ j x < ξ j Exampes Exampe 8.1. In case of ordinary poynomia spines of order n with a connection matrices equa to identity matrices we have q j (y, c) = (y c)j 1 (j 1)! j = 1,..., n

26 120 G. W. Mühbach and Y. Tang and from theorem 7.3 for ξ j x < ξ j+1 and = j k,..., j the spine weights λ k,n (x) = µ k,n D µ+ (ξ ) 1 D µ+ (ξ ) 1 rq n [ξ +1,..., x,..., x,..., ξ }{{} +k 1 ] (ξ ) n k (x) = 1 λ k,n (x) = ξ +k x. ξ +k ξ rq n [ξ +1,..., x,..., x,..., ξ }{{} +k ] (ξ ) n k 1 = x ξ ξ +k ξ, In this exampe agorithm 7.7 reduces to the de Boor agorithm computing points of B-spine curves in case of ordinary poynomia spines of order n when a connection matrices are identity matrices. Exampe 8.2. In case of Tchebycheff spines of order n with a connection matrices equa to identity matrices we have q j (y, c) = s j,n(y, c) j = 1,..., n and from theorem 7.1 for ξ j x < ξ j+1 and = j k,..., j the spine weights λ k,n j (x) = µ k,n j D µ+ (ξ j ) 1 D µ+ (ξ j) 1 (x) = 1 λ k,n j (x) rq n [ξ j+1,..., x,..., x,..., ξ }{{} j+k 1 ] (ξ j ) n k rq n [ξ j+1,..., x,..., x,..., ξ }{{} j+k ] (ξ j ) n k 1 giving new interpretations to the weights due to Lyche [10]. Exampe 8.3. For the goba ECT-system s 1,n(y, c),..., s n,n(y, c) (1.11), (1.12) on [a, b] of exampe 1.2 where a connection matrices are identity matrices the Nj n are are Chebycheff-B-spines with respect to this ECT-system. In this case the functions q 1 (, x),..., q n (, x) of theorem 5.2 are known as the rationa functions (1.4), (1.5), (1.6). This ECT-system being aso a Cauchy-Vandermonde system with respect to the poes b 1 =,..., b n 2 =, b n 1 = a ε, b n = b + ε aows to compute the spine weights of theorem 5.2 and of agorithm 7.7 expicity using the expicit expression (42) of [22] of the interpoation remainder in terms of the nodes and the poes. If ξ j x < ξ j+1, j { n + 1,..., µ} according to theorem 5.2 for n N at eve k = 1,..., n for m = j k + 1,..., j λ k,n m (x) := im ξ m ξ m 0 rq n[ξ m+1,...,x,..., x }{{} n k rq n [ξ m+1,...,x,..., x }{{} n k 1 = ξ m x b + ε ξ m+k ξ m ξ m+k b + ε x,...,ξ m+k 1 ] ( ξ m),...,ξ m+k ] ( ξ m ) if ξ j x < ξ j+1 if ξ j < x < ξ j+1 (8.1)

27 Construction of ECT-B-spines, a survey 121 and µ k,n m (x) = 1 λ k,n m (x) = ξ m+k x ξ m+k ξ m b + ε ξ m+k b + ε x. (8.2) The spine weights (8.1) and (8.2) agree with the weights given by Gresbrand [7] where spines with the ordinary smoothness conditions constructed from Cauchy- Vandermonde systems with respect to arbitrary given poes b 1, b 2,..., b n outside [a, b] are considered. This shows that aso in the simpe case of Chebycheff ECT- B-spines with one poe of order n 1 at b + ε the auxiiary ECT-B-spines are the ECT-B-spines of ower orders. More anaytica and some numerica exampes can be found in [25]. Of course, it wi depend on the appications what kind of spines a designer wi choose. The famiy of ECT-B-spines provides a rea aternative to the cassica poynomia B- spines. In fact, they aow more freedoms without increasing computationa costs too much. References [1] Barry, P.J., de Boor-Fix dua functionas and agorithms for Tchebycheffian B-spines curves, Constructive Approximation 12 (1996), [2] Barry, P.J. Dyn, N. Godman, R.N. Micchei, C.A., Identities for piecewise poynomia spaces determined by connection matrices, Aequationes Math. 42 (1991), [3] Buchwad, B., Mühbach, G., Rationa spines with prescribed poes, JCAM 167 (2004), [4] Dyn, N., Micchei, C.A., Piecewise poynomia spaces and geometric continuity of curves, Numerische Mathematik 54, [5] Dyn, N., Ron, A., Recurrence Reations for Tchebycheffian B-Spines, J. Anayse Math. 51 (1988), [6] Goodman, T.N.T., Properties of beta-spines, J. Approx. Theory 44 (1985), [7] Gresbrand, A., Rationa B-spines with prescribed poes,num. Agorithms 12 (1996), [8] Karin, S. & Studden, W.J., Tchebycheff Systems: With Appications in Anaysis and Statistics, Interscience Pubishers (1966). [9] Karin, S., Zieger, Z., Tchebysheffian spine functions, SIAM Num. 3 (1966), [10] Lyche, T.,A recurrence reation for Chebychevian B-spines,Constr. Approx. 1 (1985),