290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f

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1 Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica Aplicada, Universidad de Zaragoza, E Zaragoza, Spain Received December 4, 1992 / Revised version received March 2, 1993 Summary. We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given. Mathematics Subject Classication (1991): 65D07, 65D17, 41A15, 15A48 1. Introduction In spline spaces, it is important to choose an adequate basis whose elements have small support because this makes it possible to design algorithms for solving interpolation problems with lower computational cost. In a vector space of functions a nonzero function f is said to be minimally supported if supp f is a minimal element in the set of all nonempty supports for the partial order given by the inclusion, that is, if supp g $ supp f then g = 0. While it would be desirable to have bases consisting of minimally supported elements, even some very common spaces fail to have such bases. For example, the space of functions on [0; n] spanned by u j := (j? ) n?1 +, j = 1; : : : ; n, fails to have such a basis when n > 1. However the given sequence (u 1 ; : : : ; u n ) is a least supported basis for this space in the sense that for every basis (v 1 ; : : : ; v n ) for this space, there is some permutation so that (1:1) supp u j supp v (j) ; j = 1; : : : ; n: On the other hand, not every nite-dimensional space of functions has such a least supported basis. This is most easily seen by constructing a space which has a? Both authors were partially supported by DGICYT PS page 289 of Numer. Math. 67: 289{301 (1994)

2 290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which fails to satisfy (1.1) for any permutation, as is, e.g. the case for the particular choices, u 1 := [0;3], u 2 := [2;4], u 3 : x 7! x [0;1][[3;4] (x) (as functions on [0,4]), and (v 1 ; v 2 ; v 3 ) := (u 1? u 2 ; u 2 ; u 3 ). For, any least supported basis would have to have the same support sequence (up to permutations) as the rst basis, yet could not be least supported after all, since by construction, it would then fail to satisfy (1.1) for the second basis and any permutation. These examples may leave us uncertain as to which of these two kinds of bases to prefer. It is the main purpose of this paper to generate some preference for the least supported bases by showing that a basis is least supported if and only if it is locally linearly independent. The concept of local linear independence introduced here coincides with the concept dened in spline spaces. So, for instance, the B-spline basis of the space of polynomial splines is least supported since the functions in this basis are locally linearly independent [cf. Theorem 4.18 of Schumaker (1981)] and also in the more general case of Tchebychean spline functions, the Tchebychean B-spline basis is least supported because these functions are locally linearly independent by Lemma 6.3 of Schumaker (1976). In multivariate spline spaces it is also usual that the basic functions are not minimally supported. For this reason, in Chui and He (1988) [see also p. 138 of Chui (1988)] it is introduced the concept of a quasiminimally supported function. On the other hand, certain box-splines form a least supported basis as follows from the proof of its local linear independence in Dahmen and Micchelli (1985). Also the bases of bivariate splines obtained in Carnicer and Pe~na (1993b) are least supported because, by Theorem 3.1 and Theorem 4.4 of the same paper, they are locally linearly independent. The layout of this paper is as follows: In Sect. 2, we begin by dening an order relation among the supports of bases and, if it exists, a least supported basis will be a basis whose support is the least element for the order relation. We shall also introduce the concept of quasiminimally supported function of degree l. In Sect. 3 we study the local linear independence of functions. In Theorem 3.4 we show that the local linear independence of a basis is equivalent to saying that the basis is least supported. In Theorem 3.7, we characterize in dierent ways locally linearly independent bases in a space which has a least supported basis and show how to construct all of them. In Theorem 3.9, we give a characterization of quasiminimally supported functions of degree l in a space which has a locally linearly independent basis and we obtain all the quasiminimally supported functions of degree l in this space. In Sect. 4, we focus on spaces which have a totally positive basis [see Goodman (1989)]. This is the case for many spline spaces such us polynomial splines, Tchebychean splines, spline spaces for the generation of curves with geometric continuity, etc. We prove that the B-bases constructed in Carnicer and Pe~na (1993c) are least supported among all totally positive bases and that if there exists a totally positive basis of locally linearly independent functions, then the B-basis is also locally linearly independent. page 290 of Numer. Math. 67: 289{301 (1994)

3 Least supported bases and local linear independence Minimally supported bases Let U be a nite dimensional vector space of functions dened on a topological space. The most important particular case occurs when is a subset of R m with the induced topology, because this allows us to deal with univariate and multivariate spline spaces. For any function f 2 U, we denote by supp f := fx 2 j f(x) 6= 0g the support of f. For any basis B = (b 1 ; : : : ; b n ) of U we dene the support of a basis supp B := (supp b 1 ; : : : ; supp b n ), up to the order, that is, if C = (c 1 ; : : : ; c n ) is any other basis of U, then supp B = supp C if and only if there exists a permutation such that supp b i = supp c (i) ; i = 1; : : : ; n : We say that supp B supp C if there exists a permutation such that supp b i supp c (i) ; i = 1; : : : ; n : In other words, for some 1-1 map : B! C and for all b 2 B, supp b supp (b). Lemma 2.1. In a nite dimensional vector space, the relation is a partial order among the supports of bases. Proof. The reexive and transitive properties are obvious. For the antisymmetry let B = (b 1 ; : : : ; b n ) and C = (c 1 ; : : : ; c n ) be bases of U such that supp B supp C and supp C supp B. Then we know that there exist two permutations and of f1; : : : ; ng such that (2:1) supp b i supp c (i) ; supp c i supp b (i) ; i = 1; : : : ; n : Let be the permutation given by (i) = ((i)). Since is a permutation of some nite set, there exists k such that k is the identity and then (2:2) supp b i supp b (i) supp b 2 (i) supp b k (i) = supp b i : From (2.1) and (2.2) we derive supp b i supp c (i) supp b i and so supp b i = supp c (i), for all i = 1; : : : ; n : We want to obtain bases with supports as small as possible. If the support of a basis is a minimal element for the partial order then we say that the basis is minimally supported. The most interesting case occurs when there exists only one minimal element (in the collection of support sets): Denition 2.2. B is called a least supported basis if supp B is the minimum (or least element) for the order relation, that is, for any other basis C of U, supp B supp C. page 291 of Numer. Math. 67: 289{301 (1994)

4 292 J.M. Carnicer, J.M. Pe~na If B = (b 1 ; : : : ; b n ) is a minimally supported basis (or even a least supported basis), then any function b i cannot be generated by functions whose supports are strictly contained in supp b i. In fact, if b i can be generated by functions with strictly smaller support w j, j = 1; : : : ; r, then there exists j 0 2 f1; : : : ; rg such that w j0 is not in the subspace generated by b 1 ; : : : ; b i?1 ; b i+1 ; : : : ; b n, because otherwise b i would be also in this subspace, which contradicts that B is a basis. Thus, replacing b i by w j0 in the basis B, we obtain a basis C such that supp C < supp B, contradicting that B is minimally supported. Now we are going to construct a minimally supported basis of any nite dimensional vector space of functions U. Let us see rst that any nonempty subset S of fsupp fjf 2 Ug has minimal elements. Otherwise there would exist an innite sequence of functions (f k ) such that supp f k+1 $ supp f k. Since all the supports are dierent, (f k ) should be linearly independent, which contradicts dim U < 1. Applying this fact to the set fsupp fjf 2 Unf0gg we derive that there exists at least one minimally supported function. Let V 0 6= 0 be the vector space generated by the minimally supported functions in U. If V 0 = U, then there exists a basis of minimally supported functions, which is obviously a minimally supported basis. Unfortunately in many standard function spaces V 0 6= U and then there are no bases with all the functions being minimally supported. If V 0 6= U, there exist functions whose support is a minimal element of fsupp fjf 2 UnV 0 g, which we call quasiminimally supported (of degree 1). Let V 1 be the vector space generated by the minimally and quasiminimally supported functions of degree 1 in U. If V 1 6= U we continue this process until obtaining a chain of subspaces 0 $ V 0 $ V 1 $ $ V r = U; r n: Denition 2.3. A function f is quasiminimally supported of degree l if f =2 V l?1 and its support is a minimal element of fsupp fjf 2 UnV l?1 g, l = 1; : : : ; r. Then l is called the support{degree of f and will be denoted by sdeg (f). We shall also say that a nonzero function f is quasiminimally supported of degree 0 if f is minimally supported. Remark 2.4. Let us observe that if g is quasiminimally supported of degree l, then g 2 V l nv l?1 and for any function f whose support is strictly contained in supp g, f 2 V l?1. Our denition of quasiminimally supported function of degree 1 is similar to the one given in p. 33 of Chui (1988): \a function is said to have quasiminimal support D if it is not in the linear span of all minimally supported functions in the space whose supports lie in D but any function in this space whose support is properly contained in D must be in this linear span". But, in general, this denition does not coincide with the denition of quasiminimally supported function of degree 1 given above as it will be shown in Example 2.9. The following proposition gives some properties of the support{degree of quasiminimally supported functions. Proposition 2.5. Let f, g be quasiminimally supported functions in a nite dimensional vector space such that supp f supp g. Then sdeg (f) sdeg (g) and sdeg (f) = sdeg (g) if and only if supp f = supp g. page 292 of Numer. Math. 67: 289{301 (1994)

5 Least supported bases and local linear independence 293 Proof. If supp (f) $ supp (g) and m = sdeg (g), then by Remark 2.4 f 2 V m?1 and, since f is also quasiminimally supported, again by Remark 2.4 sdeg (f) m? 1 < sdeg (g). Let us assume now that supp f = supp g. Let k = min(sdeg (f); sdeg (g)). Then f; g =2 V k?1 and at least one of them has minimal support among the functions of UnV k?1, and so has the other since supp f = supp g. That means that sdeg (f) = k = sdeg (g). From the denition of quasiminimally supported function it is easy to obtain that if f is a quasiminimally supported function of any degree then f cannot be generated by the functions whose support is strictly contained in supp f. The converse is not true in general as we shall see in Example 2.9. However, in Theorem 3.8 we shall prove that, if there exists a least supported basis, these conditions are equivalent. A nite dimensional vector space of functions has always minimally supported bases. In the next theorem we shall give a description of some minimally supported bases. Theorem 2.6. Every nite dimensional vector space of functions dened on some domain contains a basis B which consists of quasiminimally supported elements with B \ V l a basis of V l for each l. Moreover, any such basis is necessarily minimal. Proof. The basis B can be constructed as follows. First take a basis of V 0 formed by minimally supported functions and complete this basis to a basis of V 1 with quasiminimally supported functions of degree 1. Iterating this procedure we obtain a basis B satisfying the properties. Let C = (c 1 ; : : : ; c n ) be a basis such that supp C supp B. Then for some 1-1 map : B! C and all b 2 B, supp (b) supp b. We claim that sdeg ((b)) = sdeg (b) and prove it by induction on sdeg (b). It is clear for sdeg (b) = 0 since (b) is necessarily a nonzero function. Assuming l := sdeg (b) > 0, it follows from the linear independence of C and the induction hypothesis that V l?1 is spanned by f(b 0 ) j b 0 2 B; sdeg (b 0 ) < lg and hence (b) =2 V l?1. Since supp (b) supp b this implies by Remark 2.4 that supp (b) = supp b and then (b) is quasiminimally supported. By Proposition 2.5, sdeg ((b)) = l = sdeg (b). Let us see that any basis B of quasiminimally supported elements with B \V l a basis of V l for each l is minimally supported. If C is any basis such that supp C < supp B, then C consists of quasiminimally supported functions and there exists a bijection : B! C such that sdeg ((b)) = sdeg (b). By Proposition 2.5, supp (b) = supp b, which shows that B is minimally supported. Remark 2.7. In the proof of the previous theorem it is actually shown that if B consists of quasiminimally supported elements with B \ V l a basis of V l for each l, any basis C with supp C supp B consists of quasiminimally supported elements and the number of basic elements of given support{degree in each basis coincides. Therefore C is also a basis of quasiminimally supported elements with C \ V l a basis of V l for each l. Remark 2.8. From the construction provided in the proof of Theorem 2.6 it can be deduced that for any l-quasiminimally supported function f (0 l r) there page 293 of Numer. Math. 67: 289{301 (1994)

6 294 J.M. Carnicer, J.M. Pe~na exists a minimally supported basis B satisfying the hypothesis of Theorem 2.6, such that f 2 B. Example 2.9. Let U be the vector space generated by the linearly independent functions u 1 (x; y) := 0 if x 0 and y 0, 1 otherwise, u 3 (x; y) := x if x 0, 0 otherwise, u 2 (x; y) := 1 if y 0, 0 otherwise, u 4 (x; y) := 0 if?y < x < 0, y otherwise. It is easy to see that (u 1 ; u 2 ; u 3 ) is a basis of V 0 consisting of minimally supported functions. The function u 1? u 2 2 V 0 is not minimally supported because supp (u 1? u 2 ) % supp u 3, and then it cannot be quasiminimally supported of any degree l 0. However any element f 2 U with support strictly contained in the support of u 1? u 2 is necessarily a multiple of u 3. Hence u 1? u 2 fails to be in the linear span of the elements with support strictly inside its support. Therefore u 1? u 2 is quasiminimally supported in the sense of Chui. Since dim U = dim V 0 + 1, V 1 = U. On the other hand it is easy to check that every function f in V 1 nv 0 satises supp f supp u 4 and so u 4 is quasiminimally supported of degree 1. However u 4 is not quasiminimally supported in the sense of Chui because supp u 3 $ supp (u 1? u 2 ) $ supp u 4. Finally let us observe that (u 1?u 2 ; u 2 ; u 3 ; u 4 ) is a minimally supported basis and it does not satisfy the properties of Theorem 2.6 because u 1? u 2 is not a quasiminimally supported function. Remark A nite dimensional vector space with a least supported basis enjoys some special properties. By Remark 2.7, if there exists a least supported basis, it is of the nature of the basis constructed in the Theorem 2.6. Since in that case, any minimally supported basis must also be least supported, the construction of Theorem 2.6 then provides least supported bases. By Remark 2.8, for any quasiminimally supported function f of some degree there exists a least supported basis B such that f 2 B. Therefore, taking into account that all least supported bases have the same support, we may deduce that, if B is any least supported basis and f is any quasiminimally supported function, then the support of f must coincide with the support of some basic function b 2 B. 3. Local linear independence and least supported bases Let us introduce the concept of local linear independence. Our denition is equivalent with other classical denitions of local linear independence in spaces of spline functions. Denition 3.1. A nite collection B of functions on some topological space P is locally linearly independent if, for any open set D and any 2 R B, b2b bb = 0 on D implies that b b = 0 on D for all b 2 B. page 294 of Numer. Math. 67: 289{301 (1994)

7 Least supported bases and local linear independence 295 Clearly, taking D =, if b 1 ; : : : ; b n are nonzero and locally linearly independent functions, they are linearly independent. If B is a basis of the open sets in, then any open set D in can be written as a union of elements in B. Then it is easy to show that it is sucient to use sets B 2 B instead of all the open sets D. Usually will be a subset of R n with the induced topology. In this case we can use the basis of open sets B = fb (x) \ j8 > 0; 8 x 2 g, where B (x) = fy 2 j ky? xk < g for a given norm k k. If = R m, the open connected and bounded sets form a basis of open sets, and so our denition coincides with that of Dahmen and Micchelli (1985). The following result gives simple characterizations of local linear independence. Proposition 3.2. Let u 1 ; : : : u n be functions dened on. Then the following conditions are equivalent: (i) u 1 ; : : : ; u n are locally linearly independent. P n (ii) For any linear combination f = i=1 iu i, j 6= 0 implies that supp f supp u j. (iii) For any 2 R n, supp ( P n i=1 iu i ) = S fsupp u i j i 6= 0g. P n Proof. Let us prove rst the equivalence between (i) and (ii). Let f = i=1 iu i and j 2 f1; : : : ; ng such that j 6= 0. Then supp f supp u j if and only if nsupp f nsupp u j. Since these complementary subsets are open in, the last condition is equivalent to the following condition: \for any D open in, if D nsupp f then D nsupp u j ", which is equivalent to saying that \for any D open in, f = 0 on D implies P u j = 0 on D". Therefore we have seen n that (ii) is equivalent to saying that if i=1 iu i = 0 on D, then i u i = 0 on D, for all i = 1; : : : ; n. Now let us show that (ii) and (iii) are equivalent. Clearly (iii) implies (ii). Let us assume now that (ii) holds. Then supp f supp u j for all j such that j 6= 0 and so supp f [fsupp u j j j 6= 0g. Finally, if x 0 62 [fsupp u j j j 6= 0g, then there exists an open set D in such that x 0 2 D and u i (x) = 0 8 x 2 D, for all i such that i 6= 0, and so f(x) = 0 8 x 2 D. Therefore x 0 62 supp f and so supp f [fsupp u j j j 6= 0g. Let us see now that the least supported bases coincide with the locally linearly independent bases. For this purpose let us state the following matricial result, which is a clear consequence of the denition of the determinant det A = X 2 n sign ()a 1(1) a n(n) ; where n is the symmetric group of order n and sign () is the signature of the permutation. Lemma 3.3. Let A be a nonsingular n n matrix. Then there exists a permutation matrix P such that AP has nonzero diagonal elements. Theorem 3.4. Let B be a basis of a nite dimensional vector space of functions U. Then B is locally linearly independent if and only if B is a least supported basis. page 295 of Numer. Math. 67: 289{301 (1994)

8 296 J.M. Carnicer, J.M. Pe~na Proof. First let us assume that B = (b 1 ; : : : ; b n ) is locally linearly independent and let C = (c 1 ; : : : ; c n ) be another basis of U. Let K be the matrix of change of basis (c 1 ; : : : ; c n ) = (b 1 ; : : : ; b n )K: By Lemma 3.3 there exists a permutation matrix P such that KP has nonzero diagonal elements. Let be the permutation of indices such that (c (1) ; : : : ; c (n) ) = (c 1 ; : : : ; c n )P. Then (c (1) ; : : : ; c (n) ) = (b 1 ; : : : ; b n )KP: Since KP has nonzero diagonal elements, by Proposition 3.2 supp b i supp c (i) and then supp B supp C. Let us assume now that B = (b 1 ; : : : ; b n ) is least supported. Then if f = P n i=1 ib i is any linear combination with j 6= 0, we shall see that supp f supp b j and by Proposition 3.2 b 1 ; : : : ; b n will be locally linearly independent. Let us consider the basis (b 1 ; : : : ; b j?1 ; f; b j+1 ; : : : ; b n ) of U. We know that there exists a permutation such that supp f supp b?1 (j) ; supp b i supp b?1 (i) ; i 6= j: Let k be the smallest positive integer such that k (j) = j. Then and so supp f supp b j. supp f supp b?1 (j) supp b?k (j) = supp b j Remark 3.5. Now we know that if a least supported basis exists then it is locally linearly independent and so satises (ii) and (iii) of Proposition 3.2. On the other hand, if there exists a basis of locally linearly independent functions then it is least supported and so we may apply all the properties of this kind of basis. In particular, from Remark 2.10 we may derive: a) Any quasiminimally supported function can be completed to a locally linearly independent basis. b) If B is any locally linearly independent basis and f is any quasiminimally supported function then supp f must coincide with the support of some basic function b 2 B. Let us analyze now the question if a given vector space possesses a locally linearly independent basis. From Remark 2.10 and Theorem 3.4, we know that if a basis constructed as suggested in Theorem 2.6 is not locally linearly independent then the given space has not any locally linearly independent basis. This criterium will be applied in the next example. Example 3.6. Let us recall (see de Boor and DeVore (1983), de Boor and Hollig (1982/3)) that for any given v 1 ; : : : ; v n 2 R s, the box spline B(xjv 1 ; : : : ; v n ) is dened as the unique compact supported function satisfying Z R s f(x)b(xjv 1 ; : : : ; v n ) = Z [0;1] n f(t 1 v 1 + t n v n )dt 1 dt n ; 8 f 2 C(R s ): Let b : R 2! R be the function given by B(x; yje 1 ; e 2 ; e 1 + e 2 ; e 1? e 2 ), where e 1 = (1; 0) T, e 2 = (0; 1) T and let page 296 of Numer. Math. 67: 289{301 (1994)

9 Least supported bases and local linear independence 297 B :=? b(x; y); b(x + 1; y + 1); b(x + 1; y? 1); b(x + 2; y); b(x + 1; y); b(x + 2; y + 1); b(x + 2; y? 1) : One can check that B is a system of linearly independent functions, which are translates of b(x; y) and generate a 7-dimensional vector space U of piecewise quadratic C 1 functions in R 2. Taking into account the pattern of the supports shown in Fig. 1, it is straightforward to see that the given basis consists of minimally supported functions. However B is not locally linearly independent as follows from the fact that b(x; y) + b(x + 1; y + 1) + b(x + 1; y? 1) + b(x + 2; y)? b(x + 1; y)? b(x + 2; y + 1)? b(x + 2; y? 1) = 0 ; is the intersec- for all (x; y) 2 D, where D = (x; y) 2 R 2 j 0 x y 1? x tion of the supports of all basic functions (see Fig. 1). Fig. 1. Supports of the basic functions and domain of dependence Therefore U has not any locally linearly independent basis. More generally, if we consider a criss-cross triangulation of the plane (that is, a rectangular grid rened by adding all diagonals of the cells), the box-spline b (and all its translates b(? i;? j), i; j 2 Z) is minimally supported among the C 1 and piecewise quadratic functions on the given triangulation, but the space of all spline functions supported on a suciently large domain has no locally linearly independent basis. This shows a dierence between univariate and multivariate polynomial spline spaces. Given a partition of an interval into subintervals, one can always nd a locally linearly independent basis of spline functions (the B- spline basis). However the previous example shows that, even in very simple domains of the plane with a regular partition, the corresponding spline spaces have no locally linearly independent bases. In the next theorem we give a characterization of the bases of locally linearly independent functions when we know the existence of one. Furthermore, the condition (v) will allow us to construct all such bases from a given one. Theorem 3.7. If B = (b 1 ; : : : ; b n ) is a locally linearly independent basis of U and C = (c 1 ; : : : ; c n ) is another basis of U, the following conditions are equivalent: (i) C is locally linearly independent. (ii) C is a least supported basis. page 297 of Numer. Math. 67: 289{301 (1994)

10 298 J.M. Carnicer, J.M. Pe~na (iii) supp B = supp C. (iv) C consists of quasiminimally supported elements and the number of basic elements of given support{degree in C and B coincides (v) There exists a permutation matrix P and a nonsingular matrix K = (k ij ) 1i;jn with k ij = 0 if supp b i 6 supp b j such that (c 1 ; : : : ; c n ) = (b 1 ; : : : ; b n )KP: Proof. By Theorem 3.4, B is least supported. Theorem 3.4 provides the equivalence of (i) and (ii). The equivalence of (ii) and (iii) follows immediately because B is least supported. If (iv) holds, then using Proposition 2.5 we derive (iii). Conversely, if (iii) holds for some basis C then by Theorem 2.6 we may construct a basis D as suggested in Theorem 2.6, which will be minimally supported and so supp D = supp B because B is least supported. Since supp B = supp C we obtain supp C = supp D and by Remark 2.7, (iv) follows. Let us see that (v) implies (iii). Since K is nonsingular, C is a basis of U. Let be the permutation such that (3:1) (c 1 ; : : : ; c n ) = (c (1) ; : : : ; c (n) )P: Then (3:2) (c (1) ; : : : ; c (n) ) = (b 1 ; : : : ; b n )K; which means that for any 1 j n (3:3) c (j) = nx i=1 k ij b i with k ij = 0 if supp b i 6 supp b j, and then by Theorem 3.2 supp c (j) = [ fsupp b i j supp b i supp b j g = supp b j : In consequence supp B = supp C. Finally let us see that (iii) implies (v). Let be the permutation such that supp b i = supp c (i) for all i and let P be the matrix such that (3.1) holds. If K is the matrix of change of basis satisfying (3.2) (and so is nonsingular) then (3.3) holds. By Theorem 3.2, if k ij 6= 0, supp b i supp c (j) = supp b j, and the result follows. Remark 3.8. Let us observe that condition (iv) of the previous theorem cannot be replaced by the following weaker condition: the functions in C are quasiminimally supported of some degree. In fact it is very easy to construct examples of spaces with a basis B of locally linearly independent functions, where we may choose bases C of quasiminimally supported functions which are not locally linearly independent. For instance, consider B = (b 1 ; b 2 ), C = (b 1 + b 2 ; b 2 ), where b 1, b 2 are the functions dened on [0; 2] given by b 1 = 1?x+j1?xj and b 2 = 1?j1?xj. Now we may characterize the quasiminimally supported functions of any space with a basis of locally linearly independent functions. Condition (iii) of page 298 of Numer. Math. 67: 289{301 (1994)

11 Least supported bases and local linear independence 299 the next result will allow to obtain all quasiminimally supported functions for such space. Theorem 3.9. Let U be a nite dimensional vector space which has a locally linearly independent basis B = (b 1 ; : : : ; b n ). For any nonzero function f in U the following conditions are equivalent: (i) f is quasiminimally supported of some degree. (ii) f cannot be generated by the functions g such that supp g $ supp f. (iii) f can be expressed as f = X b2b supp b=supp f b b + and b 6= 0 for some b with supp b = supp f. X b2b supp b$supp f Proof. (i) =) (ii) Let f be a quasiminimally supported function and l = sdeg (f). Any function g with supp g $ supp f must belong to the vector space V l?1. Taking into account that f 62 V l?1 (ii) follows. P (ii) =) (iii) Since B is a basis, we may write f = b2b bb. If supp b 6 supp f then, by Proposition 3.2, b = 0. Therefore f = X b2b supp b=supp f b b + X b2b supp b$supp f and by the hypothesis there exists b 6= 0 for some b with supp b = supp f. (iii) =) (i) By hypothesis there are basic functions b j1 ; : : : ; b jk 2 B such that supp b ji = supp f and all of them are quasiminimally supported of the same support{degree l by Theorem 3.7 (iv) and by Proposition 2.5. If g 2 U and supp g $ supp f = supp b ji, then g 2 V l?1 because sdeg (b ji ) = l. In order to prove that f is quasiminimally supported it remains to show that f =2 V l?1. Otherwise, since V l?1 has a basis formed by functions in B (by Remark 2.10), when expressing f in terms of the basis B, the coecients corresponding to basic elements with degree of quasiminimality higher than l? 1 would be zero, which contradicts (iii). Now, from the previous theorem, Proposition 3.2 and Remark 3.5 a), one can deduce the equivalence between Chui's denition of quasiminimally supported function and our denition of quasiminimally supported function of degree 1 when the space possesses a locally linearly independent basis. b b b b 4. An example for spaces with totally positive bases Let us recall that a matrix A is called totally positive (TP) if all its minors are nonnegative. Let U be a nite dimensional vector space of functions dened on a subset of the real line. A basis B = (b 1 ; : : : ; b n ) is called totally positive if for any t 1 < t 2 < < t n in all the collocation matrices (u j (t i )) n i;j=1 are totally positive. Many important univariate spline spaces have TP bases: for instance, polynomial splines, Tchebychean splines, spline spaces for the generation of curves page 299 of Numer. Math. 67: 289{301 (1994)

12 300 J.M. Carnicer, J.M. Pe~na with geometric continuity, etc. [cf. Goodman (1989)]. One reason for the interest in TP bases when solving interpolation problems is that the collocation matrices are TP, which implies that the linear system can be solved without row exchanges. Furthermore, any other choice of the pivots leads to less accurate solutions [see de Boor and Pinkus (1977), Gasca and Pe~na (1993)]. Besides it is numerically convenient that the TP basis has small support. In this case the collocation matrix is banded and the band structure is preserved during the elimination process. Thus it will be interesting to obtain the TP bases with least support among all TP bases. Denition 4.1. Let S be a set of bases of U. We say that a basis B 2 S is a least supported basis among all bases in S if supp B is the minimum (or least element) in the set fsupp CjC 2 Sg. On the other hand, TP bases are useful in Computer Aided Geometric Design because of their good shape preserving properties [cf. Goodman (1989), Goodman and Said (1991) and Carnicer and Pe~na (1993a)]. Let us denote by T the set of all TP bases of U. In Carnicer and Pe~na (1993c) it was shown that in any space with a TP basis there always exists a TP basis B = (b 1 ; : : : ; b n ) 2 T (which has been called B-basis) which can be characterized by the following property: (4:1) T = f(b 1 ; : : : ; b n )Kj K is a nonsingular TP matrix g: Geometrically this means that the B-basis has optimal shape preserving properties among all bases in T. In Carnicer and Pe~na (1993a) it was shown that the Bernstein basis is a B-basis for the space of polynomials of degree less than or equal to n and in Carnicer and Pe~na (1993c) it has been proved that the B-splines form a B-basis for the space of polynomial splines. The next result shows that a B-basis is least supported among all TP bases. Proposition 4.2. Let U be a nite dimensional vector space of functions which has a TP basis. Then any B-basis B = (b 1 ; : : : ; b n ) is least supported among all TP bases. Proof. Let C = (c 1 ; : : : ; c n ) be any TP basis for U. By (4.1) there exists a nonsingular TP matrix K = (k ij ) i;j=1;:::;n such that c j = nx i=1 k ij b i ; j = 1; : : : ; n: By Corollary 3.8 of Ando (1987), k ii > 0 for all i, and since the functions b i (i = 1; : : : ; n) are nonnegative supp c j supp b j ; j = 1; : : : ; n: Therefore supp B supp C for all C 2 T. Corollary 4.3. Let U be a nite dimensional vector space which has a locally linearly independent TP basis C. Then any B-basis B is also locally linearly independent. page 300 of Numer. Math. 67: 289{301 (1994)

13 Least supported bases and local linear independence 301 Proof. By Theorem 3.4 C is a least supported basis and, by Proposition 4.2, supp B supp C and then B is also a least supported basis. Again by Theorem 3.4 B is locally linearly independent. The previous corollary provides a method to determine if a space with a TP basis has a locally linearly independent TP basis (or equivalently a least supported basis which is TP). Given any TP basis C, a construction of a B- basis B for the space is given in Carnicer and Pe~na (1993c). The space will have a locally linearly independent TP basis if and only if the constructed basis B is locally linearly independent. Acknowledgement. The authors wish to thank Dr. Carl de Boor for his valuable suggestions to improve this paper. References Ando, T. (1987): Totally positive matrices. Linear Algebra Appl. 90, 165{219 de Boor, C., DeVore, R. (1983): Approximation by smooth multivariate splines. Trans. Amer. Math. Soc. 276, 775{785 de Boor, C., Hollig, K. (1982/83): B-splines from parallelepipeds. J. Analyse Math. 42, 99{115 de Boor, C., Pinkus, A. (1977): Backward error analysis for totally positive linear systems. Numer. Math 27, 485{490 Carnicer, J.M., Pe~na, J.M. (1993a): Shape preserving representations and optimality of the Bernstein basis. Advances in Comput. Math. 1, 173{196 Carnicer, J.M., Pe~na, J.M. (1993b): A Marsden's type identity for periodic trigonometric splines. J. Approx. Theory, to appear Carnicer, J.M., Pe~na, J.M. (1993c): Totally positive bases and optimality of B-splines. Computer Aided Geometric Design, to appear Chui, C.K. (1988): Multivariate splines. Regional Conference Series in Applied Mathematics 54. SIAM, Philadelphia Chui, C.K., He, T.X. (1988): On minimal and quasi{minimal supported bivariate splines. J. Approx Theory 52, 217{238 Dahmen, W., Micchelli, C.A. (1985): On the local linear independence of translates of a boxspline. Studia Mathematica 82, 243{263 Gasca, M., Pe~na, J.M. (1993): Scaled pivoting in Gauss and Neville elimination for totally positive systems. Applied Numer. Math., to appear Goodman, T.N.T. (1989): Shape preserving representations. In: T. Lyche, L.L. Schumaker, eds., Mathematical methods in CAGD, pp. 333{357. Academic Press, Boston Goodman, T.N.T., Said, H.B. (1991): Shape preserving properties of the generalized Ball basis. Computer Aided Geometric Design 8, 115{121 Schumaker, L.L. (1976): On Tchebychean spline functions. J. Approx. Theory 18, 278{303 Schumaker, L.L. (1981): Spline functions: basic theory. John Wiley, New York page 301 of Numer. Math. 67: 289{301 (1994)