New upper bound for the B-spline basis condition number II. K. Scherer. Institut fur Angewandte Mathematik, Universitat Bonn, Bonn, Germany.

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1 New upper bound for the B-spline basis condition nuber II. A proof of de Boor's 2 -conjecture K. Scherer Institut fur Angewandte Matheati, Universitat Bonn, 535 Bonn, Gerany and A. Yu. Shadrin Coputing Center, Siberian Branch, Russian Acadey of Sciences, Novosibirs, Russia For the p-nor condition nuber ;p of the B-spline basis of order we prove the upper estiate ;p < 2 : This proves de Boor's 2 -conjecture up to a polynoial factor.. Introduction It is of central iportance for woring with B-spline series that its condition nuber is bounded independently of the underlying not sequence. This fact was proved by C. de Boor in 968 for the sup-nor and in 973 for any L p -nor (see [B] for references). In the paper [B2] he gave the direct estiate ;p < 9 (:) for ;p, the worst possible condition nuber with respect to the p-nor of a B-spline basis of order, and conjectured that the real value of ;p grows lie 2 : ;p 2 ; (:2) which is seen to be far better than (.). The conjecture was based on nuerical calculations of soe related constants which oreover gave soe evidence that the extree case occurs for a not sequence without Supported by a grant fro the Alexander von Huboldt { Stiftung

2 interior nots (the so-called Bernstein nots). Maybe due to this reason, a few papers devoted to the 2 -conjecture for ;p were concerned only with the \Bernstein nots" conjecture for the extree not sequence, see [B3], [C], [Ly], [S]. These papers gave further support for de Boor's conjecture (.2), in particular T. Lyche [Ly] obtained a lower bound for ; fro which it follows [S] that ;p > c =p 2 : (:3) In the unpublished anuscript [SS] we returned to de Boor's direct approach in [B2], and considered the possibility of iproving his 9 -estiate by several odications of his ethod. In particular, a slight revision based on Kologorov's estiate for interediate derivatives had shown that ;p < ; = 6:25: In the previous paper [SS2] we developed a further approach to obtain ;p < =2 4 : In this paper using the sae approach we give a surprisingly short and eleentary proof of Theore. For all and all p 2 [; ], ;p < 2 : (:4) With respect to (.2)-(.3), this conrs C. de Boor's conjecture up to a polynoial factor. We show also that the optial factor which can be obtained in (.4) within this approach is =2 and discuss further possible approaches by which this factor could be reoved. 2. Condition nuber and related constants Let f ^N j g be the B-spline basis of order on a not sequence t = (t j ), t j < t j+, noralized with respect to the L p -nor ( p ), i.e., ^N j (x) = (=(t j+ t j )) =p N j (x); where fn j g is the B-spline basis which fors a partition of unity. Recall here that and that N j (t) = ([t j ; : : : ; t j+ ] [t j+ ; : : : ; t j+ ]) ( t) + N j (x) > 0; x 2 (t j ; t j+ ); N j (x) = 0; x =2 [t j ; t j+ ]; X Nj = : 2

3 The condition nuber of the L p -noralized basis f ^Nj g is dened as ;p;t := sup b b lp P b j ^Nj Lp sup b P b j ^Nj Lp b lp = sup b b lp P b j ^Nj Lp ; where the L p -nor is taen with respect to the sallest interval containing the not sequence (t i ). The last equality in the above denition follows fro noralization so that ^N j (x) = M =p j (x)n =q j (x); M j (x) := t j+ t j N j (x); Z M j (x) dx = ; P b j ^Nj Lp = P b j M =p j N =q j Lp ( P b p jm j ) =p ( P N j ) =q Lp = ( P b p jm j ) =p Lp = P b p jm j =p L b lp ; with equalities for b j = ((t j+ t j )=) =p. The worst B-spline condition nuber is dened then as ;p := sup ;p;t : t Its value gives a easure for the unifor stability of the B-spline basis and is iportant for nuerical calculations with B-splines. Following [B2] we introduce now related constants that are upper bounds for ;p. This has been done already in [SS2] but for convenience of the reader we state here again the relevant leas. More details can be found in [B], [B2] and [S]. and let Lea A. Let H i be the class of functions h 2 L q such that where =p + =q =. Then D ;p := sup t ) supp h [t i ; t i+ ] R 2) hnj = ij sup i n o inf (ti+ t i ) =p h q h2h i ;p D ;p : Now set i(x) := Y (x t i+ ): ( ) = 3

4 Then an easy way for obtaining h 2 H i is to set h = (g i ) () for soe appropriate sooth function g. We forulate this as Lea B. Let G i be the class of functions g such that and let G () i := f(g i ) () : g 2 G i g. Then ) g i 2 W q [t i; t i+ ]; ( 0; -fold at ti ; 2) g i = i; -fold at t i+ ; G () i H i : Cobining Leas A and B gives Corollary. ;p B ;p := sup t sup i inf g2g i n (ti+ t i ) =p (g i ) () q o : Finally, due to the local character of the quantity B ;p, it is sucient to restrict attention to the eshes of the for = (t 0 t : : : t ) ; t 0 < t : Set also and (x) := (x) = Y (x t i ) = 0 (x); (2:) ( ) N(t) = N (t) = ([t 0 ; : : : ; t ] [t ; : : : ; t ]) ( t) + : i= Lea C. For given via as in (2:), let G be the class of functions g such that ) g 2 Wq [t 0 ; t ] ( 0; -fold at t0 ; 2) g = ; -fold at t ; and let Then B ;p := sup inf (t t 0 ) =p (g ) () q : g2g ;p B ;p B ; : (2:2) Rear. Lea A is taen fro [B2, p.23] whereas Leas B and, respectively, C are soewhat ore accurate versions of what is given in [B2, Eq.(4.)]. Naely, they show 4

5 the possibility to choose a soothing function g depending on. C. de Boor's estiate of B ; resulting in (:) was based on the inequalities B ; inf sup g2g X i= (g) () inf g () X g2g i= sup ( ) ; g () sup ( ) with soe special choice of g 2 G := \G that is seen to be independent of. Notice, that in the latter su for any choice of g 2 G the ter with = is equal at least to 4 (see [B2,p.32]). 3. Proof of Theore. The idea in the previous paper [SS2] was to choose g 2 G as the indenite integral of the L -noralized B-spline, i.e., g (x) := Z x N (t) dt: t t 0 t 0 Then, the inclusion g 2 G is alost evident (see [SS2]), and thus we can ajorize the constant B ; by B ; S ; := sup (t t 0 )s () (3:) where s := g : (3:2) Notice that supp s () [t 0 ; t ], so that actually the L -nor in (3.) is taen over [t 0 ; t ]. In view of X (t t 0 ) s () (x) = N ( ) (x) ( ) (x); (3:3) = we showed in [SS2] that, for any and = ; : : : ;, N ( ) ( ) which, by Lea C and (3.)-(3.3), iplies the bound Here we iprove (3.4) by Lea. For any, and = ; : : : ; ;p < =2 4 : ; (3:4) N ( ) ( ) : (3:5) 5

6 Now, by (3.)-(3.5) and Lea C, ;p S ; X = = (2 ) < 2 which proves Theore. Rear. If has a ultiple zero := t = t + = : : : = t +p of ultiplicity p, then N ( p ) has a jup at. In this case we can dene the value N ( p +q) ( ) (p q) ( ) as a liit either fro the left or fro the right. This liit is equal to zero, if 2 (t 0 ; t ). Also this denition justies the equality (3.3). 4. Lee's forula and a lea of interpolation For arbitrary r 2 Z + and t 2 R, set r (x; t) := r (x t)r +; and dene Q (x; t) and Q 2 (x; t) as algebraic polynoials of degree with respect to x that interpolate the function (; t) on the eshes = (t 0 ; t ; : : : ; t ); 2 = (t ; : : : ; t ; t ); respectively. The following nice forula is due to Lee [L]. Lea D [L]. For any, N(t) (x) = Q (x; t) Q 2 (x; t): (4:) Proof [L]. The dierence on the right-hand side is an algebraic polynoial of degree with respect to x that is equal to zero at x = t ; : : : ; t, hence Y Q (x; t) Q 2 (x; t) = c(t) (x t i ): i= Further, since the leading coecient of the Lagrange interpolant to f on the esh ( i ) i= is equal to [ ; : : : ; ]f, we have and the lea is proved. c(t) = ([t 0 ; : : : ; t ] [t ; : : : ; t ]) (; t) =: 6 ( ) N(t);

7 We will use Lee's forula (4.) to evaluate the product N ( ) (t) ( ) (t) by taing the corresponding partial derivatives with respect to x and t in (4.) and setting x = t. Our next two leas give a bound for the values obtained in that way on the righthand side of (4.). For arbitrary p 2 N, p r, and any sequence = ( 0 : : : p ); dene, for a xed t, Q t (x) := Q(x; t) := Q(x; t; r ; ) as the polynoial of degree p with respect to x that interpolates r (; t) at. Lea 2. For any adissible p; r; t;, where the derivative is taen with respect to x. Proof. First we prove 0 Q (r) t (x) x=t (4:2) A. The case r = 0. Then Q t () is a polynoial of degree p that interpolates, for this xed t, the function (x t) 0 + := ( ; x t; 0; x < t: We have to prove that 0 Q t (x) x=t (4:3) and distinguish the following cases: A. If t = i for soe i, then (4.3) is evident. A2. If all the points of interpolation lie either to the left or to the right of t, i.e., if p < t; or t < 0 ; then Q t 0; or Q t ; respectively, and (4.3) holds. A3. If t lies between two points, i.e., for soe 0 : : : < t < + : : : p ; then in view of Q 0 t(x) = [Q t 0 (; t)] 0 (x) for x 6= t, the polynoial Q 0 t(x) has at least zeros on the left of, and at least p zeros on the right of +, which gives p zeros in total. Hence Q 0 t has no zeros in ( ; + ), so that Q t is onotone in ( ; + ), that is 0 = Q t ( ) < Q t (t) < Q t ( + ) = : 7

8 B. The case r > 0. This case is reduced to the case r = 0 by Rolle's theore. The dierence r (; t) Q t has p + zeros (counting ultiplicity), thus its r-th derivative 0 (; t) Q (r) t ust have at least p + r changes of sign. If (4.2) does not hold, then this function does not change sign at x = t, and Q (r) t is a polynoial of degree p r that interpolates 0 (; t) at p r + points all distinct fro t. But according to the Case A3 this would iply (4.2), a contradiction. Hence, (4.2) holds, and the lea is proved. Lea 3. For any adissible p; r; t;, 0 ( ) r s Q(x; t) x=t : (4:4) Proof. Let l i be the fundaental Lagrange polynoials of degree p for the esh, i.e., l i ( j ) = ij. Then Q t = Q(; t), which is the Lagrange interpolant to r (; t), can be expressed as Thus, we obtain It is readily seen that Q(x; t) = r px i=0 ( sq(x; t) = (r s) ( i t) r + l i(x): px i=0 ( i t) r s + l i (x): Q 0;t (x) := Q 0 (x; t) := ( t) is a polynoial of degree p with respect to x that interpolates r s ( ; t) = ( t)r s + (r s) at the sae esh. Now (4.4) follows fro Lea Proof of Lea We need to bound N (s) (t) ( s) (t) = N (s) (t) ( s) (x) x=t ; s = 0; ; : : : ; : Now according to Lea D N (s) (t) ( s) (x) sq (x; s Q 2 (x; t); and by Lea 3 for any 0 ( ) Q (x; t) s 8 x=t :

9 Hence, since both ters are of the sae sign and of absolute value, which proves Lea. N (s) (t) ( s) (t) ; 6. On the factor in Theore Nuerical coputations [B3] show that so a natural question is whether the factor in the bound ;p c 2 ; (7:) ;p < 2 (7:2) of Theore can be reoved. A siple exaple will show now that within the particular ethod we used in Section 3 (see (3.)), an extra polynoial factor p appears unavoidably. Naely, one can prove that for soe choice of S ; (t t 0 ) s () c =2 2 : In fact, in the case of the Bernstein nots in [0; ], i.e., for we have and obtain s () (x) = ( ) X = (x) = ( ) x (x ) ; N (x) = x ( x) ; hx ( x) i ( ) h x (x ) i ( ) : It is not hard to see that at x = the -th ter vanishes, unless = +, which gives js () ()j = ( ) With this, we tae + = b=2c to obtain js ()j = + b=2c ( ) = > c =2 2 : + : 9

10 7. Possible reneents We describe here soe further approaches that ay perit reoval of the polynoial factor in the upper bound for the sup-nor condition nuber ;.. The rst approach is to ajorize ; using the interediate estiate (2.2) with the value B ; instead of B ; used in Theore, that is ; B ; : Then the desired 2 -bound without an extra factor will follow fro the following Conjecture. For any =, there exists a function g 2 G such that sign g () (x) = sign ( ) (x); x 2 [t 0 ; t ]; = ; : : : ; : (7:) This conjecture iplies that g () ( ) L [t 0 ;t ] = Z t t 0 g () (x) ( ) (x) dx : Then observe that, because of the boundary conditions satised by g and the way g and are noralized, Z ( ) t Z g () (x) ( ) t (x) dx = g(x) 0 ( ) (x) dx = : t 0 Hence t 0 g () ( ) L [t 0 ;t ] = ; = ; : : : ; ; (7:2) and using this bound, one could show, exactly as in Section 3, that ; B ; X = = 2 : Rear. A function g satisfying (7.) should in a sense be close to the function g considered in Section 3 (though it is not necessarily unique). Moreover, g can serve as g for the polynoials with the Bernstein nots (x) = c(x t 0 ) (x t ) : Also, it loos quite probable that, even though the equality (7.2) is not valid with g = g for arbitrary, there holds g () ( ) L [t 0 ;t ] c; = ; : : : ; ; that is for the B-spline M (x) = (=(t t 0 )) N (x) we have 0

11 M ( ) ( ) L [t 0 ;t ] c: 2. Another possibility to iprove the result of Theore would be to nd a sharp bound for one of the related constants considered in [S]. In this respect it is nown, e.g., that ; E ;p (7:3) where E ;p := inf inf j X inffn j c i N i p g: c i i6=j In particular, there is equality in (7.3) for p =. The hope is to prove that the not sequence at which the value E ;p is attained for p = or p = 2 is the Bernstein one, in which case the inequalities E ; < c2 ; or E ;2 < c2 would follow. (It is nown that the Bernstein not sequence is not extree for p =, see [B3]). References [B] C. de Boor, Splines as linear cobinations of B-splines, a survey, in \Approxiation Theory II" (G. G. Lorentz, C. K. Chui and L. L. Schuaer, Eds. ), pp. {47, Acadeic Press, New Yor, 976. [B2] C. de Boor, On local linear functionals which vanish at all B-splines but one, in \Theory of Approxiation with Applications" (A. G. Law and B. N. Sahney, Eds.), pp.20{45, Acadeic Press, New Yor, 976. [B3] C. de Boor, The exact condition of the B-spline basis ay be hard to deterine, J. Approx. Theory, 60 (990), 344{359. [C] Z. Ciesielsii, On the B-spline basis in the space of algebraic polynoials, Ur. Math. J., 38 (986), 359{364. [L] E. T. Y. Lee, Marsden's Identity, Coput. Aided Geo. Design, 3 (996), 287{305. [Ly] T. Lyche, A note on the condition nuber of the B-spline basis, J. Approx. Theory, 22 (978), 202{205. [S] K. Scherer, The condition nuber of B-splines and related constants, in \Open Probles in Approxiation Theory" (B. Bojanov, Ed.), pp.80{9, SCT Publishing, Singapore, 994. [SS] K. Scherer, A. Yu. Shadrin, Soe rears on the B-spline basis condition nuber, unpublished anuscript. [SS2] K. Scherer, A. Yu. Shadrin, New upper bound for the B-spline basis condition nuber I, East J. Approx. 2 (996), 33{342.