b-splines from parallelepipeds

B - S P L I N E S F R O M P A R A L L E L E P I P E D S

By

C. DE BOOR' AND K. HOLLIG *~

O. I n t r o d u c t i o n

Following [BH~], we define the B-spline MB as the m-shadow of the polyhedral

convex body B C_ R", i.e., as the distribution on R" given by the rule

f (1) M.~b:= j ~b oP, all ~b @ C~(R"),

B

Here, P :R"--~R" :x ~(x(i))'~ is the canonical projection and fK denotes the

k-dimensional integral over K in case dim K = k, i.e., K spans a k-dimensional

fiat.

It is obvious that M~ is nonnegative, with supp MB C PB. It is easy to see that MB

is a piecewise polynomial function of degree < _ - n - m once one knows the

recurrence relation [BH~]

(2) DpzMB = - E {z l n,}M.,, l

Here,

D,f:= E y(i)D~f,

all z E R".

with D,[ the partial derivative of f with respect to its i-th argument. Further, B,

denotes the typical (n -1)-dimensional polyhedron of which the boundary of B

consists, and n, denotes the corresponding outward normal. Finally, {. [. ) denotes

the scalar product.

In principle, MB can be evaluated with the aid of the stable recurrence [BH~]

(3) (n - m ) M . ( P z ) = ~ . {b,-zln,)M.,(Pz), all z ~R" , t

with b, an arbitrary point in the fiat spanned by B,.

t Sponsored by the United States Army under Contract No. DAAG29-80-C-01NI. " Supported in part by the National Science Foundation under Grant No. MCS-7927062.

99 JOURNAL D'ANALYSE MATHI~MATIQUE, Vol, 42 (1982183)

100 C. DE BOOR AND K. HOLLIG

Cases of particular interest are: (i) the simplex spline, obtained when B is a simplex. These B-splines were

introduced in [B] following up on IS] and have already been studied intensively,

mostly by W. Dahmen and C. A. Micchelli [Mr_2], [D~], [DM~_3], but also by

Goodman & Lee [GL], Hakopian [Hk~_3], and by H611ig [Hl1_2]. (ii) the truncated powers or cone splines, obtained when B is a proper polyhedral

cone spanned by some basis for R n. These were introduced by Dahmen [D2] and

further studied by Dahmen and Micchelli in [DM3]. For example, they show that,

near an extreme point of its support, a simplex spline coincides with a truncated

power. (iii) the box spline, obtained when B is a parallelepiped. These splines were

introduced in [BD] and are the object of study of the present note. To be precise, a box spline is, by a definition slightly more general than the one in

[BD], a distribution M_ on R" given by the rule

(4) M=_: ~b ~ f 1o,11'

for some sequence -=:= (~:,)~ in R". Let

If dim (_=) = m, then

with

(E) := span -= := { ~-~ A (i)~:' ,=1 /

M_= = MB/vol, B

/ ^ ~ for the parallelepiped spanned by some linearly independent sequence (~,)1 in R"

which P~, = ~:,, all i, and volrB the r-dimensional volume of B. We would like,

though, to consider Mz also in case dim ( E ) < m.

For this, we find it convenient to enlarge the above definition of the B-spline MB

by allowing P in (1) to be an arbitrary linear map on B into R ' . Then (1) defines MB as the P-shadow of B. One checks that this leaves the recurrence relations (2) and

(3) unchanged (see Sect. 1).

In these terms, the box spline M~ defined by (4) is the P-shadow of the box [0, 1]',

with P the linear map given by

PX:= ~ ,~ (i)~,. i = 1

Here is an outline of the paper. We discuss P-shadows in Section 1. In Section 2, we give some basic information about the box spline M~, such as its recurrence

B-SPLINES FROM PARALLELEPIPEDS 101

relations, its Fourier transform, and its relationship to the forward difference

operator A~ = A~ �9 . . . . A~, and to the truncated powers. We show in Section 3 that it

is usually possible to make a partition of unity out of the box spline and certain of

its translates in many ways. We use this fact in Section 4 to show that the box spline

and its translates are usually globally linearly dependent, thus destroying all hopes

for stability or the existence of a set of dual linear functionals for such sets except in

special circumstances. One such is discussed in [BH2].

In the remaining sections, we consider the space

with

S=_:= { ~, a(I")M~ : a ERV I

M~:= M_=(-- j) , all j E V : = Z ",

and under the assumption that -= _C V and that (=_) = R '~. In Section 5, we determine

all polynomials in S_= as well as the largest k for which all polynomials of (total)

degree k or less are contained in S_=. We use this information in Section 6 to

construct a quasi-interpolant from S~ and thereby to obtain statements about the

degree of approximation obtainable from S-.h: = {X ~ f ( x /h ) : f E S=_} as h ~ 0.

We could have obtained our results concerning S_= with the aid of the general

theory of spaces spanned by translates of a fixed function developed by Fix and

Strang [FS], particularly if we had been content to discuss only L2. We chose to

derive our results directly since it seems no more effort to do this than it is to verify

that the general conditions given in [FS] are satisfied for our specific examples.

We point out in Section 2 that S_= C L i d), with

d := max{r : (~ , \Z) = R " for all Z C ~ with ]Z[ = r}

(see Section 2 for how we treat the sequence =, as a set). This raises the question of

the relationship of S_-- to the space of all pp functions in L~ ) on the same mesh and

of degree =< [E I - m. We study this difficult question in [BH2] just for m = 2 and

only for the 3-direction mesh, i.e., for ran E = {el, e2, el + e2}.

N o t a t i o n . With A C R m, we denote by [A ] the convex hull of A and by (A) its

linear span. We use x (r) for the r-th entry of the vector x. For x E R '~ and j E ZT,

the number x j is computed as

x':= x (1)'~',.. x (m y ' ' ,

as usual. We denote by 7r the class of all polynomials (on R"), and by ~'k its

subspace made up of those of total degree no larger than k. Thus


with I j l : = j ( 1 ) + . . . + j ( m ) . We also use D ' : = D~~ D g "~ and, more generally,

p (D) : = Ek a ( k ) D ~ in case p : x ~ E~ a (k)x k. Here, we use the notation D, f for the

partial derivative with respect to its i-th argument of the function f with domain in

R". We also use the notation D~f:=Y.?y( i )D, f . For a sequence of vectors in ~",

such as -= = (~:~ . . . . . so,), we use

DE : = Del �9 �9 �9 De,.

We also use Az : = Ae,.. �9 A,, and V-- : = ~ T e l , , , ~ 7 e r , with

a J : = f ( . + y ) - f , V y f : = f - f ( . - y).

1. P - s h a d o w s

As defined in the introduction, the P-shadow of a convex polyhedron B in ~" is

the distribution M on R" given by the rule

M : ~ b ~ f ~ b o P , all ~b E D ( R " ) , B

with P : R" ~ R '~ an afline map.

We claim that the recurrence relations for B-splines established in [BH1;

Theorem 2] remain valid for these more general B-splines and state this in the

following theorem, for the record. For this, we make the assumption that P is a

linear map, i.e., P0 = 0. This can always be achieved by a translation in ~". Further,

we assume that B is proper, i.e., n-dimensional. If r: = dim B < n, then this can be

achieved by restricting P to the affine hull of B and identifying this hull with R r.

Given that B is a proper convex polyhedron, its boundary is made up of

(n - 1)-dimensional convex polyhedra Bi, with corresponding outward normals ni,

and b, denotes an arbitrary point in the attine hull of B~. Further, D stands for the

first order differential operator given by the rule

Dr: = 2 ~ , D , f t = l

in case f has its domain in R', with

(q~,f) ( x ) : = x (i ) f ( x ).

Thus ( D f ) ( x ) = (Dxf ) (x ) , and the adjoint of D is - Y . D , ~ , .

T h e o r e m 1. Let B be a proper convex polyhedron in R" and let M be its

P-shadow in R m under the linear map P : R" ~ R ~. Then

(i) Dp, M = - ~., (z I vi)M,, all z ~ R ~.

(ii) (n - m ) M ( P z ) = X, ( b , - z I ,,,)M,(Pz), all z E R ~.

(iii) D M = (n - m ) M - ~ , (hi [ v~)M~.


Proof . The proof is a slight extension of the arguments for Theorem 2 of [BH~]. The proof of (i) is immediate:

B B 8B

f B~

This uses the fact that, by definition, the derivative DyM of the distribution M is

the distribution obtained by the rule f ~ M ( - D y f ) , and the standard interplay

(1) Dy (6 ~ P) = (Dey6)o p

between differentiation and a linear change of variables. This interplay also implies

that

(2) ( V 6 ) ( P x ) = (Dpxqb)(Px) = (D~ (6 o P))(x) = ( D ( 6 ~ P) ) ( x ) ,

of use in proving (iii): We have

therefore

and

Dj qb, = 1 + cI),D.

B B

f ~ D, qb,(~b o p) = n M ~ + f D(q~oP) . 1=1

B B

Here, the last integral in the first line equals the last integral in the second, by (2).

Thus

(DM)~b = (n - m)M~b - ,=1 D,d~, (cb o P). B

This settles (iii) since

f f f , = 1 , = 1 u ( i ) * , ( f o P ) = ( . I v ) f o p B aB OB

and, on the facet B,, (. I v) is constant.


Finally, to prove (ii), conclude from (i) and (iii) that, for any z with Pz = x,

0 = (D - Dez ) M ( x )

= (n - m ) M ( x ) - ~ (b, I1,,)M, ( x ) + ~ , (z Iv, )M, (x). III i i

2. Basic properties of the box spline

The box spline Ms defined in (0.4) (with r = n) is a symmetr ic function of the

sequence = = (~:,)l. In other words, M~, = M_= for any rearrangement ,~' of the

sequence E. For this reason, we find it excusable and, in any case, convenient to

treat _~ in the sequel as if it were a set, o fcardinal i ty n, rather than a sequence, even

though the set {~, : i = 1 . . . . . n} may well have fewer than n elements. Thus, we

write

or

,~A(~:)~: instead of ~ A(i)~i --" I = l

-=/~ instead of ( ~ $ ( , ) ) ~ - - 1

for the appropriate subsequence s(1) . . . . . s (n - 1) of 1 . . . . . n. In the latter example,

this abuse of notation stresses the fact that it doesn't matter which one of the

possibly several occurrences of the vector ~ in the sequence ~ is being omitted. We

also find it convenient to identify R" with R -= and, in particular, the box B = [0,1]"

with [0, l] -=. This means that the outward normal for one of the facets of B is of the

form -+ e, for some ~: E _=, with ee the corresponding unit vector.

It is clear that 34= is nonnegative and that

(1) supp M_= = { ~ A(~)~:A E[O, 1]-=].

Further, from (0.4),

(2) II M=-II = 1

as a linear functional on C(R"). Also,

(3) M=_ ~ L= iff (~) = R ' .

The recurrence relations of Theorem 1 for general B-splines simplify for the box

spline as follows.

Proposit ion 2.1 . I f y = X=_- A (~)~, then

(4) DyM=(y) = ~= A (~:) (M_.-\~ - M_=\, (. - ~:)),


O)

P r o o f .

form

(n - m ) M - ( y ) = e~__~ (h (~)M=,e (y) + (1 - h (~))M-_-\e (y - ~)).

The typical facet (i.e., (n - 1)-dimensional face) of B = [0, 1] -= has the

Be:={/* (E [0, 11-= : / , (~ :) = O}

or else the form e, + B,, for some ~ @ E. Further, Be and ee + Be have the outward

normal - e e and e,, respectively. Thus

j" - B, = I [ /z (~), B, = ee + Be,

and (4) and (5) now follow from Theorem 1.(i) and (ii). Ill

S m o o t h n e s s . We associate with _= the number

(6) d : = max{r : (_=\Z) = R" for all Z C _ ~ with IZ[ = r}

and say for short that E is d-spanning. (We take d = - 1 in case (~ , )# R' . ) The

number d is of interest here since it follows from (4) and (3) that all r-th order

derivatives of M_= are in L| as long as (.~ \ Z ) = R" for all Z C_ ,= with I Z [ = r. Thus

M=_ ~ L (f ) C C (d-l).

Obviously, d cannot be bigger than I ~ , l - m which is the total degree of the

polynomial pieces of which Ms consists. Precisely, on each connected component

of the complement of

(-J {[~\Z]+ ~' rI : H C-Z' (~ \Z}#R"}

M-- agrees with some polynomial of degree --<1~1-m.

E x a m p l e s . (i) Foi" m = 1 and {~ = el, all ~ E ~, M_= is just the forward cardinal

B-spline, i.e., M_--= M ( . ; 0, 1 . . . . . n). For m > 1 and ~ containing only el . . . . . e=

(each at least once), M-- is the tensor product of such univariate B-splines.

(ii) For m = 2 and I_=1= n --3, with d = 1, we obtain a standard linear finite

element.

(iii) For m = 2 and _= = (e,, e2, el + e2, e~ - e2), M_ is a piecewise quadratic

function first studied by Zwart [Z] and independently derived by Powell [P] and

Sabin [PSI. Its support is shown in Fig. 1 together with its "mesh" , i.e., its lines of

transition from one polynomial piece to a neighboring piece. The dotted mesh lines

occur in the above references. Our construction makes clear that they do not

appear in actuality since they do not lie in some one-dimensional image PF of some

face F of [0, 1] 4. Precisely, their boundary endpoint does not correspond to a vertex of Io, 1] 4.


Fig. 1. Support and meshlines for a C'-quadratic box spline.

(iv) Further examples for _= containing only el, e2, e, + e2 and/or el - e2 can be

found in Sablonniere's study [Sb] of smooth finitely supported pp functions on

regular meshes. The "generalized triangular splines" of Frederickson [F~_2] can now

be recognized as spanned by the box spline M~, with ,= containing each of the three

vectors e~, e2, and e3 the same number of times.

Associated difference operator. It follows from (4) that

D~M=_ = M=_x~ - M=_\~ (" - ~) = V~M=_\~,

(9)

From this we see that

(10)

n 1 - e-'<~fx ~

i <~ix >

M~_ * M z = M~_uz.

therefore

(7) DzM=_ = VzM=_xz, for Z C E.

In particular,

D=_-Mz = V=_6,

since Mo = 8 := point evaluation at 0. Therefore

(8) f M~_D~_6 = (A_=6)(0), for all ~b E CIZJ(Rr').

This close association between the box spline and the forward difference operator

brings to mind the well known association

f M ( x ; to . . . . . tn)6~")(x)/n! [ to , . . . , t~]6 dx

between the univariate B-spline and the divided difference.

The Fourier transform of M_~ is quite simple,


S y m m e t r i e s a n d l o c a l s t r u c t u r e . We pointed out earlier that M_= does not

depend on the order of the vectors in the sequence E. This is due to the fact that any linear map in R" which permutes the unit vectors leaves the box [0,1]" invariant. Multiplication of some of the unit vectors by - 1 will change [0, 1]", but [0, 1]" can be restored by a subsequent shift. Therefore

(11) M_=~ = M_-- (. - ~ ~ 1 ~ ) _ _ 2

in case ~== is obtained from ~ by multiplying each ~: E ,.~ by cr(~)E { - 1, 1}. A symmetry of a different sort occurs when =~' is the image of ~ under some invertible

linear map Q on R". Precisely,

(12) M_-- = [det O [Mo_= o O.

This implies certain symmetries for M~ in case ~ = QE . The box spline is particularly simple near an extreme point of its support.

P r o p o s i t i o n 2.2. If e is an extreme point of supp M=_, then, in a neighborhood

of e, M=_ agrees with some truncated power of degree I ,~ [ -m. In particular,

M~(. + e) is homogeneous of degree [ H [ - m near O.

P r o o f . We pointed out earlier that

supp M~. = {~z_ " h(~)~:: h E[0,1]-=}.

Thus any extreme point e of the (closed) support is necessarily of the form

e = ~

for some Z C_ ~, for which, further, there exists rl E R" so that (r/[ s c) > 0 for all

E ~ with

cr(~:):= { - 1, ~ E Z ,

1, ~ ~ \ z .

Therefore, from (11),

M_(- + e) = M ~ ,

showing that it is sufficient to consider the case that e = 0 and, for some 77 E R ' ,

e := min~n('0 [ ~ ) > 0 . In this case,

[O,1l" R.7.


for all test functions ch with support in the hyperplane

{x m :<x In)_-<

and this hyperplane contains 0 in its interior. Here, one recognizes H_= as the

shadow of a cone, i.e., as a truncated power considered in [D2], [DM3]. II]

3. Partit ion of unity

In this section, we show that appropriate translates of an appropriately scaled

version of the box spline

M: = M_=

form a partition of unity. By a standard argument, this implies that the space

spanned by these translates can, at least, approximate continuous functions (as the

mesh size is reduced by scaling).

Proposi t ion 3. Suppose E contains the basis Z (for ~"). Then

IEzz ~EZ

Proof. Since R ~" is the essentially disjoint union of the sets

{2 we find that

[0,1 in m R m

The change of variables A ~ EzA(f)ff carries this to

and this equals

Corollary.

Proof.

[0. I ] " m ~m

f ~b vol,_,. [0, 1]" " / Ide t Z I R m

If ~-C_ Z" and (E)= ~ ' , then E , ~ z - m ( . - j ) = 1.

Let Z C_ E be a basis (for R"). Then

A:={~z j(~)~:jEZz }


is a subgroup of Z" ~ Z z, and its factor group G : = Z"/A has (finite) order Idet Z I .

Therefore

M ( - - j ) = ~ ~ M ( . - g - j ) = l G l / ] d e t Z l = l . III IEZ m g E G l E A

4. Linear independence of translates

For any particular subset V of R '~, we consider the collection of translates

Mo:= M ( . - v ) , v E V, of the box spline

M: = M~.

Such a collection is always (algebraically) linearly independent: Indeed, if

f:= Y, va(v)Mo with W:= supp a a finite nonempty set, then

(supp M w ) \ U (supp M.) ~ 0 v ~ W \ w

for some w ~ W, hence f # 0.

We are interested in considering nontrivial sums of infinitely many translates.

For this, we make the assumption that V has no finite limit points. Then only

finitely many of the translates have any particular point in their support and thus,

for a E R '~ with suitably controlled growth at infinity,

[:= a(v)Mo v

defines a distribution on R m.

Assume that M is a function, i.e., that E contains a basis (for Rm). Then

S=_-.v:=span(Mo)v:={ ~,v a(v)Mv : a ERV ]

is a space of piecewise polynomial functions, possibly quite smooth, and it becomes

of interest to find out whether (Mo)v is a basis for this space or one of its subspaces.

We call (Mv)v (globally) linearly independent if the linear map

(1) a " ~ a(v)M~ v

is 1-1 on R v. Such linear independence is a first necessary condition for other

properties of interest to hold. One such property is stability of the basis (Mo)v for

S~,v, i.e., the property that the map (1) is bounded and bounded below on L(V) into L~(R "~ ). Another is the possibility o[ interpolation from S=_.v, i.e., the existence

of points p~, typically with M~ (po) ~ 0, all v E V, so that, for any function f in some

class K, there exists one and only one s E S=,v n K which agrees with f at (po)v.


We make clear below that this (global) linear independence is usually not present.

Yet, as is pointed out in [BD], if X v a ( v )Mo = 0 and a ~ O, then there exists r > 0 so

that, for all v E V, a changes sign on V N Br(v), with B,(v ) the ball around v of

radius r. Since for any nontrivial polynomial and any r > 0, there exists x ~ R ~" so

that p is of one sign on Br(x), this implies that the map (1) is 1-1 on the restriction

7try of 7r to V.

The space Sz, v = span(Mv)v becomes interesting when V is related to _=. By

assumption, M is a function, i.e., _= contains a basis Z (for Rm). Therefore,

according to Proposition 3.1, the collection Mo, v E Vz: = {Xzj(ff)~ :/" E Z z} forms

a partition of the constant 1/I det Z I. This suggests consideration of V of the form

Vz for some basis Z in _=. We go one step further, though, and consider from now

on only the following normalized situation:

(2) _= C_ V = Z".

This is the same, up to an affine transformation, as the assumption that Vz C_ V for

all bases Z in ~. We abbreviate

S~ : = S~.z~.

P r o p o s i t i o n 4. Under the assumption (2), (M~)v is linearly dependent unless

[det Z I = 1 for all bases Z G "~.

P r o o f . By assumption, ~ contains a basis Z for R", therefore ( I d e t Z [ M , ) , , z

provides a partition of unity as does (Mj)v, by Proposition 3 and its corollary. If

now [ d e t Z l ~ 1 for some basis Z in ~, then Vz~ V, yet

IdetZIM, = 1 = ~ M~. III Vz v

R e m a r k . It would be nice to know whether the converse of this proposition

holds. (Added in proof: The converse has now been proved by Jia, and by Dahmen

and Micchelli.)

5. T h e p o l y n o m i a l s in S~

In this section, we determine S_= n 7r. This information is important in the

discussion of the degree of approximation to smooth functions attainable from S=_.h.

We continue to use the abbreviations and assumptions introduced in Section 4.

L e m m a 5. 7r n ker D - = 7r n ker A.~.

P r o o f . Recall from (2.8) that

(A_=f)(x) = ( MD=_f(. + x).

B - S P L I N E S F R O M P A R A L L E L E P I P E D S 111

Therefore rr n ke rA=D 7r n ker D=_. For the converse, observe that A~f = 0 im-

plies S M(. - x)D-f = 0 for all x. Since M is nonnegative and of compact support,

this cannot hold for a polynomial f unless the polynomial D=f vanishes identically.

III We also note that (2.7) together with summation by parts gives

(1) Dz(~,a(j)Mj)=~(Vza)(j)M=_\z('-j), all Z C E .

T h e o r e m 5.

(2)

P r o o t . Let

Let E * : = { Z C _ E : ( ~ = \ Z ) ~ R " } . Then

7 r A S E = K : = n k e r D z . Z~E*

a (])M~ = : p E r t n S_=.

If Z c_ E, then, by (1), the polynomial Dzp can be written

Ozp = ~ (7za) (j)M_=\z (. - j).

If also ( ~ \ Z ) ~ R ~", then supp M=_\z has zero measure, hence the polynomial Dzp must vanish identically.

For the converse statement, we first prove by induction on k that

(3) SE D rrk N K,

it being trivially true for k = - 1 . For the induction step, we show now that

(4) p E v r k N K implies q:=p-~p(l")MjETrk_]NK.

Since p belongs to K, so does Ep0" )M . by Lemma 5.1 and (1) (making use of the

fact that ker 4_= = ker V=). Thus it remains to show that q E 7r~ ,. This is established

once we show that, for any y E R",

(Dy)kq = O.

For this, we at tempt to write D~ as a linear combination of the operators Dz with

Z E ~ and [Z ] ~ k. Our basic tool is the identity

(Dy)SDz=(Dy)~' ~_~ a(~)Dzu~ ~5 = Z

which is valid as soon as we know that y = E~_=\za(~)~, and can therefore be

applied whenever Z ~ E*. This implies that we can usually replace an expression of


the form (Dy)SDz by a linear combination of such expressions but with the

exponent of Dy reduced. We are only stopped from further reducing the exponent

of Dy when we reach such an expression with Z E E*. This justifies the claim that

(5) (Dy) k = z~.ffyz,<_ k a(Z)(Dy)k-lZlDz + • a ( Z ) D z z~=_.z~_=_*,lzl=k

for certain coefficients a (Z). Now note that each term in the first sum vanishes on q

since we already know that q E K. Therefore

(D" )kq = z~=_.z~-_.,,z,~k a ( Z ) (Dzp - ~ (Vzp )(j)M=_,z(" - j ) )

and this is zero since, for each Z C_ - with Z ~ -=*, we have E M ~ z (" - j ) = 1 by the

corollary to Proposition 3, while I Z l = k implies that Vzp = Dzp is some constant,

since p E rrk.

It is now easy to complete the induction step. If p @ 7rk n K, then, by (4),

p E.S_-- + 7rk-~ N K, hence p E S.~ by induction hypothesis.

This proves that ~ -n S_= = ~ n K. But now observe that the first sum in (5)

vanishes on K while, for all sufficiently large k, the second sum is empty. This

implies that (Dy)kf = 0 for all f E K, all y E R m, and all sufficiently large k, hence

K c III

C o r o l l a r y 1. For each k, the map T : p ~ E p (j)M~ carries 7rk a S z 1-1 onto

itself.

P r o o f . We mentioned already in Section 4 that T is 1-1 on ~r. Thus it is

sufficient to show that T carries 7rk n S_= into itself. But that is obvious since, by (4),

even (1 - T)0rk n Sz) C_ 7rk-, n S_=. 111

C o r o l l a r y 2. As in (2.6), let

d : = max{r : (_=\Z} = R" [or all Z C_ =- with [ Z I = r}.

Then rrk C_ S~_ if and only if k <= d.

P r o o t . From the theorem

~rkCS_ iff ~ r k C K : = n k e rD z . ZE~"

Further, the differential operator Dz decreases the total degree of any polynomial

by at least t z1 and, for some polynomials, by exactly I Z ] . Finally,

d + l =min IZ!. ZE.~*

This shows that ~'a C_ ker Dz for all Z E ~,*. It also shows that, for some Z E _=*,

I Z [ = d + 1, hence D z p # 0 for some p E 7rd§ III


6. Degree of approx imat ion f rom S_=

In this last section, we discuss the degree of approximation from S'~.h to a

sufficiently smooth function, as h --* 0. Here, h indicates a scaling of the mesh, i.e.,

Sz.h: ={x ~ f ( x / h ): f ~ S=_}.

T h e o r e m 6. I f k <= d (with d given by (2.6)), then there exists a linear

functional A on 7rk so that p = E,)tp(. + j)Mj for all p ~ 7rk.

Proof . By Corollary 2 of Theorem 5, 7rk C S_, while, by Corollary 1 of

Theorem 5, the map T : p ~ Ep(j)Mj is 1-1 onto 7rk. Thus

p = ~ ( T - 1 p ) ( j ) M , all p E ~'k, I

with T -~ :=(Tick) -~. Further, with T~ the shift by the vector i, i.e.,

(T ,p ) ( x ) :=p(x + i), all x,

we have

showing that T commutes with T,, hence so does T -1. This proves the theorem,

with Ap: = (T-~p)(O). III

The theorem implies statements about degree of approximation to smooth

functions from S_ in the now standard quasi-interpolant fashion: Let K be the class

of functions which belong locally to some function space K0, e.g., to L1. Extend the

linear functional A of the theorem to a continuous linear functional p. on K0, with

support in supp M. The quasi-interpolant scheme

Q : K--->S-~: f - ~, I~f(" +j)M~

then reproduces 7rE and is local. This implies that

f - O f = (f - p ) - O (f - p), for all p E ~/'k

and that

I ( Q g ) ( x ) ] - - I ~ ~ g ( ' + j)M, (x) I --< Iltz tt max{llgl,,ppM, HI : M, (x) ~ 0}.

Therefore

1([- Qf)(x) l -< distp (/, 7rk)

with p the seminorm given by


P (g): = I g (x)1 + II/x II max{ I[ g Isupp ~, I]: Mj (x) / 0}.

This shows tha t O f a p p r o x i m a t e s to f local ly as well as local

app rox ima t ion . H e r e is a pa r t i cu la r resul t a long these lines.

po lynomia l

C o r o l l a r y . Let Ix be an extension of A to a continuous linear functional on

L = ( s u p p M ) and let Qh:=o'hQO'l/h with Q : f ~ Z l x f ( ' + j ) l V l ~ and (o 'h / ) (x) : =

f ( x / h ) . Then, for f E L~ k+x), IIf - Ohfll~ = O(hk+') �9

S h a r p n e s s . The o r d e r O ( h d§ is, in genera l , bes t poss ib le for the a p p r o x i m a -

t ion f rom S_= to smoo th funct ions. To see this, let

fir II:--Ill and choose Z E ~,* (cf. T h e o r e m 5) wi th I Z I = d + 1 and a po lynomia l p E 7r,+,

with Dzp = 1. If, for some a p p r o x i m a t i n g sequence s, E S-,h, we have

Ilsh - p l l = o ( h a + ' ) ,

it fol lows f rom the s t anda rd M a r k o v inequa l i ty for p iecewise po lynomia l s that

(1) I[Oz,s. - Dz,p [I = o ( h )

for any Z ' C _ Z with I Z ' l = d . Set z = Z \ Z ' . Since the s u p p o r t of Dz(Dz,sh) is

con ta ined in hype rp lanes , (Dz,sh) is p iecewise cons tan t on l ines in the d i rec t ion z.

O n the o t h e r hand , Dz,p res t r i c ted to these lines is of the fo rm (Dz ,p ) (x + tz) =

( D z , p ) ( x ) + t which cont rad ic t s (1). III

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MATHEMATICS RESEARCH CENTER UNIVERSITY OF WISCONSIN

MADISON, WI 53706 USA (Received April 9, 1982)

b-splines from parallelepipeds

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