The minimum supremum norm of a B-spline

The minimum supremum norm of a B-spline

Citation for published version (APA):

Morsche, ter, H. G. (1978). The minimum supremum norm of a B-spline. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 78-WSK-05). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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ONDERAFDELING DER WISKUNDE

The minimum supremum norm of a B-spline

by

H.G. ter Morsche

T.H.-Report 78-WSK-OS

Abstract

In this report we investigate the problem of minimizing the supremum norm (sup norm) of a B-spline of order m with knots in the real interval [0,1]. The sup norm depends on the distribution of the knots, so the question arises to find the distributions that minimize the sup norm and to compute the corresponding value. We shall prove that the minimum sup norm is attained when the knots are almost equally distributed over the end points of [O,lJ.

1. Introduction

In [lJ, p. 174 Meinardus poses the following problem: Given a positive integer m, determine a set of knots x

O,x1, ••• ,xm with 0

= Xo

S xl ~ ••• ~ such that the corresponding B-spline Bm(t; x

O,x1" " ' xm) of order m has minimum sup norm. He conjectures that

(1. 1)

where p is a positive constant.

x

=

1

m

Before turning to this problem we list some well-known properties of B-splines

(c f • [ 3

J, [

5, p. 2 -6

J) •

(1.2) (1.3) (1. 4) (1.5) (1. 6) t € (0,1) , t ;. [O,l] <X>

f

_{Bm(t; xO,x1}

,···,X

_m)dt

=

1 •

Let x. be a knot of multiplicity r, i.e. let

~

x_i-₁ < xi

=

x_i+₁

= ...

=

x_i+_r-₁ < x

i+r (x_1 := -<X>, xm+l := ~), then

B is exactly m - 1 - r times continuously differentiable in

m

(x. l ' x, ) • ~- ~+r

On each interval (x.,x, 1) B coincides with a polynomial of degree ~ ~+ m

m-1.

If the v-th derivative B(V) (v

=

0,1, ••• ,m-2) is continuous, then m

B(v) has exactly v distinct simple zeros in (0,1). m

(1.7) Let f be m times continuously differentiable and let f(x_O'x_{1, •• ·,xm)} denote its divided difference of order m based on the knots

- 2

-00

f(x_O'X₁'···'xm)

=

m\

J

_00

In the next section we give some examples of B-splines corresponding to specific distributions of the knots; these examples will be needed later.

2. Some exasPles of B-splines

The restrictions

Xo

=

0 and xm

=

1 imply that the multiplicity of a knot in the interior of [O,lJ does not exceed m - 1, hence B is always

m

continuous in (0,1). If there is a knot of multiplicity m - 1 in (0,1), then Bm is not continuously differentiable in (0,1). Moreover, if

Xo

or xm has multiplicity m, then Bm is not defined in

Xo

or x_m' We then define, B (x_O)

=

lim B (t) c.q. B (x) lim B (t)

m tvO m m m ttl m

We proceed by giving three examples of B-splines and their corresponding sup norms. (a) B (t· m ' IIBm II (b)

_Xo

= ... =

= X E (0,1) I m-l

r~)

xO,x1""'xm) _{1-t m-l} m(-) 1-x := _{sup B (t· xO'x!, ••• ,xm)} t m ' 0

,

_xl = x EO (0,1)

,

x 2 m m-l - t x ' m x m-l m-2

=

m(x/(l - (1 - x)m=2» = m

.

x3

= .•. =

0 x m $: x < x _m t t 1 1 • $: x

,

:-::; ₁

.

.

x : ' : ; t $ l .

(c) (2. 1) liB U=L(m,k) m

= ... =

x :;: 1 1 :S: k :S: m • m

o

< t < 1 •

Stirling's formule easily provides the limiting behaviour of L(m,k) as

We consider the following two cases.

(i) If k is fixed, then

e-(k-1) (k_l)k-1 m

L (m, k) ..., (k _ 1) !

(ii) If lim kim

=

a with 0 < a < 1, then

m-+<x>

(2.2) _{L (m, k) '"}

_-;:::==1

==

f21Ta (1 - a)

.;ro,

m-T OO •

We note that relation (2.2) already proves conjecture (1.1). In what follows we shall compute the minimum sup norm for a fixed value of m. First we pay special attention to the case of equally spaced knots.

3. Cardinal B-splines

When the knots x

O,x1, ••• ,xm are equally spaced on [O,lJ, the corresponding B-spline is simply related to the so-called cardinal B-spline (cf. [5, p.ll]).

In fact, we have

1

B (t;O, , ••• , 1)

m m

where the cardinal B-spline M is the B-spline function of order m with

4

-m --m m

knots -

2'

:r

+ 1""'2 •

In view of [5, p. 12J the Fourier transform of the function M is equal to

m (3. 1 ) m __ (2 sin w/2) 1/1 (w) m w Inverting (3.1) we obtain II B II = m M (0) m m m = -21T -00

Taking into account formula (4.2.4) of [2, p. 65J one has

(3.2)

Comparing (3.2) and (2.2) it follows that for sufficiently large m the sup norm corresponding to equally spaced knots is larger than the sup norm corresponding to a distribution of knots as given in example 2(c) with k

=

[(m + 1)/2], the largest integer not exceeding (m + 1)/2. The results of section 5 ascertain that this inequality holds for all integers m (m ~ 3). To prove this we need a recurrence relation for B-splines; this is the subject of the next section.

4. A recurrence relation

Let B 1 be the B-spline of order m - 1 (m ~ 2) and knots x_O,x

1, ••• ,xm_1

m-with 0

=

_Xo ~ _Xl ~

...

~ _x

=

1. We add an arbitrary knot x m-l

the given set of knots to obtain the B-spline B of m following relation holds.

Theorem 4.1.

(4.1) B (t) = t - x B • (t) + _m_~ B (t)

m m - 1 m m - 1 m-l •

at all points t, where B is differentiable. m

order m.

IS [0,1] to

Proof.

We need the Fourier transform of the spline function B , i.e.

m

00

~m

(Ul) :=

f

B _m (t)e -iOlt d t .

-00

According to property (1.7) of section 1 ~ (Ol) is equal to the divided

m

difference, based on the knots of B I of the function

m

m! -iOlt

e

The divided difference, Qlm(z) say, of e zt can be represented (cf. [4, p. 206J)

as the solution of the homogeneous linear differential equation

with initial conditions

Clearly, <p (0) m Ql' m (0)

= ...

= (m-l) (0) Ql_m m! ~ (Ol) = -....;..;..;.- <p (-iOl) • m (_iOl)m m

o ,

<p (m) (0) = m 1 •

We note that the function <p' (z) - x<p (z) solves the differential equation

m m

with initial conditions

f(O)

=

f ' (0) = ••• = f(m-2) (0)

=

a ,

f(m-l) (0) = 1 •

Hence the Fourier transforms of Band B 1 satisfy the relation

m 6 m

-(4.2) ~ (w) = --w t' (w) - -ixw ~ (w) + t 1 (w) •

m m m m m

m-Inversion of (4.2) yields (4.1).

o

Remark. Recurrence relation (4.1) can also be proved by noting that

where get)

=

(t - x)f(t) , and using property (1.7).

Relation (4.1) can be considered as a differential equation for the function a • The solution of this equation, which can easily be obtained, is given in

m

the next theorem.

Theorem 4.2. (4.3.a) t

a

_{m- 1 (T)} (t) m-l

_f

dT

a

= m(x - t) m m 0 (x - T)

o

s t < x , (4.3.b) 1

a

_{m- 1 (T)}

a

(tl m (t - x) m-1

_f

d. m m t (. - x) x < t s 1 , (4.3.cl

a

(x) m 1

a

_m_ 1 (x)

=

.

m m -Assuming that

a

m-1 is continuous in lR, integration by parts in (4.3.a) and (4.3.b) shows that

a'

can be expressed in terms of

a

1 as follows.

m m-Theorem 4.3. (4.4.a) (4.4.b)

a'

(t) m B' (t) m m(x met -t) m-2

f

0 x) m-2

J

t t dB m_1 (.) m-l

,

(x - .)

o

s t < X I 1 _dB m_1 (.) m 1 x < t

s

1 , (. - xl

(4.4.c) (m - 2)B'(x)

m m B' 1 m- (x) ,

if B is differentiable at t

=

x.

m

5. The minimum sup norm of B

- m

In order to obtain a preliminary result, we consider the following problem. As in section 4, let Bm-1 be the B-spline of order m - 1 (m ~ 2) with knots x

O,x1,···,xm-1 satisfying 0

= Xo

~ xl ~ ••• ~ xm-1 1. Adding an arbitrary

knot x E [O,lJ to the set x

O,x1, ••• ,xm-1 we obtain the B-spline Bm of order m. Evidently the sup norm of B depends on Xi the problem now is to

m

investigate this dependence. We assume that Bland B are continuous in

m- m

m,

and that B is continuously differentiable in (0,1).

*

*

m

*

*

Let x 1'x E{0,1)besuchthatB 1{x l}=IIB 1I1andB(x)=IIBII.The

m- m m- m- m- m m m

*

*

following result shows how the location of x

=

x (x) depends on x.

m m Lemma 5.1.

*

*

*

I f x > x

,

then x < x < x

,

m-1 m-l m

*

*

*

if x < x

,

then x < x < xm-l

,

m-1 m

*

*

*

if x = _{xm- 1} then x _m

=

x _m-1 " Proof.

If x > x;_1' then in view of (4.4.a) we have

*

B'(x 1) =m m

m-*

*

Hence x_m-₁ < xm

*

o

*

x 1

*

m-2

f

m- (x - x m-1) ---~-m--~1-dBm-l (,) > 0 • (x - ,)

Now let x < x 1" Using (4.4.b) we obtain

m 8 m

-*

m-2 1 B' (x 1) - m

J

m m- -

*

(x 1 - x)

--m---m-_-

₁O:-- dB m-1 (T) < 0 • (T - x) xm':"l

*

*

Hence x_m- ₁ > xm

Using (4.4.c) we likewise deduce:

*

B' (x)

*

Also

I f x > x

m-l' then m < 0 and thus x > x m

·

*

B' (x)

*

Finally,

if x < x

m-l' then m > 0 and thus x < x m

·

*

_*

if x ==

xm- 1 ' then B' (x) m == 0 and thus x == x m

·

A combination of these results yields the lemma.

Noting that the cases x ~ x * _m-l and x :;; x * 1 are similar, we only treat the

m-case x ~ x* l' As Bf (x*) == 0 we obtain from (4.1)

m- m m II B II == m m

*

1 B 1 (x ) • m - m- m

*

Since B_m-₁ is decreasing on (x m-1' 1) where inf x* l:;;x:;;1 m-liB II m

*

x == -m

*

sup x 1:;;x:;;1

m-*

x (x) m

*

In order. to obtain x we need the following lemma. -m Lemma 5.2.

*

If x m-1 < y < z :;; 1, then x (y)

*

m < x (z) m

*

Proof.

it follows that for x > x

*

m-l x (x) From (4.4.a) m (5. 1)

J

1 dB m-l (T) == 0

.

m-l 0 (T - x)

o

*

Now let x _m-l < y < z $ 1. Then, by a mean-value theorem

*

*

_*

x (y) x m-l x (y) m m

J

1 dB 1 (T) = (

J

J

) 1 _(T + m-1 m- m-1 T 0 (T - z) 0 x* (, - y) m-1 * * x (y) x m m-l '1 - Y m-l

f

1 ' - y m-1

J

2 = _(, ) _dB m_1 (T) + (T - z) - z m-l 1 (T - Y) 2 0 _x m-1

*

x (y) m

,

- y m-l , -y m-l

f

« 2 _ ( 1 ) 1 ::; ₎ dB m_1 ('n

,

'2 - z '1 -z m-l x* (T - y) m-l * * *

where '1 E (o,xm_1) and '2 E (xm_1'xm(y» •

m-l As ( ' -_T y) _Z i _S d _ecreas~ng . _on ( 0 ) _{,y we} h _ave

*

x

(y) m (5.2) _sgn(

J

o

(-1) m-1 • (T - z) The function ~(t) in fact we have

changes sign on [O,z) only at t

{ (_1)m-1 , sgn (ql(t» =

*

o

< t < x (z) m (-1) m x (z)

_*

< t < z • m

*

*

So from (5.2) it follows that x (y) < x (z) •

m m Corollary 5. 3. (5.3.a) liB II = m m

*

-m--'"""!-l B m-l (xm (1) ) m-l - y) _dB m_1 (T)

=

- Z 1 1 dB 1 (T) = m- m-(T - y)

*

= x (z); m [l

10 -(5.3.b) min II B II = O~x~x* m m-1 m 1 B 1(X*(0» m - m- m Proof.

Assertion (5.3.a) follows immediately from lemma 5.2. Assertion (5.3.b)

*

follows from the observation that the cases x ~ x 1 and x ~ x 1 are

m-

m-essentially similar.

Our main result is given in the next theorem.

Theorem 5.4.

o

Let Bm{t; xO,x1""'x

m) be the B-spline of order m (m

=

1,2, ••• ) with knots x_O,x 1,···,xm where 0 == Xo ~ Xl ~ ••• ~ xm == 1. Then L(m,n) m 2n , min liB

II

== m m = 2n + 1 •

Moreover, in case m ~ 3 the distribution of the knots for which the minimum sup norm is attained is given by

where k ;:: Proof. m + 1 2 m

2"

or m is odd , ~ + 1 2 m is even • - ••• - x = 1 , m

The case m == 1 is trivial. Furthermore, II B211 = 2 independent of the position of Xl' Let now Bm be a B-spline of order m (m~ 3) I continuously differentiable

in (0,1) and continuous in IR, and let us assume that there is a knot x. of J. B

m in the interior of [ 0, 1] • By deleting this knot we get the B-spline Bm_1

of order m - 1 with knots B 1 is

m-con Unuous, ,then by corollary 5.3 we have that

II

B

II

> m min

x

1

,···,x

m

_

1

II

B

II.

m

If B _m-1 is not continuous, we may assume that the distribution of knots of B is as given in example (b) of section 2. A simple computation shows

m

that in this case

II

B

II

> L{m,2).

m

If B is not continuous in JR, then the distribution of knots is given in

m

example (c) of section 2 with k == 1 or k:::: m and we have IIBmll = m > L(m,2). If B is not continuously differentiable in (0,1), then the distribution of

m

knots is given in example (a) of section 2 and again we have liB _m II

=

m > L(m,2). Hence, for the computation of min II B II, we may restrict ourselves

m x

1,···,xm_1

to the distribution of knots as described in example (c) of section 2. What remains is the computation of min{L(m,k); k == 2, ••• ,m - 1}. Taking into account (2.1) the theorem will now be completely proved if

m+1 we show that min{L(m,k); k :::: 2, ••• ,m - 1}

=

L(m, [-2-])'

This can be done as follows. In view of (2.1) and writing t :::: m - k we have

L(m,k + 1) L(m,k)

k-1

(1 + 1/k-1)

(1 + 1/t_l)t-1

Since (1 + l/k)k is increasing it follows that L(m,k+ 1)/L(m,k) < 1 if

m

k < ,\l" hence k <

'2 •

D

Remark 1. With respect to the supremum of the sup norm of B our analysis

m

shows that sup II B II:; m.

m

Remark 2. An elementary computation shows that the sequences

P2n+l :~ L{2n + l,n + l)/12n + 1 and p := L(2n,n)/12:n are both decreasing

II

2n

with lim P2n :::: lim P_2n+₁

I;

and, moreover, P2n-l < P_2n < P_2n-₃ (n 2,3, ••• ).

n-+«> n-+«>

Hence (cf. (1.1» for m 3,4, •••

inf IIBmll S;

~

rm

I

- 12

-8

where the constant

9 -

0,8889 cannot be replaced by a smaller number. On the other hand, for m = 1,2, ••••

where the constant

I;

=

0,7979 cannot be replaced by a larger number.

Acknowledgement. The author wishes to thank Dr. F. Schurer and Dr. F.W. Steutel of Eindhoven University of Technology for helpful criticism on an earlier

draft of this report.

References

[1] Bohmer, K., G. Meinardus and W. Schempp. Spline-Funktionen. B.I.-Wissenschaftsverlag, Mannheim-Wien-Zurich, 1974.

[2 ] Bruijn de, N.G., Asymptotic methods in analysis. North-Holland Publishing Co., Amsterdam-Groningen, 1961.

[3] Curry, H.B. and I.J. Schoenberg, On Polya frequency functions IV: The fundamental spline functions and their limits. J. d'Analyse Math.

12

(1966), 71-107.

[4] Karlin, S., C.A. Micchelli, A. Pintus and I.J. Schoenberg, Studies in spline functions and approximation theory. Academic Press Inc., New York-San Francisco-London, 1976.

[5] Schoenberg, I.J., Cardinal spline interpolation, CBMS Vol. 12. Siam, Philadelphia, 1973.