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On Cardinal Spline Smoothing
Approximation Theory and Functional Analysis J.B. Prolla (Ed.)
@North-Holland Publishing Company, 1979
ON CARDINAL SPLINE SMOOTHING
I. J. SCHOENBERG Mathematics Research Center University of Wisconsin Madison, Wisconsin 53706, USA.
INTRODUCTION The present paper describes the methods, with some changes be mentioned below, whereby I solved the numerical problem
to
assigned
to me at the Ballistics Research Laboratories in Aberdeen, Maryland, during the second World War (see [ 4
]
1.
The problem was to smooth very
extended equidistant tables of drag functions (or drag coefficients) by approximating them by very smooth functions that were easily computable with their first and second derivatives. The first question to be answered was this: When may a " m o w h g awefiage" o p e f i a t i o n L e g i t i m a t e L y b e c a L l e d a n m o o t h i n g 6ofimuLa ?
An
answer is given in Section 1 of Part I. That it is a reasonable
one
is shown in Section 2 of Part I. These ideas were later greatly generalized by Fritz John in his important work on parabolic differential equations (see [ 3 1 ) . The connection is briefly
mentioned
in
Section 2 of Part I. The second essential ingredient of our results of Part
I1
is
the process of c a f i d i n a e h p t i n e intehpolation (see [ 9 1 ). The required results are described in Section 3 of Part I. The changes from my war-time approach are developed completely in Part 11. This is the new contribution of the paper, and it arises 383
384
SCHOENBERG
i f w e a d a p t t h e i d e a o f E. T. Whittaker (see [lo 1 and [ 7
1 ) to
the
problem of smoothing a b i - i n f i n i t e sequence of e q u i d i s t a n t d a t a . The r e s u l t i s t h e cahd.inaL 4 p L i n e s m o o t h i n g phOces4. L e t m e thank P r o f e s s o r I l i o G a l l i g a n i f o r a s k i n g m e
to
visit
Rome i n May of 1 9 7 7 , where I gave a s h o r t 10-hours c o u r s e o n C a r d i n a 1 Spline Interpolation a t h i s I s t i t u t o per l e applicazioni d e l Calcolo "Mauro Picone".
PART I .
PRELIMINARIES
1. WHAT IS A SMOOTHING FORMULA?
The s e v e r a l smoothing ( o r gradua
-
t i o n ) methods d i s c u s s e d i n [lo] are a l l o f t h e "moving a v e r a g e " t y p . By t h i s w e mean t h a t w e are g i v e n a sequence o f r e a l and
symmetric
weights
which w e a p p l y t o t h e d a t a (x,)
t o produce t h e i r smoothedversicln (y,)
,
by t h e formula
yn
--
m
C
v=-co
an-Vxy
, (n=O,
f 1,
...) .
I n o t h e r words, w e c o n v o l u t e t h e sequences (a,)
and (x,)
,
an o p e r a
-
t i o n s y m b o l i c a l l y d e s c r i b e d by
(3)
By d e r i v i n g a formula ( 2 ) a c c o r d i n g t o a d e f i n i t e i d e a , s u c h a s a l e a s t s q u a r e s formula, o r W h i t t a k e r ' s formulae (see [ 1 0 , XI
1
),
Chapter
we may r e a s o n a b l y e x p e c t t h a t it w i l l t r a n s f o r m a g i v e n
se-
quence ( x ) i n t o a smoother sequence (y,). T h i s p o i n t becomes doubtn f u l l however, i n c a s e t h e formula ( 2 ) was o t h e r w i s e o b t a i n e d , e . g . ,
385
ON CARDINAL SPLINE SMOOTHING
by some a p p r o x i m a t i n g p r o c e d u r e t h a t i s n o t s t r i c t l y
interpolatory.
An example of such a p r o c e d u r e w i l l be o u r c a r d i n a l s p l i n e smoothing o f P a r t 11. A c r i t e r i o n f o r ( 2 ) t o be r e g a r d e d a s a b m o o t h i n g ~0hmlLea w a s proposed by t h e a u t h o r i n [ 4 , pp. 50
- 54 1
and p r o c e e d s a s f o l l w s .
To b e g i n w i t h , w e assume t h e L a u r a n t series
(4)
I
t o converge i n some r i n g c o n t a i n i n g t h e u n i t c i r c l e
z j =1. S e t t i n g
i u , we c a l l t h e even f u n c t i o n z = e
d(u)
(5)
iu
= F(e
)
=
a
0
+
2al cos u
+
2a2 c o s 2u
+
...
t h e chahactehintic ( u n c t i o n of t h e formula ( 2 ) . The r e g u l a r i t y of F(z) on
I
IIm
UI
z
I
= 1, i m p l i e s t h a t
$(u)
is regular
in
a
certain
strip
< a , whence t h e v a l i d T a y l o r e x p a n s i o n m
T h i s e x p a n s i o n a l l o w s us t o e x p r e s s e a s i l y an i m p o r t a n t p r o p e r t y ( 2 ) : I t s hephoductive p o w e f ~ o , r deghee
06
exactnebb.
We s a y t h a t ( 2 ) h a s t h e d e g r e e o f e x a c t n e s s i s a n a t u r a l number, p r o v i d e d t h a t ( 2 ) r e p r o d u c e s
yn
- xn
values
f o r a l l n , a sequence (x,) P(n) of
,
of
where
2m-1,
exactly,
m
i. e.
,
i f t h i s sequence r e p r e s e n t s t h e
a polynomial of d e g r e e
2m-1,
b u t d o e s n o t have
t h i s p r o p e r t y f o r any h i g h e r d e g r e e . T h a t t h e d e g r e e of e x a c t n e s s i s always odd, f o l l o w s from t h e symmetry c o n d i t i o n ( 1 ) . I n t e r m s of
(6)
w e have t h e f o l l o w i n g e a s i l y e s t a b l i s h e d p r o p o s i t i o n : T h e doamuta ( 2 ) hub t h e degkee
id t h e e x p a n n i o n ( 6 ) i n
06
the 6otm
06
eXaCtflebb
2m-1,
i6 and onLy
SCHOENBERG
386
$(u) = 1
(7)
-x
.,.
+
U2m
(A
#
0).
An example. L e t (1) be t h e sequence a
0
= 5/8,
al = 1 / 4 ,
a2 =
-
1/16,
an = 0
if
n > 2,
when t h e c o n v o l u t i o n ( 2 ) assumes t h e form
From (5) w e o b t a i n
(9)
$ ( u ) = ( 5 + 4 cox u
-
and ( 7 ) i s s e e n t o h o l d w i t h mula ( 8 ) has t h e d e g k e e
06
cos 2u)/8 = 1
rn = 2 , X
=
1/16.
- 7
u4
+
...
I t follows that the f o r -
exactness 3 .
May ( 8 ) be r e g a r d e d as a smoothing formula ? The g e n e r a l
cri-
t e r i o n i s d e s c r i b e d by t h e f o l l o w i n g .
DEFINITION:
kJe 4ay t h a t ( 2 ) Lo a s m o o t h i n g botmuLa, p h o v i d e d t h a t La2
chahactchihtic dunction
d(u) 4atiAdieb
This condition evidently implies t h a t t h e coed6icien-t
peaking i n ( 7 ) ,
A ,
ap-
io p o s i t i v e . The c h a r a c t e r i s t i c f u n c t i o n ( 9 ) i s e a s -
i l y s e e n t o s a t i s f y (lo), and so ( 8 ) i s a smoothing f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n of t h e t e r m . The c r i t e r i o n (10) raises t h e f o l l o w i n g k i n d o f e q u a t i o n :
Do
t h e numerous smoothing formulae a s g i v e n i n 110 1 s a t i s f y our criterion
301
ON CARDINAL SPLINE SMOOTHING
(lo)? S e e G r e v i l l e ' s p a p e r [ l 1 where s o m e of t h e s e q u e s t i o n s a r e answered a f f i r m a t i v e l y . I n [ 4 , p p . 50
-
54 ] good a r g u m e n t s i n s u p p o r t of
( 1 0 ) are p r e s e n t e d . Even m o r e c o n v i n c i n g r e a s o n s
the criterion
w e r e given i n
[5,
P a r t I ] . I n t h e n e x t s e c t i o n we r e p r o d u c e t h e s e a r g u m e n t s a s p r e s e n t e d i n [ 6 , pp. 2 0 0 - 2 0 4 1 .
2 . THE BEHAVIOR OF THE ITERATES O F A MOVING AVERAGE FORMULA:
THE
PROBLEM OF DE FOREST. A c o n c l u s i v e argument i n s u p p o r t o f t h e n e c e s s i t y of o u r c o n d i t i o n ( 1 . 1 0 )
i s f u r n i s h e d by t h e s o l u t i o n o f t h e f o l -
l o w i n g problem f i r s t s t a t e d a n d a t t a c k e d
by
Erasmus
L. D e
(1834-1888). I f we s u b j e c t t h e g i v e n s e q u e n c e ( x u ) n times
Forest
i n suc
-
c e s s i o n t o t h e same t r a n f o r m a t i o n ( 1 . 2 ) , we o b t a i n a l i n e a r transformation
which i s t h e n - f o l d 06
the
i t e r a t e of ( 1 . 2 ) . W h a t i n R h e a o y m p t o t i c
c a e d 6 i c i e n t n 0 6 (1) an
n
+
m ?
T h i s q u e s t i o n was
behaviah
answered
by D e F o r e s t a n d by G. B. D a n t z i g ( f o r r e f e r e n c e s see151 ) f o r t h e c a s e when a l l c o e f f i c i e n t s of that
m=l
( 1 . 2 ) a r e non - n e g a t i v e , h e n c e n e c e s s a r i l y
i n ( 1 . 7 ) . A g e n e r a l s o l u t i o n i s as follows.
L e t ( 1 . 2 ) be s u c h t h a t ( 1 . 1 7 )
hence t h a t
,
( l . l O ) , and (1.71, are s a t i s f i e d ,
X > 0. L e t
-V
which i s t h e normal f r e q u e n c y f u n c t i o n
2m
cos vx d v ,
SCHOENBERG
388
if
m =1, o t h e r w i s e (m = 2,3,.
.
.)
Gm(x) i s an e n t i r e f u n c t i o n
having
i n f i n i t e l y many zeros, a l l r e a l . T h e caeddicients a d (1) satis6q t h e a s y m p t o t i c h e L a t i o n h
--
--
1 1 --1 a ( n ) = ( i n ) 2m Gm(v(hn) 2m ) + , o ( n 2m)
(4)
a4
V
where t h e " l i t t e e
n +
m ,
v.
or' dymbok? hoed4 unidaamly d o h n l L i n t e g e h d
For a proof see ( 5 , P a r t I ] , where i t i s a l s o shown byexamples (1.10),
t h a t ( 4 ) no l o n g e r h o l d s i f t h e e q u a l i t y s i g n i s a l l o w e d i n and t h a t t h e c o e f f i c i e n t n = 2k
aAn) d i v e r g e s e x p o n e n t i a l l y t o
t e n d s t o i n f i n i t y t h r o u g h even v a l u e s , i f
( 1 . 1 0 ) are r e v e r s e d anywhere i n t h e i n t e r v a l
the
0 < u < 2.rr
+
m
,
as
inequalities
.
The f o l l o w i n g d i s c u s s i o n , w h i l e n o t d i r e c t l y r e l a t e d
to
our
s u b j e c t of smoothing, w i l l show t h e c o n n e c t i o n of t h e a s y m p t o t i c rel a t i o n ( 4 ) w i t h t h e w i d e r f i e l d of p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s . Observe t h a t ( 2 ) i m p l i e s t h a t
(5)
--1
--1
U(x,t) = t 2m G m ( x t 2m) =
-
-tv
2m
+ ixv
d v , ( t > 0).
The f u n c t i o n under t h e i n t e g r a l s i g n i s immediately s e e n t o s a t i s f y for all v
, the
d i f f e r e n t i a l equation
which r e d u c e s t o t h e f a m i l i a r h e a t e q u a t i o n i f
also -plane
m = l . I t follows t h a t
U ( x , t ) , d e f i n e d by ( 5 ) , is a s o l u t i o n of (6) i n t h e upper h a l f t > 0 . On t h e o t h e r hand, a p p l y i n g t o ( 2 ) F o u r i e r ' s inversion
formula and s e t t i n g
v = O r we f i n d that
ON CARDINALSPLINE SMOOTHING
These r e m a r k s imply t h e f o l l o w i n g : 'I 6
1x 1
say, a s
+
LA a b o t u , t i o n
a,
06
f (x)
389
cantinuow and a ( I X I - * )
,
then
t h e d i , j d e ~ e n . t i a l e q u a t i o n ( 6 ) Aattin6qLng t h e boundmy
condition
This p a r t i c u l a r s o l u t i o n
u ( x , t ) may now a l s o be
approximated
by t h e f o l l o w i n g n u m e r i c a l p r o c e d u r e : Draw i n t h e ( x , t ) - p l a n e
the
rectangular lattice of p o i n t s
(WAX,
n At)
(w
= 0, k 1
, ...
;
D e f i n e on it a l a t t i c e f u n c t i o n
n = 0,1,2
u
v
, ...) .
by s t a r t i n g w i t h
uw ,o = f ( v Ax) ,
and computing t h e v a l u e s a l o n g e a c h h o r i z o n t a l l i n e from those on the l i n e below i t , by means o f t h e t r a n s f o r m a t i o n (1.2). T h i s
evidently
amounts t o i t e r a t i n g (1.21, a n d a f t e r n s t e p s w e o b t a i n
(10)
For any g i v e n x a n d
t > 0,
( 1 0 ) w i l l go o v e r i n t o ( 8 ) i f w e
f o l l o w i n g : We 6 h A t c o n n e c t the. m e o h - n i z e b
Ax and
A t
do the
by ,the h d a t i a n
SCHOENBERG
380
A t = X (Ax) 2m.
(11)
Id t h e i n t e g e k b
n ahe buch t h a t
v and
VAX
+
x,
and
n A t
.+
t
an
Ax
0.
+
then
U
v,n
T h i s follows r e a d i l y from r e l . a t i o n (4): ( 1 0 ) d i f f e r s
U(X,t).
+
1 0 ) and ( 8 1 , i n view o f t h e
asymptotic
from a Cauchy-Riemann sum f o r tk integral
( 8 1 , by a q u a n t i t y t h a t t e n d s t o z e r o due t o t h e u n i f o r m i t y i n
v of
t h e error t e r m o f ( 4 ) . I t i s i n t e r e s t i n g t o n o t e t h a t it d o e s n o t matter which
for-
mula ( 1 . 2 ) w e u s e i n t h i s c o n s t r u c t i o n , as l o n g as it i s o f the degree of exactness
2m-1,
i.e.,
i t s a t i s f i e s (1.71, and above a l l t h a t i t
s a t i s f i e s t h e s t a b i l i t y c o n d i t i o n (1.10)
,
t h e t e r m "stabi1ity"meaning
h e r e s t a b i l i t y on i t e r a t i o n . F o r t h e g e n e r a l t h e o r y of F. J o h n , which t h e e q u a t i o n ( 6 ) i s a s p e c i a l example, see [ 3 1
.
of
I n t h i s s e c t i o n w e d e a l t e x c l u s i v e l y w i t h f o r m u l a e ( 1 . 2 ) which s a t i s f y t h e symmetry r e l a t i o n . I n [ 2 ]
T.
N.
E. G r e v i l l e d e a l t
with
t h e more d i f f i c u l t c a s e o f unsymmetric f o r m u l a e .
3 . CARDINAL SPLINE INTERPOLATION (see [ 9 , L e c t u r e s 1
l e m o f caadinal intehpolation i s t o f i n d s o l u t i o n s
-
4 1 ) . T h e prob-
f ( x ) of t h e i n -
t e r p o l a t i o n problem
(1)
f ( v ) = Y"
,
for all i n t e g e r s
v
,
where ( y v ) are t h e d a t a . A f o r m a l s o l u t i o n i s f u r n i s h e d b y t h e series
391
ON CARDINAL SPLINE SMOOTHING
i n v e s t i g a t e d i n 1 9 0 8 by de l a V a l l G e P o u s s i n , also l a t e r
by
E. T.
W h i t t a k e r , who c a l l e d i t t h e cahdinad b e h i e b . The d i f f i c u l t y w i t h ( 2 )
“i: :y
i s t h e s l o w decay o f t h e f u n c t i o n
as
x
-. A
+
s o l u t i o n of (1) i s t h e p i e c e w i b e l i n e a h i n t e h p o t h z t
much s i m p l e r g i v e n by
S1(x)
m
(3)
where
M2(x) i s t h e roof f u n c t i o n d e f i n e d by
in
M2(x) = x + l
,
[-1,01
M (x) = 1
2
-x
i n [ O , l l and%(x)
=o
The p u r p o s e o f cahdinad b p l i n e i n t e h p o & z t i o n i s t o b r i d g e between t h e p i e c e w i s e l i n e a r
if 1x1 ’1.
the
gap
S1(x) d e f i n e d by (31, a n d t h e c a r d i n a l
series ( 2 ) . I t a i m s a t r e t a i n i n g s o m e of t h e s t u r d i n e s s a n d s i n p l i c i t y of ( 3 ) , a t t h e same t i m e c a p t u r i n g some of t h e s m o o t h n e s s a n d s o p h i s t i c a t i o n of
(2).
Le-t m be a natuhad numbeh, and d e b
(4)
S2rn-l
b e t h e cLadb
06
= {S(X)3
cahdinad b p d i n e d
S(x)
0 6 deghee
2m-1
dedined
by
the two conditionb:
(5)
The h e s t h i c t i o n whete
v
i d
04
S ( x ) -to e u e h y u n i t i n t e n v a l
a n i n X e g e h , i b apolynomia!.
(v ,v
0 6 deghee 2
2m
+11,
-
1.
392
SCHOENBERG
For
m =1
we f i n d
S1
t o be i d e n t i c a l w i t h t h e c l a s s ( 3 )
c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s . Observe t h a t t h e c l a s s o f p o l y n o m i a l s of d e g r e e s n o t e x c e e d i n g The r o l e o f t h e r o o f - f u n c t i o n t h e s o - c a l l e d centha.! B-npLine
M 2 m ( ~ ) : Waiting
SZmml c o n t a i n s 2m-1.
of (3)
M2(x)
of
x+
i s t a k e n o v e r by = max ( x , O ) ,
it
may be d e d i n e d b y
Clearly port
M2m(~)
€
S2m-l; w e also f i n d t h a t
M2m(~) > 0
i n its
sup-
- m < x < m. The B - s p l i n e s h o u l d be f a m i l i a r i n view of the fun-
damental i d e n t i t y
which a l s o shows t h a t
IM2,(x)dx
= 1 if
w e choose
f (x) = x
The r e p r e s e n t a t i o n ( 3 ) a l s o g e n e r a l i z e s , and eueAny S ( x )
2m E
. SZm-l
admitb a unique hepheoentation m
S(x) = c c
~
M*m(X
--m
whehe t h e
-
v)
I
c v ahe c o n n t a n t n . T h i s i s t h e s o - c a l l e d ntandahd heptebefl-
t a t i o n . The c o n v e r s e i s clear: Every series (8) f u r n i s h e s an e l e m e n t of
SZm-1
ments o f
.
W e now t r y t o s o l v e t h e i n t e r p o l a t i o n problem (1) b y e l e -
S2m-l.
I n t h i s d i r e c t i o n t h e r e are t w o d i f f e r e n t k i n d s o f
results.
A. T h e d a t a (y,) s e q u e n c e (y,)
ahe
06 poweh
g h o w t h (See [ 8
i s of p o w e h g n o w t h , and w r i t e
1). W e s a y t h a t t h e
ON CARDINAL SPLINE SMOOTHING
(y,)
(9)
393
E PG,
provided t h a t
y,
(10)
v
+
E
PG,
= ~ ( l v l y ) as
f
m,
f o r some
y
2
0.
y1
2
Similarly, we w r i t e
f(x)
(11)
provided t h a t
f ( x ) = O ( l x l y l ) as
x
+
f o r some
f m,
Below w e e x c l u d e t h e t r i v i a l c a s e when
m=l,
l e m i s s o l v e d by ( 3 ) w i t h o u t any r e s t r i c t i o n on t h e
THEOREM 1:
16 t h e heqUenCt
(y,)
i h
0.
s i n c e o u r prob
-
(y,,).
a d pawet g t u w t h , t h e n t h e i n t e h -
palation p t a b l e m
huh a u n i q u e h o l u t i o n
S(x)
huch t h a t
The a s s u m p t i o n ( 9 ) o f Theorem 1 i s a rough one; i t admits,e.g., a l l bounded s e q u e n c e s ( y V ) , w i t h
y = 0 i n ( 1 0 ) . The s e c o n d assump-
t i o n t o which w e now p a s s , i s much more s e l e c t i v e , and
takes
a c c o u n t t h e f i n e r s t r u c t u r e of t h e sequence: i n f a c t i t a d m i t s a narrow subclass of t h e s e q u e n c e s of
PG.
into only
As u s u a l , w i t h strongeras-
sumptions, s t r o n g e r c o n c l u s i o n s are p o s s i b l e : The i n t e r p o l a n t w i l l e x h i b i t a n i m p o r t a n t extremum p r o p e r t y .
S(x)
SCHOENBERG
394
m
B. T h e c a n e when
IAmyv12 <
C
-m
(See [ 9 , L e c t u r e
m
6] )
.
We
i n t r o d u c e t h e classes o f s e q u e n c e s and f u n c t i o n s a s follows:
(14)
(15)
L:={f(x);
Li
Of c o u r s e
f,
...,f (m-l)
and
W e may also d e s c r i b e
ments of L :
a r e a b s o l u t e l y continuous, f(")(x) ELZ@)
are t h e f a m i l i a r
L;
l;
L2 and
1.
Lzl respectively.
as t h e c l a s s o f s e q u e n c e s o b t a i n e d f r a n e l e -
L2 by n s u c c e s s i v e summation. S i m i l a r l y t h e e l e m e n t s o f
are o b t a i n e d from t h o s e o f
THEOREM 2.
L2
by
n
successive integrations.
76
t h e n t h e i n t e h p o l a t i o n phobLem
has a u n i q u e n o L u t i o n n u c h t h a t
Jhio dolution f(x)
(19)
and
i b
S(x)
han t h e 6oLLowing exthemum p h o p e h t y :
an a t b i t a a h y 6unction ouch t h a t
76
396
ON CARDINAL SPLINE SMOOTHING
f ( v ) = y,
v ,
hat all
then
(21)
1-U7
f (x) =
UnLebh
J-m
x.
d o t aLL keaL
S(X)
I n words: If (Y,)
€
R2
t h e n t h e s p l i n e i n t e r p o l a n t S ( x ) mini-
r
mizes t h e i n t e g r a l
(22)
among a l l s u f f i c i e n t l y smooth i n t e r p o l a n t s o f If
y, = P ( v ) f o r a l l
v , where P ( x ) E
(y,).
7
,~t h e -n
~
m P(x) ~ ES2nrlnL2,
and t h e r e f o r e
S ( x ) = P ( x ) by t h e u n i c i t y of t h e s o l u t i o n i n Theorem m 2 . However, h e r e I(S) = 0 . I n t h e g e n e r a l c a s e of (y,) E L 2 wemay
therefore say t h a t
S ( x ) i s among a l l i n t e r p o l a n t s o f
t h a t " i s most n e a r l y " a p o l y n o m i a l o f d e g r e e If P(X)
y,
E SZm-1
= P ( u ) , where
n PG, and so
P
(XI
E
IT^^-^ ,
-
1.
P(x)
9
2 m
but
~
~
the
-
~
f ( x ) such t h a t
I (f) <
m
How d o we a c t u a L L y c o n n t h u c t t h e
9 :l
.
S(x)
04
E
ll ,
AOLUtiOnb
hence a f o r t i o r i
(y,)
E
L2
.
T h i s i n s u r e s t h e c o n t i n u i t y of t h e p e r i o d i c f u n c t i o n m
T(u) =
C -m
y,e
ivu
is
.
assume t h a t
(y,)
r again t h a
There
these
t e h p o l a t i a n p k o b l e m n ? To answer t h i s q u e s t i o n l e t u s f o r t h e
(23)
one
S ( x ) = P ( x ) i s t h e unique s o l u t i o n o f The-
orem 1. Theorem 2 does n o t a p p l y h e r e b e c a u s e (y,) no i n t e r p o l a n t
(y,),
in-
moment
SCHOENBERG
366
which w e c a l l t h e g e n e h a t i n g
6uncLLon of t h e sequence ( y v ) . Here and
below w e d e n o t e t h e r e l a t i o n s h i p between a sequenceand its g e n e r a t i n g f u n c t i o n s y m b o l i c a l l y by w r i t i n g
We a l s o r e q u i r e t h e g e n e r a t i n g f u n c t i o n o f t h e sequence ( M 2 m ( ~ ) ) r which
is
Z2,(u)
m- 1
=
C
v=- (m-1)
ivu MZm(v)e
T h i s i s a c o s i n e polynomial o f o r d e r
I
x
I 2
m-1,
m . I t i s r e a d i l y e v a l u a t e d by ( 7 )
@,(u) = 1 , p14(u) = -1 ( ~ + c o s u), 3
Z,(U) =
I
.
because
MZm(x) = 0
if
and w e f i n d t h a t
1 ~ ( 3 3 + 2 6cos
... .
U + C O S ~ U ) ,
It a l s o has the property t h a t
(27)
0 < d 2 m ( ~5) d2,.,,(u)
5
Z2m(0) = 1
for all
u.
I t f o l l o w s t h a t i t s r e c i p r o c a l h a s an expansion
w i t h real c o e f f i c i e n t s
w
,W-v
= W v # t h a t decay e x p o n e n t i a l l y .
Let
us f i n d t h e s t a n d a r d r e p r e s e n t a t i o n
o f t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem ( 1 7 ) , which r e q u i r e s that
ON CARDINAL SPLINE SMOOTHING
-
C c . M2m(v j J
(30)
j) = y,
397
v.
for all
Furthermore l e t
b e t h e a s y e t unknown g e n e r a t i n g f u n c t i o n o f t h e ( c . ) . S i n c e t h e con-
I
v o l u t i o n o f two s e q u e n c e s h a s a g e n e r a t i n g f u n c t i o n t h a t i s t h e produ c t o f t h e g e n e r a t i n g f u n c t i o n s o f t h e two s e q u e n c e s , w e see by (241, (26) , and (31) , t h a t t h e r e l a t i o n s
( 3 0 ) are e q u i v a l e n t t o t h e rela
-
tion
* (wv)
Now ( 2 8 ) shows t h a t ( c v ) = (y,)
c
(33)
V
c y j wv-j
=
and t h e r e f o r e
v.
for all
j
T h e b e ake t h e c o e 6 d i c i e n A b a d t h e intekpaLating s p l i n e ( 2 9 ) .
EXAMPLES:
1. 16 m = l ,
we o b t a i n
c v = y,
Section
v
f o r all
.
0
= 1 , wv = O ( v # 0 )
,
and
16 m = 2 , w e f i n d ( S e e [ 9 , L e c t u r e 4 ,
51 ) t h a t
W
V
XIv1,
=
2. I f w e choose
shows t h a t
(34)
t h e n $,(u) = 1 , hence w
cv = w
V
.
y,
-
where
6" , w h e r e
X
= -2
+
47
=
-.26795.
6 o = 1, 6 v = O ( V
Therefore t h e s p l i n e
# O),then (33)
388
SCHOENBERG
i s t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem
L*m-l(4
(35)
=
6v
I
for a l l
v.
The f u n c t i o n ( 3 4 ) i s t h e dundamental & u n c t i o n o f t h e p r o c e s s , and t h e S(X)
A O h t i O M
o f t h e g e n e h U l p k o b l e m (17) LO g i v e n b y m
T h i s c a r d i n a l i n t e r p o l a t i o n f o r m u l a b r i d g e s t h e gap between the linear i n t e r p o l a n t ( 3 ) a n d t h e c a r d i n a l series ( 2 ) . I n f a c t , n o t i c e t h a t i f
m = 1 t h e n ( 3 6 ) r e d u c e s t o ( 3 ) , w h i l e w e have
l i m S2m-1(~)= m+m
(37)
Also every d e r i v a t i v e
sin
TIX
TIx
(k) (x) c o n v e r g e s t o t h e corresponding derivaS2m-l
t i v e of t h e r i g h t s i d e of
(37) I uniformly f o r a l l real
x
I n o u r d i s c u s s i o n w e have assumed t h a t ( 2 3 ) h o l d s .
the
tULatbJMd
. However,
( 3 3 ) , ( 2 9 ) , and ( 3 6 ) a k e v a l i d d o % b o t h T h e o t e m d 1 and
2, undeh t h e i t t e n p e c t i w e a d d u m p t i a n d .
PART 11.
THE CARDINAL SMOOTHING SPLINE
1. STATEMENT OF THE PROBLEM:
We assume now t h a t
(1)
and r e s t r i c t o u r s e l v e s t o r e a l - v a l u e d d a t a and f u n c t i o n s .
We
also
r e c a l l t h e d e f i n i t i o n s ( 3 . 1 4 ) and ( 3 . 1 5 ) o f P a r t I , o f t h e c l a s s e s ly and
L:
.
I n view o f t h e i n c l u s i o n r e l a t i o n s
ON CARDINAL SPLINE SMOOTHING
( S e e [ 9 , p. 1 0 4 1 )
,
399
w e o b s e r v e t h a t (1) i m p l i e s t h a t (y,)
satisfiesthe
a s s u m p t i o n s o f Theorem 2 f o r a l l m .
We a t e g i v e n m and a n m o o t h i n g patrameteh
THE PROBLEM:
E < 0.
Among
aLL 6unctionn
we w i b h t o d i n d t h e . b o l u t i o n
I
m
J(f) =
(4)
E
*
06
t h e phobLem
2
m
+
( f ( m ) (x)) d x
C
-m
-03
-
y V l 2 = minimum.
I n bOlVing t h e minimum p t o b l e m ( 4 ) we may tenLkiot t h e choice
LEMMA 1: 06
(f (v)
adminnible dunctianb
f ( x ) t o t h e eeementb
06
(5)
PROOF:
If
f ( x ) is such t h a t
J(f) <
m
,
then ( f ( v )
a p p l y Theorem 2 t o t h e s e q u e n c e ( f ( v ) ), and l e t
be such t h a t
s(v)
= f
(v) for a l l
v
.
But t h e n
-
yv) E L2
.
We
4w
SCHOENBERG
and so
in view of the extremum property of
S ( x )
as expressed by (3.21)
Theorem 2. Therefore, for any
f ( x ) , the spline
f (x), produces a value
J(f).
Let
U6
J(s)
of
s(x) that interpolates
thehedote d i n d t h e nolutian
o d t h e m i n i m u m ptobLem
I, m
J(S) =
(8)
E
m
(S(m))2dx+ C
-w
(S(v)
-
Y,)~ = minimum.
Here we need another
LEMMA 2:
7 6 ( 7 ) o a t i d d i e n S(x)
E
L2 (R),
(S(m)(x))2dx = -00
PROOF: From (7) we find that
hence aLno (c.) E L2 , t h e n
C yj-” c . c j, v
3
,
ON CARDINAL SPLINE SMOOTHING
401
i s t h e e v e n s e q u e n c e d e f i n e d by
where (y,)
where, t o s i m p l i f y n o t a t i o n s w e dropped t h e s u b s c r i p t 2m o f M 2 m ( ~ ) . I n t e g r a t i o n s by p a r t s show t h a t
(-ilm-' M(2m-1) x)
Observe t h a t
Jm
-m
M i ( X ) M ( ~ ~ -( x~ )- r ) d x
.
i s a s t e p f u n c t i o n assuming i n c o n s e c u t i v e
u n i t i n t e r v a l s t h e v a ue s
... 1 0 ,
(14)
011,
-
(2m-1 1
)
I
(2m-1 2 )
...
I
1 - 1 1
0, 0,
... .
T h i s sequence h a s t h e g e n e r a t i n g f u n c t i o n
except f o r a s h i f t f a c t o r
eiuk which w e d i s r e g a r d . N o w ( 1 3 ) indicates
t h a t ( y ) i s t h e c o n v o l u t i o n o f t h e sequence ( 1 4 ) w i t h t h e sequence
r
However, i n (13) t h e s e q u e n c e Cu avbv-r
.
If
(yr)
appears
as a sum o f
w e pass from ( a v ) t o t h e r e v e r s e d s e q u e n c e
o b t a i n a genuine c o n v o l u t i o n
Cva-vbv-r
.
L e t us t h e r e f o r e
the
form
(a_"),
we
reverse
t h e f i r s t s e q u e n c e (14). As w e o b t a i n t h e g e n e r a i n g f u n c t i o n o f t h e r e v e r s e d sequence by c h a n g i n g
u into
- u i n its o r i g i n a l generating
function, we f i n d the generating function of factor
eiuk) t h e p r o d u c t
(yr
t o b e ( u p t o a shift
402
-ium
= e
S i n c e (y,)
(2 sin
u 2m 7)
Z2,(u).
i s an e v e n s e q u e n c e , i t s g e n e r a t i n g f u n c t i o n m u s t b e e v e n ,
and t h e r e f o r e
e s t a b l i s h i n g (10).
2 . SOLUTION OF THE PROBLEM:
From ( a ) ,
( 9 1 , and ( 7 ) w e f i n d t h a t
L e t us minimize t h i s f u n c t i o n of t h e (c,).
tions, we differentiate
-a 2 ack
J(S) =
E
To o b t a i n t h e normalequa-
J(S) obtaining
Z yj-kcj+
C { Z c . M ( w - j ) - y w ) M ( w - k ) = O (kEZ).
j
v
j
7
I f w e sum w i t h i n t h e double-sum o n l y w i t h r e s p e c t t o
where
v , we obtain
ON CARDINAL SPLINE SMOOTHING
(3)
(au)
403
2
.
(d2m(u))
+
The normal e q u a t i o n s t h u s become
or
(4)
C j
+
(clj-k
E
Y,
yj-k)~j=
M2m(~
-
k)
(k
E
However, by ( 3 ) a n d ( 1 . 1 0 ) w e f i n d
and w r i t i n g
(6)
(c,,)
+
C(u),
(y,)
+
T(u)
,
w e f i n d t h e normal e q u a t i o n s (4) t o b e e q u i v a l e n t t o t h e r e l a t i o n
i ( p ~ , ~ ( u +) )E~( 2 s i n + ) 2 m
whence
This e s t a b l i s h e s
~ , , ( u ) ) c ( u ) = ~ ( u 4) 2 m ( u ) t
if).
SCHOENBERG
404
THEOREM 3:
I n tehmh
06
whehe t h e c a e d d i c i e n t n dicientn (c.) 7
04
t h e expannion
w"(E)
=
w-"(E)
d e c a y e x p o n e n t i a L L y , t h e coed-
t h e naLution
0 6 t h e minimum p h o b l e m ,
ahe
W e c a l l t h e s o l u t i o n ( 9 ) t h e cahdinad smoothing n p l i n e .
3.
A dew p h o p e h t i e b A.
06
t h e cahdinad nmoothing n p d i n e
W e have assumed above t h a t
E
S(X) =s(x;E).
> 0 . However, i f w e s e t
E
=O
i n (2.81, i t becomes
and a comparison w i t h t h e e x p a n s i o n ( 3 . 2 8 ) o f P a r t I , w v ( 0 ) = wv
for all
v : T h i n nhawn t h a t S (x)
inteapolafing caadinal npline B.
What
n t h e eddect
a n t h e ahiginai? dada sequence
(S( v )
,
(y,) ?
06
06
shows
that
S ( x ; O ) = S ( x ) &educed A0 t h e
Theohem
2.
t h e nmoathing n p l i n e
S(x) = S(x;
E)
T h i s w e answer by determining the "smoothed"
t o compare i t w i t h ( y , ) .
By ( 2 . 9 ) and (2.10)we find
406
ON CARDINAL SPLINE SMOOTHING
a n d t h e r e f o r e , by ( 2 . 7 )
I
I n terms o f t h e e x p a n s i o n
1 (2 sin
(3)
=
u 2m
c
eivu
uv(E)
V
Z2,(U)
+
( 2 ) shows t h a t t h e s e q u e n c e ( S ( V ; E ) ) a h i n e n d h o m t h e d a t a (y,)
by t h e
n m o o t h i n g dohmuea
Observe t h a t by ( 2 . 8 ) a n d ( 3 ) t h e c o e f f i c i e n t s i n terms of
W v ( ~ )
by
u"(E)
= C MZm (v
j
-
j)
U ~ ( E )
are e x p r e s s e d
W.(E).
3
Is (4) a s m o o t h i n g f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n o f P a r t I , S e c t i o n l ? T h a t it i s o n e w e see i f w e i n s p e c t i t s c h a r a c t e r i s t i c
function
K(u;E)
(5)
1 u 2m (2 sin T )
=
!d2m(u)
+
f o r it is evident t h a t
0 < K(u;E)
(6)
C. cheabing
< K(O;E)
T h e b m o o t h i n g poweh E
.
06
=
1
for
0 < u < 2r
.
t h e 6ohmuLa ( 4 ) i n c h e a b e n w i t h
I n [ 4 , D e f i n i t i o n 2 , p . 5 3 1 w e g a v e good r e a s o n s
in-
for
406
SCHOENBERG
t h e f o l l o w i n g d e f i n i t i o n : Of two d i f f e r e n t smoothing formulae h a v i n g the characteristic functions
d ( u ) and
$(u)
, we
s a y t h a t t h e second
h a s g r e a t e r smoothing power, p r o v i d e d t h a t
(7)
/J(U)
However, i f
I 5
0 <
E
Id(u)I
<
El
for a l l
u, excluding e q u a l i t y f o r a l l
u.
i t i s c l e a r by ( 5 ) t h a t
and t h e c r i t e r i o n ( 7 ) i s s a t i s f i e d .
D.
The deghee
06
eXUCtneAA
0 6 ,the nmootking 6omonuRa
(4) A = h - l .
T h i s f o l l o w s from ( 1 . 7 ) o f P a r t I , b e c a u s e ( 5 ) shows t h a t w e h a v e t h e e x p a n s i o n i n powers o f
E.
06
u
I f w e d r o p o u r a s s u m p t i o n (1.1) , and assume o n l y t h a t ( y v )
poweh ghowth, Rhen
b y t h e dohmutae(2.8)
,
#in conb&ucfi#n
06 t h e bmvotking
(2.10) , and (2.9) , hemaind appticabLe.0f murse, J ( S ) , of
i t s e a r l i e r connection with t h e funtional holds. I n f a c t we w i l l f i n d t h a t sumably, it i s s t i l l t r u e t h a t o u r t h a t (y,)
npfine S ( x ) = S ( x ; E )
J(S) =
m
( 1 . 8 ) , no l o n g e r
f o r a l l s p l i n e s S . Pre-
S ( X ; E ) minimizes
J ( S ) , provided
s a t i s f i e s the condition
o f Theorem 2 . However, t h i s I was n o t a b l e t o e s t a b l i s h . I n any c a s e I recommend t h e c a r d i n a l smoothing s p l i n e ( S ( X ; E ) ) , which r e p r e s e n t s t h e m o d i f i c a t i o n , found more t h a n 30 y e a r s l a t e r , o f may war-time approach t o t h e problem o f c a r d i n a l smoothing.
407
ON CARDINALSPLINE SMOOTHING
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T. N.
E.
GREVILLE, On s t a b i l i t y o f l i n e a r s m o o t h i n g
[ 21
T. N.
E.
GREVILLE, On a p r o b l e m of E .
SIAM J. N u m . A n a l y s i s , 3 ( 1 9 6 6 ) , p p . 1 5 7 - 1 7 0 .
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,
L.
De Forest i n iterated
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376
FRITZ J O H N , On i n t e g r a t i o n o f p a r a b o l i c e q u a t i o n s by m e t h o d s , Corn. on P u r e a n d Appl. Math.
[ 41
formulas,
I.
J.
,
- 398. difference
5 ( 1 9 5 2 ) ,pp.155-211.
SCHOENBERG, C o n t r i b u t i o n s t o t h e p r o b l e m o f approximation
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4 ( 1 9 4 6 ) , P a r t A, p p . 45 - 9 9 ,
P a r t B , pp. 1 1 2 - 1 4 1 .
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of
s m o o t h i n g , C o u r a n t A n n i v e r s a r y volume ".Sfidh% and en nay^", New York, 1 9 4 8 , p p .
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[ 71
I . J . SCHOENBERG, S p l i n e f u n c t i o n s a n d t h e p r o b l e m o f g r a d u a -
f u n c t i o n s , B u l l . Amer. Math. SOC., 59 ( 1 9 5 3 ) , p p . 1 9 9
t i o n , Proc. N a t . [ 81
Acad. S c i . 5 2 ( 1 9 6 4 ) , pp.
- 230.
947 - 9 5 0 .
I . J . SCHOENBERG, C a r d i n a l i n t e r p o l a t i o n and s p l i n e
functions
11. I n t e r p o l a t i o n o f d a t a o f power g r o w t h , J . Approx. The-
o r y , 6 ( 1 9 7 2 ) , pp. 4 0 4 [ 9
1
- 420.
I . J . SCHOENBERG, C a k d i n a b h p l i n e . i n t e t p o e a t i o n ,
Reg.
Conf.
Monogr. NQ 1 2 , 1 2 5 p a g e s , SIAM, P h i l a d e l p h i a , 1 9 7 3 .
[lo]
E . T . WHITTAKER a n d G.
ROBINSON, T h e caecueun o d o b n e n v a t i a n n ,
B l a c k i e a n d Son, London, 1924.
D e p a r t m e n t of M a t h e m a t i c s U n i t e d S t a t e s M i l i t a r y Academy
West P o i n t , N e w York 1 0 9 9 6
@North-Holland Publishing Company, 1979
ON CARDINAL SPLINE SMOOTHING
I. J. SCHOENBERG Mathematics Research Center University of Wisconsin Madison, Wisconsin 53706, USA.
INTRODUCTION The present paper describes the methods, with some changes be mentioned below, whereby I solved the numerical problem
to
assigned
to me at the Ballistics Research Laboratories in Aberdeen, Maryland, during the second World War (see [ 4
]
1.
The problem was to smooth very
extended equidistant tables of drag functions (or drag coefficients) by approximating them by very smooth functions that were easily computable with their first and second derivatives. The first question to be answered was this: When may a " m o w h g awefiage" o p e f i a t i o n L e g i t i m a t e L y b e c a L l e d a n m o o t h i n g 6ofimuLa ?
An
answer is given in Section 1 of Part I. That it is a reasonable
one
is shown in Section 2 of Part I. These ideas were later greatly generalized by Fritz John in his important work on parabolic differential equations (see [ 3 1 ) . The connection is briefly
mentioned
in
Section 2 of Part I. The second essential ingredient of our results of Part
I1
is
the process of c a f i d i n a e h p t i n e intehpolation (see [ 9 1 ). The required results are described in Section 3 of Part I. The changes from my war-time approach are developed completely in Part 11. This is the new contribution of the paper, and it arises 383
384
SCHOENBERG
i f w e a d a p t t h e i d e a o f E. T. Whittaker (see [lo 1 and [ 7
1 ) to
the
problem of smoothing a b i - i n f i n i t e sequence of e q u i d i s t a n t d a t a . The r e s u l t i s t h e cahd.inaL 4 p L i n e s m o o t h i n g phOces4. L e t m e thank P r o f e s s o r I l i o G a l l i g a n i f o r a s k i n g m e
to
visit
Rome i n May of 1 9 7 7 , where I gave a s h o r t 10-hours c o u r s e o n C a r d i n a 1 Spline Interpolation a t h i s I s t i t u t o per l e applicazioni d e l Calcolo "Mauro Picone".
PART I .
PRELIMINARIES
1. WHAT IS A SMOOTHING FORMULA?
The s e v e r a l smoothing ( o r gradua
-
t i o n ) methods d i s c u s s e d i n [lo] are a l l o f t h e "moving a v e r a g e " t y p . By t h i s w e mean t h a t w e are g i v e n a sequence o f r e a l and
symmetric
weights
which w e a p p l y t o t h e d a t a (x,)
t o produce t h e i r smoothedversicln (y,)
,
by t h e formula
yn
--
m
C
v=-co
an-Vxy
, (n=O,
f 1,
...) .
I n o t h e r words, w e c o n v o l u t e t h e sequences (a,)
and (x,)
,
an o p e r a
-
t i o n s y m b o l i c a l l y d e s c r i b e d by
(3)
By d e r i v i n g a formula ( 2 ) a c c o r d i n g t o a d e f i n i t e i d e a , s u c h a s a l e a s t s q u a r e s formula, o r W h i t t a k e r ' s formulae (see [ 1 0 , XI
1
),
Chapter
we may r e a s o n a b l y e x p e c t t h a t it w i l l t r a n s f o r m a g i v e n
se-
quence ( x ) i n t o a smoother sequence (y,). T h i s p o i n t becomes doubtn f u l l however, i n c a s e t h e formula ( 2 ) was o t h e r w i s e o b t a i n e d , e . g . ,
385
ON CARDINAL SPLINE SMOOTHING
by some a p p r o x i m a t i n g p r o c e d u r e t h a t i s n o t s t r i c t l y
interpolatory.
An example of such a p r o c e d u r e w i l l be o u r c a r d i n a l s p l i n e smoothing o f P a r t 11. A c r i t e r i o n f o r ( 2 ) t o be r e g a r d e d a s a b m o o t h i n g ~0hmlLea w a s proposed by t h e a u t h o r i n [ 4 , pp. 50
- 54 1
and p r o c e e d s a s f o l l w s .
To b e g i n w i t h , w e assume t h e L a u r a n t series
(4)
I
t o converge i n some r i n g c o n t a i n i n g t h e u n i t c i r c l e
z j =1. S e t t i n g
i u , we c a l l t h e even f u n c t i o n z = e
d(u)
(5)
iu
= F(e
)
=
a
0
+
2al cos u
+
2a2 c o s 2u
+
...
t h e chahactehintic ( u n c t i o n of t h e formula ( 2 ) . The r e g u l a r i t y of F(z) on
I
IIm
UI
z
I
= 1, i m p l i e s t h a t
$(u)
is regular
in
a
certain
strip
< a , whence t h e v a l i d T a y l o r e x p a n s i o n m
T h i s e x p a n s i o n a l l o w s us t o e x p r e s s e a s i l y an i m p o r t a n t p r o p e r t y ( 2 ) : I t s hephoductive p o w e f ~ o , r deghee
06
exactnebb.
We s a y t h a t ( 2 ) h a s t h e d e g r e e o f e x a c t n e s s i s a n a t u r a l number, p r o v i d e d t h a t ( 2 ) r e p r o d u c e s
yn
- xn
values
f o r a l l n , a sequence (x,) P(n) of
,
of
where
2m-1,
exactly,
m
i. e.
,
i f t h i s sequence r e p r e s e n t s t h e
a polynomial of d e g r e e
2m-1,
b u t d o e s n o t have
t h i s p r o p e r t y f o r any h i g h e r d e g r e e . T h a t t h e d e g r e e of e x a c t n e s s i s always odd, f o l l o w s from t h e symmetry c o n d i t i o n ( 1 ) . I n t e r m s of
(6)
w e have t h e f o l l o w i n g e a s i l y e s t a b l i s h e d p r o p o s i t i o n : T h e doamuta ( 2 ) hub t h e degkee
id t h e e x p a n n i o n ( 6 ) i n
06
the 6otm
06
eXaCtflebb
2m-1,
i6 and onLy
SCHOENBERG
386
$(u) = 1
(7)
-x
.,.
+
U2m
(A
#
0).
An example. L e t (1) be t h e sequence a
0
= 5/8,
al = 1 / 4 ,
a2 =
-
1/16,
an = 0
if
n > 2,
when t h e c o n v o l u t i o n ( 2 ) assumes t h e form
From (5) w e o b t a i n
(9)
$ ( u ) = ( 5 + 4 cox u
-
and ( 7 ) i s s e e n t o h o l d w i t h mula ( 8 ) has t h e d e g k e e
06
cos 2u)/8 = 1
rn = 2 , X
=
1/16.
- 7
u4
+
...
I t follows that the f o r -
exactness 3 .
May ( 8 ) be r e g a r d e d as a smoothing formula ? The g e n e r a l
cri-
t e r i o n i s d e s c r i b e d by t h e f o l l o w i n g .
DEFINITION:
kJe 4ay t h a t ( 2 ) Lo a s m o o t h i n g botmuLa, p h o v i d e d t h a t La2
chahactchihtic dunction
d(u) 4atiAdieb
This condition evidently implies t h a t t h e coed6icien-t
peaking i n ( 7 ) ,
A ,
ap-
io p o s i t i v e . The c h a r a c t e r i s t i c f u n c t i o n ( 9 ) i s e a s -
i l y s e e n t o s a t i s f y (lo), and so ( 8 ) i s a smoothing f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n of t h e t e r m . The c r i t e r i o n (10) raises t h e f o l l o w i n g k i n d o f e q u a t i o n :
Do
t h e numerous smoothing formulae a s g i v e n i n 110 1 s a t i s f y our criterion
301
ON CARDINAL SPLINE SMOOTHING
(lo)? S e e G r e v i l l e ' s p a p e r [ l 1 where s o m e of t h e s e q u e s t i o n s a r e answered a f f i r m a t i v e l y . I n [ 4 , p p . 50
-
54 ] good a r g u m e n t s i n s u p p o r t of
( 1 0 ) are p r e s e n t e d . Even m o r e c o n v i n c i n g r e a s o n s
the criterion
w e r e given i n
[5,
P a r t I ] . I n t h e n e x t s e c t i o n we r e p r o d u c e t h e s e a r g u m e n t s a s p r e s e n t e d i n [ 6 , pp. 2 0 0 - 2 0 4 1 .
2 . THE BEHAVIOR OF THE ITERATES O F A MOVING AVERAGE FORMULA:
THE
PROBLEM OF DE FOREST. A c o n c l u s i v e argument i n s u p p o r t o f t h e n e c e s s i t y of o u r c o n d i t i o n ( 1 . 1 0 )
i s f u r n i s h e d by t h e s o l u t i o n o f t h e f o l -
l o w i n g problem f i r s t s t a t e d a n d a t t a c k e d
by
Erasmus
L. D e
(1834-1888). I f we s u b j e c t t h e g i v e n s e q u e n c e ( x u ) n times
Forest
i n suc
-
c e s s i o n t o t h e same t r a n f o r m a t i o n ( 1 . 2 ) , we o b t a i n a l i n e a r transformation
which i s t h e n - f o l d 06
the
i t e r a t e of ( 1 . 2 ) . W h a t i n R h e a o y m p t o t i c
c a e d 6 i c i e n t n 0 6 (1) an
n
+
m ?
T h i s q u e s t i o n was
behaviah
answered
by D e F o r e s t a n d by G. B. D a n t z i g ( f o r r e f e r e n c e s see151 ) f o r t h e c a s e when a l l c o e f f i c i e n t s of that
m=l
( 1 . 2 ) a r e non - n e g a t i v e , h e n c e n e c e s s a r i l y
i n ( 1 . 7 ) . A g e n e r a l s o l u t i o n i s as follows.
L e t ( 1 . 2 ) be s u c h t h a t ( 1 . 1 7 )
hence t h a t
,
( l . l O ) , and (1.71, are s a t i s f i e d ,
X > 0. L e t
-V
which i s t h e normal f r e q u e n c y f u n c t i o n
2m
cos vx d v ,
SCHOENBERG
388
if
m =1, o t h e r w i s e (m = 2,3,.
.
.)
Gm(x) i s an e n t i r e f u n c t i o n
having
i n f i n i t e l y many zeros, a l l r e a l . T h e caeddicients a d (1) satis6q t h e a s y m p t o t i c h e L a t i o n h
--
--
1 1 --1 a ( n ) = ( i n ) 2m Gm(v(hn) 2m ) + , o ( n 2m)
(4)
a4
V
where t h e " l i t t e e
n +
m ,
v.
or' dymbok? hoed4 unidaamly d o h n l L i n t e g e h d
For a proof see ( 5 , P a r t I ] , where i t i s a l s o shown byexamples (1.10),
t h a t ( 4 ) no l o n g e r h o l d s i f t h e e q u a l i t y s i g n i s a l l o w e d i n and t h a t t h e c o e f f i c i e n t n = 2k
aAn) d i v e r g e s e x p o n e n t i a l l y t o
t e n d s t o i n f i n i t y t h r o u g h even v a l u e s , i f
( 1 . 1 0 ) are r e v e r s e d anywhere i n t h e i n t e r v a l
the
0 < u < 2.rr
+
m
,
as
inequalities
.
The f o l l o w i n g d i s c u s s i o n , w h i l e n o t d i r e c t l y r e l a t e d
to
our
s u b j e c t of smoothing, w i l l show t h e c o n n e c t i o n of t h e a s y m p t o t i c rel a t i o n ( 4 ) w i t h t h e w i d e r f i e l d of p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s . Observe t h a t ( 2 ) i m p l i e s t h a t
(5)
--1
--1
U(x,t) = t 2m G m ( x t 2m) =
-
-tv
2m
+ ixv
d v , ( t > 0).
The f u n c t i o n under t h e i n t e g r a l s i g n i s immediately s e e n t o s a t i s f y for all v
, the
d i f f e r e n t i a l equation
which r e d u c e s t o t h e f a m i l i a r h e a t e q u a t i o n i f
also -plane
m = l . I t follows t h a t
U ( x , t ) , d e f i n e d by ( 5 ) , is a s o l u t i o n of (6) i n t h e upper h a l f t > 0 . On t h e o t h e r hand, a p p l y i n g t o ( 2 ) F o u r i e r ' s inversion
formula and s e t t i n g
v = O r we f i n d that
ON CARDINALSPLINE SMOOTHING
These r e m a r k s imply t h e f o l l o w i n g : 'I 6
1x 1
say, a s
+
LA a b o t u , t i o n
a,
06
f (x)
389
cantinuow and a ( I X I - * )
,
then
t h e d i , j d e ~ e n . t i a l e q u a t i o n ( 6 ) Aattin6qLng t h e boundmy
condition
This p a r t i c u l a r s o l u t i o n
u ( x , t ) may now a l s o be
approximated
by t h e f o l l o w i n g n u m e r i c a l p r o c e d u r e : Draw i n t h e ( x , t ) - p l a n e
the
rectangular lattice of p o i n t s
(WAX,
n At)
(w
= 0, k 1
, ...
;
D e f i n e on it a l a t t i c e f u n c t i o n
n = 0,1,2
u
v
, ...) .
by s t a r t i n g w i t h
uw ,o = f ( v Ax) ,
and computing t h e v a l u e s a l o n g e a c h h o r i z o n t a l l i n e from those on the l i n e below i t , by means o f t h e t r a n s f o r m a t i o n (1.2). T h i s
evidently
amounts t o i t e r a t i n g (1.21, a n d a f t e r n s t e p s w e o b t a i n
(10)
For any g i v e n x a n d
t > 0,
( 1 0 ) w i l l go o v e r i n t o ( 8 ) i f w e
f o l l o w i n g : We 6 h A t c o n n e c t the. m e o h - n i z e b
Ax and
A t
do the
by ,the h d a t i a n
SCHOENBERG
380
A t = X (Ax) 2m.
(11)
Id t h e i n t e g e k b
n ahe buch t h a t
v and
VAX
+
x,
and
n A t
.+
t
an
Ax
0.
+
then
U
v,n
T h i s follows r e a d i l y from r e l . a t i o n (4): ( 1 0 ) d i f f e r s
U(X,t).
+
1 0 ) and ( 8 1 , i n view o f t h e
asymptotic
from a Cauchy-Riemann sum f o r tk integral
( 8 1 , by a q u a n t i t y t h a t t e n d s t o z e r o due t o t h e u n i f o r m i t y i n
v of
t h e error t e r m o f ( 4 ) . I t i s i n t e r e s t i n g t o n o t e t h a t it d o e s n o t matter which
for-
mula ( 1 . 2 ) w e u s e i n t h i s c o n s t r u c t i o n , as l o n g as it i s o f the degree of exactness
2m-1,
i.e.,
i t s a t i s f i e s (1.71, and above a l l t h a t i t
s a t i s f i e s t h e s t a b i l i t y c o n d i t i o n (1.10)
,
t h e t e r m "stabi1ity"meaning
h e r e s t a b i l i t y on i t e r a t i o n . F o r t h e g e n e r a l t h e o r y of F. J o h n , which t h e e q u a t i o n ( 6 ) i s a s p e c i a l example, see [ 3 1
.
of
I n t h i s s e c t i o n w e d e a l t e x c l u s i v e l y w i t h f o r m u l a e ( 1 . 2 ) which s a t i s f y t h e symmetry r e l a t i o n . I n [ 2 ]
T.
N.
E. G r e v i l l e d e a l t
with
t h e more d i f f i c u l t c a s e o f unsymmetric f o r m u l a e .
3 . CARDINAL SPLINE INTERPOLATION (see [ 9 , L e c t u r e s 1
l e m o f caadinal intehpolation i s t o f i n d s o l u t i o n s
-
4 1 ) . T h e prob-
f ( x ) of t h e i n -
t e r p o l a t i o n problem
(1)
f ( v ) = Y"
,
for all i n t e g e r s
v
,
where ( y v ) are t h e d a t a . A f o r m a l s o l u t i o n i s f u r n i s h e d b y t h e series
391
ON CARDINAL SPLINE SMOOTHING
i n v e s t i g a t e d i n 1 9 0 8 by de l a V a l l G e P o u s s i n , also l a t e r
by
E. T.
W h i t t a k e r , who c a l l e d i t t h e cahdinad b e h i e b . The d i f f i c u l t y w i t h ( 2 )
“i: :y
i s t h e s l o w decay o f t h e f u n c t i o n
as
x
-. A
+
s o l u t i o n of (1) i s t h e p i e c e w i b e l i n e a h i n t e h p o t h z t
much s i m p l e r g i v e n by
S1(x)
m
(3)
where
M2(x) i s t h e roof f u n c t i o n d e f i n e d by
in
M2(x) = x + l
,
[-1,01
M (x) = 1
2
-x
i n [ O , l l and%(x)
=o
The p u r p o s e o f cahdinad b p l i n e i n t e h p o & z t i o n i s t o b r i d g e between t h e p i e c e w i s e l i n e a r
if 1x1 ’1.
the
gap
S1(x) d e f i n e d by (31, a n d t h e c a r d i n a l
series ( 2 ) . I t a i m s a t r e t a i n i n g s o m e of t h e s t u r d i n e s s a n d s i n p l i c i t y of ( 3 ) , a t t h e same t i m e c a p t u r i n g some of t h e s m o o t h n e s s a n d s o p h i s t i c a t i o n of
(2).
Le-t m be a natuhad numbeh, and d e b
(4)
S2rn-l
b e t h e cLadb
06
= {S(X)3
cahdinad b p d i n e d
S(x)
0 6 deghee
2m-1
dedined
by
the two conditionb:
(5)
The h e s t h i c t i o n whete
v
i d
04
S ( x ) -to e u e h y u n i t i n t e n v a l
a n i n X e g e h , i b apolynomia!.
(v ,v
0 6 deghee 2
2m
+11,
-
1.
392
SCHOENBERG
For
m =1
we f i n d
S1
t o be i d e n t i c a l w i t h t h e c l a s s ( 3 )
c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s . Observe t h a t t h e c l a s s o f p o l y n o m i a l s of d e g r e e s n o t e x c e e d i n g The r o l e o f t h e r o o f - f u n c t i o n t h e s o - c a l l e d centha.! B-npLine
M 2 m ( ~ ) : Waiting
SZmml c o n t a i n s 2m-1.
of (3)
M2(x)
of
x+
i s t a k e n o v e r by = max ( x , O ) ,
it
may be d e d i n e d b y
Clearly port
M2m(~)
€
S2m-l; w e also f i n d t h a t
M2m(~) > 0
i n its
sup-
- m < x < m. The B - s p l i n e s h o u l d be f a m i l i a r i n view of the fun-
damental i d e n t i t y
which a l s o shows t h a t
IM2,(x)dx
= 1 if
w e choose
f (x) = x
The r e p r e s e n t a t i o n ( 3 ) a l s o g e n e r a l i z e s , and eueAny S ( x )
2m E
. SZm-l
admitb a unique hepheoentation m
S(x) = c c
~
M*m(X
--m
whehe t h e
-
v)
I
c v ahe c o n n t a n t n . T h i s i s t h e s o - c a l l e d ntandahd heptebefl-
t a t i o n . The c o n v e r s e i s clear: Every series (8) f u r n i s h e s an e l e m e n t of
SZm-1
ments o f
.
W e now t r y t o s o l v e t h e i n t e r p o l a t i o n problem (1) b y e l e -
S2m-l.
I n t h i s d i r e c t i o n t h e r e are t w o d i f f e r e n t k i n d s o f
results.
A. T h e d a t a (y,) s e q u e n c e (y,)
ahe
06 poweh
g h o w t h (See [ 8
i s of p o w e h g n o w t h , and w r i t e
1). W e s a y t h a t t h e
ON CARDINAL SPLINE SMOOTHING
(y,)
(9)
393
E PG,
provided t h a t
y,
(10)
v
+
E
PG,
= ~ ( l v l y ) as
f
m,
f o r some
y
2
0.
y1
2
Similarly, we w r i t e
f(x)
(11)
provided t h a t
f ( x ) = O ( l x l y l ) as
x
+
f o r some
f m,
Below w e e x c l u d e t h e t r i v i a l c a s e when
m=l,
l e m i s s o l v e d by ( 3 ) w i t h o u t any r e s t r i c t i o n on t h e
THEOREM 1:
16 t h e heqUenCt
(y,)
i h
0.
s i n c e o u r prob
-
(y,,).
a d pawet g t u w t h , t h e n t h e i n t e h -
palation p t a b l e m
huh a u n i q u e h o l u t i o n
S(x)
huch t h a t
The a s s u m p t i o n ( 9 ) o f Theorem 1 i s a rough one; i t admits,e.g., a l l bounded s e q u e n c e s ( y V ) , w i t h
y = 0 i n ( 1 0 ) . The s e c o n d assump-
t i o n t o which w e now p a s s , i s much more s e l e c t i v e , and
takes
a c c o u n t t h e f i n e r s t r u c t u r e of t h e sequence: i n f a c t i t a d m i t s a narrow subclass of t h e s e q u e n c e s of
PG.
into only
As u s u a l , w i t h strongeras-
sumptions, s t r o n g e r c o n c l u s i o n s are p o s s i b l e : The i n t e r p o l a n t w i l l e x h i b i t a n i m p o r t a n t extremum p r o p e r t y .
S(x)
SCHOENBERG
394
m
B. T h e c a n e when
IAmyv12 <
C
-m
(See [ 9 , L e c t u r e
m
6] )
.
We
i n t r o d u c e t h e classes o f s e q u e n c e s and f u n c t i o n s a s follows:
(14)
(15)
L:={f(x);
Li
Of c o u r s e
f,
...,f (m-l)
and
W e may also d e s c r i b e
ments of L :
a r e a b s o l u t e l y continuous, f(")(x) ELZ@)
are t h e f a m i l i a r
L;
l;
L2 and
1.
Lzl respectively.
as t h e c l a s s o f s e q u e n c e s o b t a i n e d f r a n e l e -
L2 by n s u c c e s s i v e summation. S i m i l a r l y t h e e l e m e n t s o f
are o b t a i n e d from t h o s e o f
THEOREM 2.
L2
by
n
successive integrations.
76
t h e n t h e i n t e h p o l a t i o n phobLem
has a u n i q u e n o L u t i o n n u c h t h a t
Jhio dolution f(x)
(19)
and
i b
S(x)
han t h e 6oLLowing exthemum p h o p e h t y :
an a t b i t a a h y 6unction ouch t h a t
76
396
ON CARDINAL SPLINE SMOOTHING
f ( v ) = y,
v ,
hat all
then
(21)
1-U7
f (x) =
UnLebh
J-m
x.
d o t aLL keaL
S(X)
I n words: If (Y,)
€
R2
t h e n t h e s p l i n e i n t e r p o l a n t S ( x ) mini-
r
mizes t h e i n t e g r a l
(22)
among a l l s u f f i c i e n t l y smooth i n t e r p o l a n t s o f If
y, = P ( v ) f o r a l l
v , where P ( x ) E
(y,).
7
,~t h e -n
~
m P(x) ~ ES2nrlnL2,
and t h e r e f o r e
S ( x ) = P ( x ) by t h e u n i c i t y of t h e s o l u t i o n i n Theorem m 2 . However, h e r e I(S) = 0 . I n t h e g e n e r a l c a s e of (y,) E L 2 wemay
therefore say t h a t
S ( x ) i s among a l l i n t e r p o l a n t s o f
t h a t " i s most n e a r l y " a p o l y n o m i a l o f d e g r e e If P(X)
y,
E SZm-1
= P ( u ) , where
n PG, and so
P
(XI
E
IT^^-^ ,
-
1.
P(x)
9
2 m
but
~
~
the
-
~
f ( x ) such t h a t
I (f) <
m
How d o we a c t u a L L y c o n n t h u c t t h e
9 :l
.
S(x)
04
E
ll ,
AOLUtiOnb
hence a f o r t i o r i
(y,)
E
L2
.
T h i s i n s u r e s t h e c o n t i n u i t y of t h e p e r i o d i c f u n c t i o n m
T(u) =
C -m
y,e
ivu
is
.
assume t h a t
(y,)
r again t h a
There
these
t e h p o l a t i a n p k o b l e m n ? To answer t h i s q u e s t i o n l e t u s f o r t h e
(23)
one
S ( x ) = P ( x ) i s t h e unique s o l u t i o n o f The-
orem 1. Theorem 2 does n o t a p p l y h e r e b e c a u s e (y,) no i n t e r p o l a n t
(y,),
in-
moment
SCHOENBERG
366
which w e c a l l t h e g e n e h a t i n g
6uncLLon of t h e sequence ( y v ) . Here and
below w e d e n o t e t h e r e l a t i o n s h i p between a sequenceand its g e n e r a t i n g f u n c t i o n s y m b o l i c a l l y by w r i t i n g
We a l s o r e q u i r e t h e g e n e r a t i n g f u n c t i o n o f t h e sequence ( M 2 m ( ~ ) ) r which
is
Z2,(u)
m- 1
=
C
v=- (m-1)
ivu MZm(v)e
T h i s i s a c o s i n e polynomial o f o r d e r
I
x
I 2
m-1,
m . I t i s r e a d i l y e v a l u a t e d by ( 7 )
@,(u) = 1 , p14(u) = -1 ( ~ + c o s u), 3
Z,(U) =
I
.
because
MZm(x) = 0
if
and w e f i n d t h a t
1 ~ ( 3 3 + 2 6cos
... .
U + C O S ~ U ) ,
It a l s o has the property t h a t
(27)
0 < d 2 m ( ~5) d2,.,,(u)
5
Z2m(0) = 1
for all
u.
I t f o l l o w s t h a t i t s r e c i p r o c a l h a s an expansion
w i t h real c o e f f i c i e n t s
w
,W-v
= W v # t h a t decay e x p o n e n t i a l l y .
Let
us f i n d t h e s t a n d a r d r e p r e s e n t a t i o n
o f t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem ( 1 7 ) , which r e q u i r e s that
ON CARDINAL SPLINE SMOOTHING
-
C c . M2m(v j J
(30)
j) = y,
397
v.
for all
Furthermore l e t
b e t h e a s y e t unknown g e n e r a t i n g f u n c t i o n o f t h e ( c . ) . S i n c e t h e con-
I
v o l u t i o n o f two s e q u e n c e s h a s a g e n e r a t i n g f u n c t i o n t h a t i s t h e produ c t o f t h e g e n e r a t i n g f u n c t i o n s o f t h e two s e q u e n c e s , w e see by (241, (26) , and (31) , t h a t t h e r e l a t i o n s
( 3 0 ) are e q u i v a l e n t t o t h e rela
-
tion
* (wv)
Now ( 2 8 ) shows t h a t ( c v ) = (y,)
c
(33)
V
c y j wv-j
=
and t h e r e f o r e
v.
for all
j
T h e b e ake t h e c o e 6 d i c i e n A b a d t h e intekpaLating s p l i n e ( 2 9 ) .
EXAMPLES:
1. 16 m = l ,
we o b t a i n
c v = y,
Section
v
f o r all
.
0
= 1 , wv = O ( v # 0 )
,
and
16 m = 2 , w e f i n d ( S e e [ 9 , L e c t u r e 4 ,
51 ) t h a t
W
V
XIv1,
=
2. I f w e choose
shows t h a t
(34)
t h e n $,(u) = 1 , hence w
cv = w
V
.
y,
-
where
6" , w h e r e
X
= -2
+
47
=
-.26795.
6 o = 1, 6 v = O ( V
Therefore t h e s p l i n e
# O),then (33)
388
SCHOENBERG
i s t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem
L*m-l(4
(35)
=
6v
I
for a l l
v.
The f u n c t i o n ( 3 4 ) i s t h e dundamental & u n c t i o n o f t h e p r o c e s s , and t h e S(X)
A O h t i O M
o f t h e g e n e h U l p k o b l e m (17) LO g i v e n b y m
T h i s c a r d i n a l i n t e r p o l a t i o n f o r m u l a b r i d g e s t h e gap between the linear i n t e r p o l a n t ( 3 ) a n d t h e c a r d i n a l series ( 2 ) . I n f a c t , n o t i c e t h a t i f
m = 1 t h e n ( 3 6 ) r e d u c e s t o ( 3 ) , w h i l e w e have
l i m S2m-1(~)= m+m
(37)
Also every d e r i v a t i v e
sin
TIX
TIx
(k) (x) c o n v e r g e s t o t h e corresponding derivaS2m-l
t i v e of t h e r i g h t s i d e of
(37) I uniformly f o r a l l real
x
I n o u r d i s c u s s i o n w e have assumed t h a t ( 2 3 ) h o l d s .
the
tULatbJMd
. However,
( 3 3 ) , ( 2 9 ) , and ( 3 6 ) a k e v a l i d d o % b o t h T h e o t e m d 1 and
2, undeh t h e i t t e n p e c t i w e a d d u m p t i a n d .
PART 11.
THE CARDINAL SMOOTHING SPLINE
1. STATEMENT OF THE PROBLEM:
We assume now t h a t
(1)
and r e s t r i c t o u r s e l v e s t o r e a l - v a l u e d d a t a and f u n c t i o n s .
We
also
r e c a l l t h e d e f i n i t i o n s ( 3 . 1 4 ) and ( 3 . 1 5 ) o f P a r t I , o f t h e c l a s s e s ly and
L:
.
I n view o f t h e i n c l u s i o n r e l a t i o n s
ON CARDINAL SPLINE SMOOTHING
( S e e [ 9 , p. 1 0 4 1 )
,
399
w e o b s e r v e t h a t (1) i m p l i e s t h a t (y,)
satisfiesthe
a s s u m p t i o n s o f Theorem 2 f o r a l l m .
We a t e g i v e n m and a n m o o t h i n g patrameteh
THE PROBLEM:
E < 0.
Among
aLL 6unctionn
we w i b h t o d i n d t h e . b o l u t i o n
I
m
J(f) =
(4)
E
*
06
t h e phobLem
2
m
+
( f ( m ) (x)) d x
C
-m
-03
-
y V l 2 = minimum.
I n bOlVing t h e minimum p t o b l e m ( 4 ) we may tenLkiot t h e choice
LEMMA 1: 06
(f (v)
adminnible dunctianb
f ( x ) t o t h e eeementb
06
(5)
PROOF:
If
f ( x ) is such t h a t
J(f) <
m
,
then ( f ( v )
a p p l y Theorem 2 t o t h e s e q u e n c e ( f ( v ) ), and l e t
be such t h a t
s(v)
= f
(v) for a l l
v
.
But t h e n
-
yv) E L2
.
We
4w
SCHOENBERG
and so
in view of the extremum property of
S ( x )
as expressed by (3.21)
Theorem 2. Therefore, for any
f ( x ) , the spline
f (x), produces a value
J(f).
Let
U6
J(s)
of
s(x) that interpolates
thehedote d i n d t h e nolutian
o d t h e m i n i m u m ptobLem
I, m
J(S) =
(8)
E
m
(S(m))2dx+ C
-w
(S(v)
-
Y,)~ = minimum.
Here we need another
LEMMA 2:
7 6 ( 7 ) o a t i d d i e n S(x)
E
L2 (R),
(S(m)(x))2dx = -00
PROOF: From (7) we find that
hence aLno (c.) E L2 , t h e n
C yj-” c . c j, v
3
,
ON CARDINAL SPLINE SMOOTHING
401
i s t h e e v e n s e q u e n c e d e f i n e d by
where (y,)
where, t o s i m p l i f y n o t a t i o n s w e dropped t h e s u b s c r i p t 2m o f M 2 m ( ~ ) . I n t e g r a t i o n s by p a r t s show t h a t
(-ilm-' M(2m-1) x)
Observe t h a t
Jm
-m
M i ( X ) M ( ~ ~ -( x~ )- r ) d x
.
i s a s t e p f u n c t i o n assuming i n c o n s e c u t i v e
u n i t i n t e r v a l s t h e v a ue s
... 1 0 ,
(14)
011,
-
(2m-1 1
)
I
(2m-1 2 )
...
I
1 - 1 1
0, 0,
... .
T h i s sequence h a s t h e g e n e r a t i n g f u n c t i o n
except f o r a s h i f t f a c t o r
eiuk which w e d i s r e g a r d . N o w ( 1 3 ) indicates
t h a t ( y ) i s t h e c o n v o l u t i o n o f t h e sequence ( 1 4 ) w i t h t h e sequence
r
However, i n (13) t h e s e q u e n c e Cu avbv-r
.
If
(yr)
appears
as a sum o f
w e pass from ( a v ) t o t h e r e v e r s e d s e q u e n c e
o b t a i n a genuine c o n v o l u t i o n
Cva-vbv-r
.
L e t us t h e r e f o r e
the
form
(a_"),
we
reverse
t h e f i r s t s e q u e n c e (14). As w e o b t a i n t h e g e n e r a i n g f u n c t i o n o f t h e r e v e r s e d sequence by c h a n g i n g
u into
- u i n its o r i g i n a l generating
function, we f i n d the generating function of factor
eiuk) t h e p r o d u c t
(yr
t o b e ( u p t o a shift
402
-ium
= e
S i n c e (y,)
(2 sin
u 2m 7)
Z2,(u).
i s an e v e n s e q u e n c e , i t s g e n e r a t i n g f u n c t i o n m u s t b e e v e n ,
and t h e r e f o r e
e s t a b l i s h i n g (10).
2 . SOLUTION OF THE PROBLEM:
From ( a ) ,
( 9 1 , and ( 7 ) w e f i n d t h a t
L e t us minimize t h i s f u n c t i o n of t h e (c,).
tions, we differentiate
-a 2 ack
J(S) =
E
To o b t a i n t h e normalequa-
J(S) obtaining
Z yj-kcj+
C { Z c . M ( w - j ) - y w ) M ( w - k ) = O (kEZ).
j
v
j
7
I f w e sum w i t h i n t h e double-sum o n l y w i t h r e s p e c t t o
where
v , we obtain
ON CARDINAL SPLINE SMOOTHING
(3)
(au)
403
2
.
(d2m(u))
+
The normal e q u a t i o n s t h u s become
or
(4)
C j
+
(clj-k
E
Y,
yj-k)~j=
M2m(~
-
k)
(k
E
However, by ( 3 ) a n d ( 1 . 1 0 ) w e f i n d
and w r i t i n g
(6)
(c,,)
+
C(u),
(y,)
+
T(u)
,
w e f i n d t h e normal e q u a t i o n s (4) t o b e e q u i v a l e n t t o t h e r e l a t i o n
i ( p ~ , ~ ( u +) )E~( 2 s i n + ) 2 m
whence
This e s t a b l i s h e s
~ , , ( u ) ) c ( u ) = ~ ( u 4) 2 m ( u ) t
if).
SCHOENBERG
404
THEOREM 3:
I n tehmh
06
whehe t h e c a e d d i c i e n t n dicientn (c.) 7
04
t h e expannion
w"(E)
=
w-"(E)
d e c a y e x p o n e n t i a L L y , t h e coed-
t h e naLution
0 6 t h e minimum p h o b l e m ,
ahe
W e c a l l t h e s o l u t i o n ( 9 ) t h e cahdinad smoothing n p l i n e .
3.
A dew p h o p e h t i e b A.
06
t h e cahdinad nmoothing n p d i n e
W e have assumed above t h a t
E
S(X) =s(x;E).
> 0 . However, i f w e s e t
E
=O
i n (2.81, i t becomes
and a comparison w i t h t h e e x p a n s i o n ( 3 . 2 8 ) o f P a r t I , w v ( 0 ) = wv
for all
v : T h i n nhawn t h a t S (x)
inteapolafing caadinal npline B.
What
n t h e eddect
a n t h e ahiginai? dada sequence
(S( v )
,
(y,) ?
06
06
shows
that
S ( x ; O ) = S ( x ) &educed A0 t h e
Theohem
2.
t h e nmoathing n p l i n e
S(x) = S(x;
E)
T h i s w e answer by determining the "smoothed"
t o compare i t w i t h ( y , ) .
By ( 2 . 9 ) and (2.10)we find
406
ON CARDINAL SPLINE SMOOTHING
a n d t h e r e f o r e , by ( 2 . 7 )
I
I n terms o f t h e e x p a n s i o n
1 (2 sin
(3)
=
u 2m
c
eivu
uv(E)
V
Z2,(U)
+
( 2 ) shows t h a t t h e s e q u e n c e ( S ( V ; E ) ) a h i n e n d h o m t h e d a t a (y,)
by t h e
n m o o t h i n g dohmuea
Observe t h a t by ( 2 . 8 ) a n d ( 3 ) t h e c o e f f i c i e n t s i n terms of
W v ( ~ )
by
u"(E)
= C MZm (v
j
-
j)
U ~ ( E )
are e x p r e s s e d
W.(E).
3
Is (4) a s m o o t h i n g f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n o f P a r t I , S e c t i o n l ? T h a t it i s o n e w e see i f w e i n s p e c t i t s c h a r a c t e r i s t i c
function
K(u;E)
(5)
1 u 2m (2 sin T )
=
!d2m(u)
+
f o r it is evident t h a t
0 < K(u;E)
(6)
C. cheabing
< K(O;E)
T h e b m o o t h i n g poweh E
.
06
=
1
for
0 < u < 2r
.
t h e 6ohmuLa ( 4 ) i n c h e a b e n w i t h
I n [ 4 , D e f i n i t i o n 2 , p . 5 3 1 w e g a v e good r e a s o n s
in-
for
406
SCHOENBERG
t h e f o l l o w i n g d e f i n i t i o n : Of two d i f f e r e n t smoothing formulae h a v i n g the characteristic functions
d ( u ) and
$(u)
, we
s a y t h a t t h e second
h a s g r e a t e r smoothing power, p r o v i d e d t h a t
(7)
/J(U)
However, i f
I 5
0 <
E
Id(u)I
<
El
for a l l
u, excluding e q u a l i t y f o r a l l
u.
i t i s c l e a r by ( 5 ) t h a t
and t h e c r i t e r i o n ( 7 ) i s s a t i s f i e d .
D.
The deghee
06
eXUCtneAA
0 6 ,the nmootking 6omonuRa
(4) A = h - l .
T h i s f o l l o w s from ( 1 . 7 ) o f P a r t I , b e c a u s e ( 5 ) shows t h a t w e h a v e t h e e x p a n s i o n i n powers o f
E.
06
u
I f w e d r o p o u r a s s u m p t i o n (1.1) , and assume o n l y t h a t ( y v )
poweh ghowth, Rhen
b y t h e dohmutae(2.8)
,
#in conb&ucfi#n
06 t h e bmvotking
(2.10) , and (2.9) , hemaind appticabLe.0f murse, J ( S ) , of
i t s e a r l i e r connection with t h e funtional holds. I n f a c t we w i l l f i n d t h a t sumably, it i s s t i l l t r u e t h a t o u r t h a t (y,)
npfine S ( x ) = S ( x ; E )
J(S) =
m
( 1 . 8 ) , no l o n g e r
f o r a l l s p l i n e s S . Pre-
S ( X ; E ) minimizes
J ( S ) , provided
s a t i s f i e s the condition
o f Theorem 2 . However, t h i s I was n o t a b l e t o e s t a b l i s h . I n any c a s e I recommend t h e c a r d i n a l smoothing s p l i n e ( S ( X ; E ) ) , which r e p r e s e n t s t h e m o d i f i c a t i o n , found more t h a n 30 y e a r s l a t e r , o f may war-time approach t o t h e problem o f c a r d i n a l smoothing.
407
ON CARDINALSPLINE SMOOTHING
REFERENCES
[ 11
T. N.
E.
GREVILLE, On s t a b i l i t y o f l i n e a r s m o o t h i n g
[ 21
T. N.
E.
GREVILLE, On a p r o b l e m of E .
SIAM J. N u m . A n a l y s i s , 3 ( 1 9 6 6 ) , p p . 1 5 7 - 1 7 0 .
s m o o t h i n g , SIAM J . Math. A n a l . [ 31
,
L.
De Forest i n iterated
5(1974) , pp.
376
FRITZ J O H N , On i n t e g r a t i o n o f p a r a b o l i c e q u a t i o n s by m e t h o d s , Corn. on P u r e a n d Appl. Math.
[ 41
formulas,
I.
J.
,
- 398. difference
5 ( 1 9 5 2 ) ,pp.155-211.
SCHOENBERG, C o n t r i b u t i o n s t o t h e p r o b l e m o f approximation
o f e q u i d i s t a n t d a t a by a n a l y t i c f u n c t i o n s , Q u a r t . o f Appl. Math., [ 51
4 ( 1 9 4 6 ) , P a r t A, p p . 45 - 9 9 ,
P a r t B , pp. 1 1 2 - 1 4 1 .
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