SlXn REMAINDERS: FUNCTIONS OF SEVERAL VARIABLES,

REMAINDERS: FUNCTIONS OF SEVERAL VARIABLES,

B y A R T H U R S A R D 1

of FLUSHING, INT. Y .

1. I n t r o d u c t i o n . M a n y of t h e r e m a i n d e r s R ( x ) ~ R x in t h e t h e o r y of ap- p r o x i m a t i o n s r e l a t e d t o a f u n c t i o n x ~ x(s), 0 ~< s <_ l, are f u n c t i o n a l s which are linear on C n or Cn_ 1 a n d zero w h e n e v e r x is a p o l y n o m i a l in s of degree n - - 1 . (Defi- nitions are given in sections 2,3 below.) A c c o r d i n g l y t h e following k n o w n t h e o r e m s [5; 2 ; 7 ; l l ] are of i m p o r t a n c e .

M a s s t h e o r e m . S u p p o s e that R x is a f u n c t i o n a l linear on C n a n d zero f o r degree n - - 1 . T h e n there is a f u n c t i o n fl(s) c o]2 such that

(. ,1

R x = ..toX"(s)dfl(s) , x e C~ ^I

H e r e xn(s ) s t a n d s for t h e n t h d e r i v a t i v e of x. A fuller s t a t e m e n t is t h e o r e m (4:3) below.

K e r n e l t h e o r e m . S u p p o s e that R x is a f u n c t i o n a l linear ^{o n} Cn_l, n > 1, a n d zero f o r degree n - - 1 . T h e n there is a f u n c t i o n f ( s ) ~ o)) such that

SlXn

R x = (s)f(s)ds , x ~ C~ . A fuller s t a t e m e n t is t h e o r e m (4:15).

T h e p r e s e n t p a p e r e x t e n d s t h e a b o v e t h e o r e m s to f u n c t i o n a l s on spaces of f u n c t i o n s of several variables. T h e o r e m s (5:11), (6:11), (9:5), a n d ( 9 : l l ) a f f o r d d i r e c t access to i n t e g r a l f o r m s of r e m a i n d e r s in t e r m s of p a r t i a l d e r i v a t i v e s of o r d e r n, and, f u r t h e r m o r e , c o m p l e t e l y c h a r a c t e r i z e t h e cases in which t h e i n t e g r a l f o r m s

1 T h e a u t h o r g r a t e f u l l y a c k n o w l e d g e s f i n a n c i a l s u p p o r t received f r o m t h e Office of N a v a l R e s e a r c h , u n d e r c o n t r a c t w i t h Q u e e n s College, F l u s h i n g , N. Y.

are valid. T h e integral f o r m s lead, a m o n g o t h e r things, t o appraisals of r e m a i n d e r s a n d to criteria of best a p p r o x i m a t i o n (section 8).

T h e c h a r a c t e r of o u r results is i n d i c a t e d b y t h e following t h e o r e m , which is a c o n s e q u e n c e of t h e o r e m (6:11) a n d l e m m a (6:3). T h e space Bp, q of f u n c t i o n s x = x(s, t) is defined in section 6.

Kernel t h e o r e m . Let p, q be positive integers and a, b fixed numbers, 0 ~ a, b ~ 1. Put n = p + q . Suppose that R x is a functional linear on Bp_l,q_ 1 and zero whenever x is a polynomial in (s, t) of degree n - - 1 . Then there are functions: fi(t), i < p; g(s, t); hi(s), j < q, all in o f , such that

61 ~.I 1

R x =--- --~" foxi. n_i(a, t)fi(t)dt~-I I,, xp'q(s' t)g(s, t)dsdt

i ~ p o

,1

4- . ~ ~foxn-j'j(s' b)hJ(s)ds , x E Bp, q .

) < q

T h e t e r m s in this e q u a t i o n c a n n o t be c o m b i n e d since t h e r e are f u n c t i o n s x ~ Bp, q for which one a n d only one of t h e n ~ - I t e r m s will be d i f f e r e n t f r o m zero.

An i l l u s t r a t i o n is given in section 7.

T h r o u g h o u t we shall consider spaces of f u n c t i o n s defined on t h e u n i t h y p e r - s q u a r e U " : t h e set of points in euclidean m-space w i t h c o o r d i n a t e s all b e t w e e n 0 a n d l, end values included. Our t h e o r e m s t r a n s f o r m in t h e n a t u r a l w a y to spaces of f u n c t i o n s defined on a h y p e r - r e c t a n g l e with sides parallel to t h e axes. T h e f a c t t h a t o u r t h e o r e m s refer to f u n c t i o n s d e f i n e d on U m r a t h e r t h a n on a general c o m p a c t m e t r i c space leads to t h e following a d v a n t a g e : t h e relations defining t h e kernels a n d t h e masses are simpler t h a n t h e y otherwise would be [6; 10].

T h e f u n c t i o n a l s R x to which our t h e o r e m s a p p l y n e e d n o t i n v o l v e all of t h e u n i t s q u a r e Urn; for e x a m p l e , R x m i g h t be d e f i n e d on a space of f u n c t i o n s x d e f i n e d on a subset of U m, for t h e n R x is defined, a f o r t i o r i , on a space of f u n c t i o n s defined o n U m.

T h e spaces B(p) a n d A(p) are i n t e r e s t i n g in t h e m s e l v e s , as t h e c o m p l e t e core of a f u n c t i o n in e i t h e r space is a u n i q u e c h a r a c t e r i z a t i o n of t h e f u n c t i o n i n t e r m s of i n d e p e n d e n t elements.

2. Linear f u n c t i o n a l s . R i e s z ' s t h e o r e m . A functional Rx, defined on a space X of e l e m e n t s x, is a c o r r e s p o n d e n c e f r o m X t o t h e real n u m b e r s . Suppose t h a t X

R e m a i n d e r s : F u n c t i o n s o f S e v e r a l V a r i a b l e s 9 3 2 1

is a n o r m e d linear space. T h e f u n c t i o n a l R x is said t o b e l i n e a r if it is a d d i t i v e : R ( x ~ - y ) -~ R x - ~ R y , x, y ~ X ,

a n d c o n t i n u o u s a t one p o i n t , s a y t h e o r i g i n :

R x - + O w h e n e v e r [ l x l [ - > 0 , x E X .

O b s e r v e t h a t t h e a d d i t i v i t y implies t h a t R 0 ---- 0. I n e a c h p a r t i c u l a r s p a c e X t h a t we consider, we define t h e n o r m ]lxl[ explicitly. A l i n e a r f u n c t i o n a l R x is c o n t i n o u s a t e v e r y p o i n t , a n d h o m o g e n e o u s :

R ( c x ) -~ c R x , c a r e a l n u m b e r , x ~ X .

T h e a b o v e u s a g e of t h e w o r d " l i i i e a r " is t h a t of F. R i e s z a n d B a n a c h . [1, pp. 23, 26-27, 36-37].

All spaces X t h a t we consider a r e spaces of f u n c t i o n s of real v a r i a b l e s ; a d d i t i o n a n d s c a l a r m u l t i p l i c a t i o n of e l e m e n t s of t h e s p a c e a r e t o be u n d e r s t o o d as a d d i t i o n a n d s c a l a r m u l t i p l i c a t i o n of t h e f u n c t i o n s .

T h e s p a c e C~ is t h e s p a c e of f u n c t i o n s x = x ( s l . . . sin) c o n t i n u o u s o n t h e u n i t s q u a r e U m, w i t h n o r m

[Ixl] = Ilxllc~' = m a x ]x(s 1 .. . . . sin) I .

(8) ~ U m

A f u n c t i o n y ~- y ( s l , . . . , sin) is of bounded v a r i a t i o n on U m if it h a s t h e following p r o p e r t i e s : i) F o r all s u b d i v i s i o n s of U ~ i n t o a finite n u m b e r of r e c t a n g l e s w i t h sides p a r a l l e l t o t h e axes, ~ ' ] A 1 . . . A ~ [ is b o u n d e d , ii) F o r one p a r t i c u l a r v a l u e s o of e a c h a r g u m e n t sv, ~ is a f u n c t i o n of b o u n d e d v a r i a t i o n in t h e r e m a i n i n g m - - 1 v a r i a b l e s . Conditions i) a n d ii) i m p l y t h a t for a n y f i x e d v a l u e s o of s v, v ---- 1 . . . m,

is a f u n c t i o n of b o u n d e d v a r i a t i o n in its r e m a i n i n g m - - 1 v a r i a b l e s . W e use t h e s y m b o l c/) t o d e n o t e t h e class of f u n c t i o n s of b o u n d e d v a r i a t i o n in t h e i r v a r i a b l e s , w i t h o u t r e g a r d to t h e n u m b e r of v a r i a b l e s .

(2:1) R i e s z ' 8 t h e o r e m . S u p p o s e that R x i s a f u n c t i o n a l l i n e a r o n C~. P u t

9 t !

(2: 2) r ( s l , . 9 s ~ ) ~ = , ,

ROsi(sl)...08m(Sm) i f S l . . . S r ~ > 0 , where

(2: 3) 08, =

O,,(s) = !

^{1 if 8 =<}

8 t

! 0 if s > s ' .

21 -- 642138 A c t a m a t h e m a t i c a . 84

T h e n 7 9 ~fi, a n d

f I

(2:4) R x . . . x ( s l , . . . , s,~)dT(s 1 . . . . ,Sm), X 9 C ' ~ .

0 0

C o n v e r s e l y , i f 7(s, . . . . , s i n ) 9 (2:4) d e f i n e s a f u n c t i o n a l w h i c h is l i n e a r o n C ~ .

T h e definition (2: 2) involves a n e x t e n s i o n of t h e d e f i n i t i o n of R, since t h e a r g u m e n t of R is n o t a c o n t i n u o u s f u n c t i o n . Such a n e x t e n s i o n can be m a d e in m a n y w a y s ; one such way, which is d i r e c t a n d c o n s i s t e n t w i t h t h e e x t e n s i o n of

(2:4) as a L e b e s g u e - S t i e l t j e s integral, is t h e following. L e t

(2:5)

O~,,~(s) , k = l , 2 . . . .

be, for each s', a sequence of c o n t i n u o u s f u n c t i o n s which c o n v e r g e m o n o t o n e l y to 0~,(s). T h e n t h e sequence RO~,~, ~ 1 ( s l ) . . . 0/~, k,,(sm) converges as ( / c l , . . . , k~) -+

! t

(cx~ .. . . , ~ ) ; its limit is t a k e n as t h e d e f i n i t i o n of 7 for s l . . . s m > 0. W i t h this definition 7 will be c o n t i n u o u s f r o m a b o v e for positive a r g u m e n t s :

(2:6) 7 ( s l q - O . . . Sm~-O) : 7 ( s l , . . ., sin), s 1 . . .sm > 0 .

T h e following conditions d e t e r m i n e 7 in (2 : 4) u n i q u e l y : i) ~, 9 ~ l / a n d vanishes when- e v e r one of its a r g u m e n t s vanishes, ii) (2:4) a n d (2:6) hold. [8; 9; 4, pp. 262-271;

1, p. 61; 3].

3. C o n v e n t i o n s a n d f u r t h e r n o t a t i o n . W e say t h a t f u n c t i o n s a ( s 1 . . . s.~) a n d f l ( s l , . . . , sin) are e q u a l w i t h c o u n t a b l e e x c e p t i o n s , a n d we w r i t e t h i s :

06 ~ - f l , w . c . e . ,

if ~ a n d / ~ are e q u a l e x c e p t w h e n s, is one of a c o u n t a b l e set of values, v ---- 1, 2 . . . or m . (Thus ~ a n d fl are equal e x c e p t on a c o u n t a b l e n u m b e r of ( m - - 1 ) - p l a n e s p e r p e n d i c u l a r to t h e axes.) " C o u n t a b l e " is t o be u n d e r s t o o d as " c o u n t a b l y infinite, finite, or z e r o " .

E q u a t i o n s m a r k e d *, such as (4 : 8), are to be u n d e r s t o o d as follows. T h e f u n c t i o n or fl defined in t h e e q u a t i o n is zero w h e n e v e r a n y one of its a r g u m e n t s is zero;

t h e f u n c t i o n is as w r i t t e n w h e n e v e r all its a r g u m e n t s are positive.

W e s a y t h a t a f u n c t i o n a l R x is zero f o r degree n - - 1 if R x = 0 w h e n e v e r x is a p o l y n o m i a l of degree n - - 1 in all its variables.

E x p o n e n t s will fl iicate, n o t powers, b u t powers d i v i d e d b y t h e factorial of

Remainders: Functions of Several Variables. 323 t h e e x p o n e n t :

s ~ 1, s 1 - = s , s ~ = s ' s / 2 . . . . , s ~ - ~ s m - l " s / m . . .

I n d i c e s are n o n - n e g a t i v e integers, e x c e p t w h e r e o t h e r w i s e i n d i c a t e d .

4. F u n c t i o n s of o n e v a r i a b l e . Let C~ b e t h e s p a c e of f u n c t i o n s x = x ( s )

w i t h c o n t i n u o u s n t h d e r i v a t i v e , s e U 1, t h e n o r m b e i n g

Itxll

= Ilxlle~ = m a x Ixi(s)].

s ~ U 1 i = 0 , . . . , n

F o r c o n s i s t e n c y of n o t a t i o n w i t h l a t e r sections, we define t h e spaces B n, A n as follows :

B n = C n ; A n = B n _ 1 - - - C n _ 1, n > 1 . L e t a b e a f i x e d n u m b e r 0 < a __< 1.

(4:1) T a y l o r ' s f o r m u l a o n B n. I f x e B n, n > 1, then

8

x(s) = ~ ( s - - a ) i x i ( a ) + ~ ( 8 - - 8 ) n - l x n ( 8 ) d ~ , 8 e U 1 9

i < n r

More g e n e r a l l y (4: 1) holds if x h a s a n a b s o l u t e l y c o n t i n u o u s ( n - - 1 ) t h d e r i v a t i v e . T h e core of a f u n c t i o n x in Bn is t h e n t h d e r i v a t i v e x~(s); t h e complete core is t h e core t o g e t h e r w i t h t h e n n u m b e r s xi(a ), i < n. T a y l o r ' s f o r m u l a (4: 1) m a y be u s e d to e x p r e s s all t h e d e r i v a t i v e s of x of o r d e r less t h a n n in t e r m s of t h e c o m p l e t e core of x. Hence [[X][Bn a n d t h e m a x i m u m of t h e a b s o l u t e v a l u e s of t h e e l e m e n t s in t h e c o m p l e t e core of x a r e e q u i v a l e n t n o r m s in t h e sense t h a t e a c h is a t m o s t a c o n s t a n t t i m e s t h e o t h e r .

T h e c o m p l e t e core of x e B~ m a y be t h o u g h t of as t h e i n d e p e n d e n t p a r t of x a n d as a n i n d e p e n d e n t v a r i a b l e .

(4:2) L e m m a . B n c A n = B~_I, n > I . IlXilB~ > ilxtlA~, x e B n . A f u n c t i o n a l linear on A N is a fortiori linear on B,,.

(4 : 3) M a s s t h e o r e m o n B,,. S u p p o s e that R x is a f u n c t i o n a l linear on B n a n d zero f o r degree n - - 1 . T h e n there is a f u n c t i o n fl(s) ~ Q]) such that

1

(4: 4) R x ~- IoXn(s)dfl(s) , x e B n .

2 1 " - 6 4 2 1 3 8 A c t a mathematica. 84

Conversely, given a f u n c t i o n fl ~ 9 , (4: 4) defines a f u n c t i o n a l which is linear on B,~ a n d zero f o r degree n - - 1 .

T h e mass fl m a y be defined as follows:

(4:5) fl(s') = R (s--~)~-lOo,(~)d~, n _>- 1 , * ,

a

(4:6) fl(s') = ROs,(s ) , n = O, *

The asterisks * indicate t h a t the formulas (4: 5), (4: 6) a p p l y for s' > 0 a n d t h a t fl(0) --~ 0. W e call fl a mass even t h o u g h fl m a y be decreasing.

P r o o f . The t h e o r e m is a corollary to Riesz's t h e o r e m (2:1) for functions of one variable. F o r n ---- 0, t h e t h e o r e m is Riesz's t h e o r e m itself. F o r n ~ 1, Taylor's f o r m u l a (4: 1) implies t h a t

( 4 : 7 ) R x = R I : ( s - - 8 ) n - l x n ( 8 ) d ~ , z E B n ,

since R x is zero for degree n - - 1 . N o w t h e second m e m b e r of (4:7) is defined a n d a d d i t i v e for x n e Co 1. F u r t h e r m o r e I[xn]Iv~ -+ 0 implies t h a t

is (8--8)n-lxn(8)d'8 Bn --> 0 a

a n d hence t h a t R x --+ O. Hence t h e second m e m b e r of (4: 7) is a linear f u n c t i o n a l on C~ for x~ e Co 1. Hence b y Riesz's t h e o r e m (4:4) a n d (4:5) hold.

As in Riesz's t h e o r e m t h e relations (4:5), (4:6) involve a n extension of R x o n t o a space t h a t includes the a r g u m e n t of R in (4: 5), (4: 6). The extension m a y be m a d e in m a n y w a y s ; for preciseness we u n d e r s t a n d (4:5) as a definition b y mono- t o n e limits :

(4: 8) fl(s') = lim R (S--'~)n-~08,, k('~)cl~ , * ,

k .---~ eQ Ca

where 08, ,k(s) is a sequence (2:5). The limit (4:8) exists a n d fl(s'+O) = fl(s'), s ' > O.

The relation (4: 6) is u n d e r s t o o d similarly. This completes t h e proof, as t h e converse p a r t is i m m e d i a t e .

I f t h e f u n e t i o n a l R x is linear on B~_~, n ~ 1, (henee on Bn), t h e relation (4:5) m a y be used w i t h o u t (4:8), since t h e a r g u m e n t of R in (4:5) is an element of Bn_ P The relations (4: 5) a n d (4: 8) will t h e n be eonsistent, since

I8 I8 Bn-1

(4:9) lim (s--~)n-lo,,(~)d~ - (s--i)n-~os,,k(~)d~ = 0 .

k - - > oo a a

Remainders: Functions of Several Variables. 325 F u r t h e r m o r e (4: 5) implies t h a t

(4: 10) fl(s') = R ( s - - s ' ) ' % , ( s ) .

For, a n asterisk is n o t needed a f t e r (4: 10), since t h e a r g u m e n t of R in (4: 10), when s' ~- 0, is shOo(s) -~ 0 for all s > 0. Also,

S

(4: 11) (s--~)n-lOs,(g)d~ = (s--s')~Os,(s)-{-[(s--a)'~--(s--s')'~]O.,(a), n > 1 , a

8 8 8 t

a s m a y b e s e e n bY w r i t i n g I = f ~ - i ~ S i n c e t h e l a s t t e r m of (4: 1 l) is a p o l y - a 8" a

nomial in s of degree n - - 1 , (4: 5) does indeed i m p l y (4: 10).

(4: 12) A n e x t e n s i o n of R . Once t h e f u n c t i o n 8 has been defined, we m a y e x t e n d the R of t h e o r e m (4::3) as follows. P u t

(4: 13) R + x -~ t Xn(S)dS(s ) , x e B + ,

J u 1

where Bn + is t h e space of functions x with n t h derivative Lebesgue-Stieltjes inte- grable relative to 8 and, if n > 1, w i t h absolute b y continuous ( n - - 1 ) t h derivative.

Then B n c B + a n d R x ~ R + x for x e B~. F u r t h e r m o r e

S

(4: 14) 8(s') ~-- R + (8~-8)n-lOs,(i)di , n > 1, w . c . e . , * a

For, t h e a r g u m e n t of R + in (4: 14) has t h e n t h derivative 08,(s ) for s # s' a n d there- fore is a n element of B + except w h e n s ~- s' is a d i s c o n t i n u i t y of 8(s). Hence b y 13), t h e second m e m b e r of (4:14) equals I 08,(s)dS(s) = 8(s') whenever s' is a (4:

d U 1

c o n t i n u i t y of 8, t h a t is, w i t h only countable exceptions. B y (4: 11), (4: 14) m a y be w r i t t e n

8 ( s ' ) = R + ( s - - s ' ) ~ O A s ) , w . c. e . , *

(Here t h e case n ---- 0 is valid also).

(4:15) K e r n e l t h e o r e m , S u p p o s e that R x is a f u n c t i o n a l l i n e a r o n A,~ ~ B n _ 1, n > 1, a n d zero f o r degree n - - 1. T h e n there is a f u n c t i o n f ( s ) e G]) s u c h that

(4: 16) R x : I Xn(S)f(s)ds , x e B ++,

U 1

w h e r e B ++ i s the s p a c e o f f u n c t i o n s x w i t h a b s o l u t e l y c o n t i n u o u s ( n - - 1 ) t h derivative.

Conversely, given a f u n c t i o n f(s) e ~ (4: 16) defines a f u n c t i o n a l which can be extended so as to be linear on A n a n d zero for degree n - - 1 .

T h e kernel f(s) m a y be defined as follows:

(4: 17) f(~) = R+(s--~)n-l~a,~(~), w . c . e . , * ,

where

1 if a < = ~ < s , (4:18) Wa,~(~) ---- O~(a)--O~(s) ---- --1 if s __< g < a ,

0 otherwise ;

O~ is defined i n (2:3), a n d R + is the extension (4: 12) with n replaced by n - - 1 . Observe t h a t Bn c B ++ c An; so t h a t , in particular, (4: 16) holds f o r x e B n.

A c t u a l l y B~ ++ is n o w t h e same space as B + defined in (4: 12), b u t this f a c t is n o t used in t h e proof or t h e application of t h e present theorem.

The relation (4: 17) m a y be w r i t t e n in t h e following a l t e r n a t i v e forms, often useful for calculation:

(4: 19) f(~) -~ --R+(s--~)n-lO~(s) --- R+(s--~)n-x[1--O~(s)] , w. c. e., * , since R + x is zero for degree n - - 1 .

I f R x is linear on An_l, n ~ 2, t h e relations (4:17 a n d 19) hold with R + replaced b y R a n d w i t h w. c. e. deleted.

P r o o f : One proof is to apply t h e mass t h e o r e m (4:3) on Bn_ 1 a n d t h e r e a f t e r integrate b y p a r t s [7; 11]. Here we follow P e a n o ' s original suggestion [5] as this leads to t h e simplest proof in t h e case of several variables.

B y (4: 18) Taylor's f o r m u l a (4: 1) m a y be w r i t t e n

x(s) : ~ (s--a)ixi(a) + I (8--8)n-l~l)a,s(8)Xn(8)ds , X e B ++ .

i < n r U 1

Hence

(4: 20) R x = R I u I d 8 ( 8 - - 8 ) n l~a,s(8)Xn(8) . . . . . ' x e B ++ ⁰ N o w

1

(4: 21) R x --- fodfl(S)Xn_l(s) , x e Bn_ 1 ,

where fl ~c/), b y the mass t h e o r e m (4:3) w i t h n replaced b y n - - 1 . A p p l y (4:21) to (4: 20). T h e n

1 dsy)a,s(8)Xn(8 ) = Iuld~xn(~)i~ldfl(s)wa,,(~), x e B~ ++ , R x = fodfl(s) i w

Remainders: Functions of Several Variables. 327 b y F u b i n i ' s t h e o r e m since Xn(8)~)a,s(8 ) i s i n t e g r a b l e d~d~(s). H e n c e

Irjld~xn ++

R x = (~)f(~) , x e B , , where

(4:22) f(s) : f l uI[O~(a) - - O~(8)]dfl(8) = - - i U 1 O~(s)dfl(s)= - f l ( ~ + O ) , 1

since I dfl(s) ~ 0 b y (4:21) w i t h x e q u a l to t h e p o l y n o m i a l 8 n-1. Since f l e c]~,

f ~ p

d 0

also. This p r o v e s t h e first p a r t of t h e t h e o r e m .

L e t R + be t h e e x t e n s i o n (4: 12) of R w i t h n r e p l a c e d b y n - - l :

I U l x n - 1

R+x ---- (s)dfl(s) , x e B~_ l + .

N o w ~0~,,(~) as a f u n c t i o n of s is c o n s t a n t e x c e p t w h e n s = ~ , b y (4:18). H e n c e ( 8 - - 8 ) n - - l ~ a 8(8) , ~ Bn_ 1 + e x c e p t w h e n ~ is a d i s c o n t i n u i t y of ft. H e n c e

R+(8--8)n-ly~a,s(~) = I ~)a,*(k)dfl(s) -~ f(~) , w. c. e. , ,1 U 1

b y (4:22). This establishes (4:17),

To p r o v e t h e converse, suppose t h a t f ( s ) e ~1;, Define t h e f u n c t i o n a l R as R x = f(s)dx,~_~(s) -~- f(1)x~_~(1)--f(O)Xn_~(O ) + (s)df(8) .

U z

This f u n c t i o n a l is linear on An, reduces to (4: 16) for x e B~ , a n d vanishes for ++

degree n - - 1.

This completes t h e p r o o f of t h e t h e o r e m .

(4 : 23) R e m a r k . Suppose t h a t a f u n c t i o n a l R * x is linear on B n or A m b u t n o t necessarily zero for degree n - - 1 . One can c o n s t r u c t a r e l a t e d f u n c t i o n a l R x which will be zero for degree n - - 1 as follows:

R x = R * x - - ~ ~ c~xi(a ) , i < n

where c~ = R * ( s - - a ) i, i < n. T h a t R x is zero for degree n - - 1 follows f r o m t h e f a c t t h a t x(s) = ~ (s--a)~xi(a) w h e n e v e r x is a p o l y n o m i a l of degree n - - 1 . N o w R x is

i ~ n

linear on B~ or A~ w h e n e v e r R * x is. T h u s t h e mass or kernel t h e o r e m s m a y be

a p p l i e d t o R x t o give a f o r m for R ' x :

R * x : R x + ~ cixi(a ) .

i<~ n

5. F u n c t i o n s o f t w o v a r i a b l e s . T h e s p a c e Bp, q. L e t a, b be f i x e d n u m b e r s , 0 _< a, b ~ 1 ; a n d let 19, q b e f i x e d n o n - n e g a t i v e integers, P u t

n ~ p + q .

W e define t h e space Bp, q as t h e space of f u n c t i o n s x ~ x(s, t) whose d e r i v a t i v e s

(5:1)

xi, n_i(a,t), i < 1 9 ,

% ,

q(s, t),

x._j,j(s, b) , j < q ,

exist a n d are c o n t i n u o u s in t, (s, t), s, respectively, (s, t) 9 U s. H e r e x~,j s t a n d s for t h e p a r t i a l d e r i v a t i v e Oi+Jx/OsiOt j a c c o r d i n g t o t h e following c o n v e n t i o n as t o t h e o r d e r of d i f f e r e n t i a t i o n :

(5 : 2) Xi,j ~--

Oi-Pxp,j/O8 ~-p if i > p, j ~ q , OJ-qx i q/otJ-q if i g p, j > q , 0 *+3"

;-nXp,

q/Osi-POt j-q if i > p, j > q

t h e o r d e r of d i f f e r e n t i a t i o n being otherwise u n r e s t r i c t e d . T h u s in t h e last case of (5: 2), t h e d i f f e r e n t i a t i o n s in xp, q m a y be in a n y order, a n d t h e last i ~ - j - - n d i f f e r e n t i a t i o n s m a y be in a n y order. Also, t h e o r d e r in x~,j is u n r e s t r i c t e d if i ~ p, j ~ q. W e shall describe (5:2) as t h e c o n v e n t i o n of Bp, q.

T h e core of a f u n c t i o n x in Bp, q is t h e set of d e r i v a t i v e s (5: 1); t h e complete core of x is t h e core t o g e t h e r w i t h t h e n u m b e r s x~,j(a, b), i ~ - j < n. T h u s t h e c o m p l e t e core consists of 1 f u n c t i o n of t w o v a r i a b l e s ; n f u n c t i o n s of one variable, a n d n ( n + 1)/2 c o n s t a n t s . I n o r d e r f o r t h e core of x 9 Bp, q t o exist c e r t a i n d e r i v a t i v e s of x of lower o r d e r m u s t exist a n d be c o n t i n u o u s ; we say t h a t t h e l a t t e r d e r i v a t i v e s are covered b y t h e core.

T h u s f o r Bp, q, lowing:

(5: 3)

t h e d e r i v a t i v e s in t h e core or c o v e r e d b y t h e core are t h e fol- x~,j(s, t) , i ~ p , j ~ q ;

xi, j(a, t), i + j ~ n , j > q ; xij(s, b), i + j ~ n , i > p .

Remainders: Functions of Several Variables. 329 The

norm

of x in

Bp, q,

d e n o t e d b y IlXllBp, q, i s defined as the m a x i m u m of the absolute values of the derivatives (5:3), (s, t) e U ~.

(5:4) T a y l o r ' s f o r m u l a o n

Bp, q. I f x e Bp, q,

t

x(s, t) = ~ (s--a)i(t--b)Jxi,j(a, b ) ~ - ~ (s--a) i I (t--t)n-i-lxi'n-i(a' t)dt

i + j < n i < p b

8

~- ~ (t--b)J I (8--8)n-j-lxn-j'j(8' b)d~+T, (s, t) E U ~ ,

j < q a

where

s t

l ( 8 - - 8 ) p - 1 d 8 I ( t - - t ) q - l x p q(8, t ) d t ,

a b '

i"

T = (s,~)P-lXp, o(~, t)d~ ,

~a

ftb(t--t)q-lXo, q(s, t)dt , Xo, o(S, t),

p > : l , q _ > _ l ; p _ _ _ l , q = 0 ; p----0, q ~ : l ;

p : - q : - O .

P r o o f . Suppose t h a t p ~ 1, q >: 1;

x E B p , q.

B y Taylor's formula (4:1),

t

(5: 5)

x(s, t) = ~ (t--b)Jxo, j(s, b)+ fb(t--t)q-lXo, q(S, t)dt,

j < q

l"

(5:6)

Xo,j(s, b) = ~ (s--a)ixi j(a, b)+

( 8 - - 8 ) n - j - - l X n

j j(i, b)d~ , j < q,

~<n--j ' va -- '

(5: 7)

xo, q(s, ~) = ~ ' (s--a)%i,q(a, ~)+ ~)di .

i < p

W e will s u b s t i t u t e (5:6), (5:7) in (5:5). This will lead a m o n g other things to terms involving

t

I (t--t)q-lxi, q(a, t)dt, i < p .

b

N o w this q u a n t i t y vanishes at t = b, t o g e t h e r with its derivatives as to t of order less t h a n q. I t s derivative of order

j, j >=_ q,

is

xi,j(a, t).

Accordingly, b y Taylor's formula (4:1),

t j t

(5: 8)

I (t--t)q-lx~, q(a, t)dt = 2 " (t--b)Jxi,j(a, b) ~- Ib (t--t)n-i-lXi, n_i(a, t)dt, i < p .

b q<=j<n-i

W e now obtain (5 : 4) b y s u b s t i t u t i n g (5 : 6), (5 : 7) in (5 : 5) and using (5 : 8).

The other cases of (5:4) are established similarly and more simply.

T a y l o r ' s f o r m u l a (5 : 4) expresses x in t e r m s of its c o m p l e t e core. Applied to t h e d e r i v a t i v e s c o v e r e d b y t h e core, (5:4) a n d (4: l) express these d e r i v a t i v e s also in t e r m s of t h e c o m p l e t e core. H e n c e I[XlIBp, q a n d t h e m a x i m u m of t h e a b s o l u t e values of t h e e l e m e n t s in t h e c o m p l e t e core of x are e q u i v a l e n t n o r m s in t h e sense t h a t each is a t m o s t a c o n s t a n t times t h e other.

F o r each x e Bp, q t h e r e is a c o m p l e t e core. Conversely, g i v e n n ( n ~ - 1)/2 c o n s t a n t s : c i'j, i ~ - j < n; p c o n t i n u o u s f u n c t i o n s o f t: z i' n-i(t), i < p ; q c o n t i n u o u s f u n c t i o n s of s:

zn-J'J(s), j < q; a n d 1 c o n t i n u o u s f u n c t i o n of (s, t): z p' q(s, t); there is a u n i q u e f u n c t i o n x ~ Bp, q h a v i n g these e l e m e n t s as its c o m p l e t e core:

xl,j(a, b) -~ c i'j , i - ~ j < n , Xi, n_i( a,t) = z i'n-i(t) , i < p , (5: 9)

%,q(s,

t) = zP' q(s, t) ,

x,~_j,j(s, b) = z ~ J ' J ( s ) , j < q .

This m a y be established as follows. P u t t h e given elements into t h e second m e m b e r of (5:4), using (5:9). T h e second m e m b e r of (5:4) t h e n defines t h e f u n c t i o n x. T h e c o n v e n t i o n as t o t h e o r d e r of d i f f e r e n t i a t i o n e n t e r s here. T o i l l u s t r a t e t h e point, consider a t y p i c a l t e r m :

.t

y -= y ( s , t) = ( s - - a ) p-I I (t--t)qzP-1'q+l(t) d~ , p ~ 2, q ~ 0 ^m

.) b

T h e n Yo~ q+l( s, t) ~ ( s - - a ) P - l z p-l'q+i(t). Since z p-l'q+l is m e r e l y c o n t i n u o u s no f u r t h e r d i f f e r e n t i a t i o n s as to t are necessarily possible here. H o w e v e r yp_2,q+~(a, t), for e x a m p l e , exists. F o r

yp_~,q+~(a, t) ~-- ~ y p _ 2 , q ( a , t) - - ~ ( s - - a ) zP-l'q+l(t)dt : 0 .

b ~ s = a

T h e r e a d e r m a y v e r i f y t h a t y e Bp, q a n d t h a t t h e c o m p l e t e core of y vanishes, e x c e p t for yp_l,q+l(a, t) which equals zP-~'q+l(t).

An essential p o i n t here is t h a t each i n t e g r a l in (5:4) is differentiable at least p times as to s a n d q t i m e s as to t.

T h u s t h e c o m p l e t e core of x e Bp, q m a y be t h o u g h t of as t h e i n d e p e n d e n t p a r t of x a n d as an i n d e p e n d e n t variable.

(5:10) L e m m a . Bp, q c Bp_l,q, p ~ 1 .

llxllB~,~ >= IIx[IB~_~,q ,

x e Bp, q .

R e m a i n d e r s : F u n c t i o n s of S e v e r a l V a r i a b l e s . 331

A functional linear on Bp_s is afortiori linear on Bp, q.

Likewise for Bp, q-l, q ~ 1.

T h e c o n v e n t i o n s as to d i f f e r e n t i a t i o n in

Bp, q

a n d

Bp_l, q

are c o n s i s t e n t with one a n o t h e r ; b o t h arc to be followed a t t h e same t i m e w h e r e necessary.

(5: 11) M a s s t h e o r e m o n Bp, q.

Suppose that R x is a functional linear on Bp, q and zero for degree n--1. Then there are functions fli,] E '~]) such that

(5: 12)

R x =.~__~ SlXl, n~i(a, t)dfli'n~i(t)-~- ll SloXp, q(s, t)dfl~',(s, t)

1

+- .--'~ ~loxn-j'J(S' b)dfl'~-J'J(s) , x Bp, q .

J < q

Conversely, given masses fli, j e cl? '

( 5 : 1 2 ) d e f i n e s

a functional which is linear on Bp, q and zero for degree n--1.

The masses fli, j may be defined as follows:

t

fl~'~--i(t') = R(s--a)~l (t--t)n-~-lOt,(t)d-t, i < p , *

b

i"

fln-J,J(s') = R(t--b)J (s--i)'~-J-lO~,(i)di, j < q, *,

a

(5:13)

R I

(s-~)'-%,(~)d~I (t--~),-%(~)d~, p > 1, q > 1, *,

Ca db

ROt,(t ( 8 - - 8 ) p - 1 0 s , ( 8 ) d s , p ~ 1, q = O , *

[3p' q(s', t') = o

t

RO~,(s) \ (t--t)q-lOt,(t)dt , p = O, q ~ 1,

b

ROs,(s)Ot,(t ) , p = q = O, *

T h e asterisks i n d i c a t e t h a t t h e relations (5: 13) hold for positive a r g u m e n t s only, a n d t h a t each mass fli' j is zero if one of its a r g u m e n t s is zero. A t h e o r e m of C. A.

Fischers's [2] is r e l a t e d to t h e p r e s e n t t h e o r e m for p = q = 1 a n d R e m a r k (6: 18) below.

P r o o f . T h e proof is similar t o t h a t of t h e mass t h e o r e m on

B n,

Consider t h e case p >__ 1, q > 1. B y T a y l o r ' s f o r m u l a (5:4),

f: (t--t)n-i-lx i n~i(a, I: !: (t--t)q-lxp, ,(~,

R x = ~ , R(s--a) i t)dt-b R (s--8)p-ld8 t)dt

(5: 14)

_~_ /x, R ( t _ b ) j f 8 (s__~)n

J--lXn_j ' j(~,

b)dg, x ~ Bp, q.

2"<q a

Since the core of x ~ Bp, q m a y be a n y continuous functions, the t e r m s of (5 : 14) are defined a n d additive on t h e spaces of continuous functions of t, (s, t), a n d s, respec- tively, t h e i n d e p e n d e n t variables being the core of x. F u r t h e r m o r e each t e r m of (5:14) is continuous on C~ or C~, since R x is continuous on Bp, q. Hence Riesz's t h e o r e m applies to each t e r m , a n d (5:12), (5:13) are established.

The other cases are t r e a t e d similarly.

As in (4:5 a n d 6), we u n d e r s t a n d t h e definitions (5: 13) as definitions b y mono- t o n e limits. F o r example,

I~( P-IO~,, I t -q-~ot,,t--

(5:15)

flP'q(s',t')=- lim R s--Y) k(~)d~ (t--t) (t)dt,

k, 1 --> (oo, co) ~a b

p > l , q > l , * where 0,, k(s) is a sequence (2:5).

I f t h e f u n c t i o n a l R x is linear on Bp_l, q, p > 1, (hence on Bp, q), those relations (5: 13) in which t h e a r g u m e n t of R is a n element of Bp_~,q ( t h a t is, t h e first two relations of (5: 13)) m a y be used directly, consistently with their i n t e r p r e t a t i o n as definitions b y m o n o t o n e limits. This is proved as was the similar f a c t for B~_I a n d B n (cf. (4:9)). Likewise for Bp, q_~, q > 1. I f R x is linear on Bv_~, q_~, p > 1, q > 1, all t h e relations (5: 13) m a y , equivalently, be used directly. I n these cases t h e relations (5: 13) m a y be t r a n s f o r m e d b y t h e use of (4:11). I n particular, t h e masses fll, j t h a t are functions of a single variable t a k e on t h e simpler f o r m :

fl~' ~ ( t ' ) = R(s--a)~(t--t')~-iOt,(t), i < p , (5: 16)

fl~-J,a'(s' ) -- R(s--s')~-J(t--b)JO~,(s), j < q .

(5: 17) A n e x t e n s i o n o f / t . Once t h e masses /5 i'j h a v e been defined we m a y e x t e n d t h e R of t h e o r e m (5: l l ) as follows. P u t

(5:18) = z f S / P q(s, t)

+ 2," f x,~i, j(s, b)dfl n-j' J(s), x e By, q,

j < q U

where B~, q is t h e space of functions x for which Taylor's formula (5:4) holds w i t h integrable n t h derivatives a n d whose derivatives in ( 5 : 1 8 ) a r e Lebesgue-Stie]tjes integrable relative to their corresponding masses fli, j. T h e n Bp, q c B~, q, a n d R x = R + x for x ~ Bp, q. F u r t h e r m o r e t h e relations (5:13), w i t h R replaced b y R +, are valid with countable exceptions [13]; t h e relations (5: 13) t h u s modified m a y be t r a n s f o r m e d b y t h e use of (4:11).

Remainders: Functions of Several Variables. 333 6. F u n c t i o n s o f t w o v a r i a b l e s . T h e s p a c e .Zip, q, p ~_~ 1, q >: 1.

I n order to o b t a i n a kernel t h e o r e m w i t h converse, we i n t r o d u c e t h e space Av, q, 19 > 1, q > 1, defined as follows: Av, q is t h e space of functions x = x(s, t)

whose derivatives

Xi, n_i_](a, t) , g < 19--1 ,

(6 : 1) Xp_l, q_1(8, t) , n -~ 19+q,

x~_j_~,j(8, b), j < q--1 ,

exist a n d are continuous in t, (s, t), s, respectively, (s, t) e U 2. The order of differen- t i a t i o n in xi, j shall be restricted according to t h e c o n v e n t i o n of Bp_l, q_~.

The core of a f u n c t i o n x in Ap, q is t h e set of derivatives (6 : 1) ; t h e complete core of x is t h e core t o g e t h e r w i t h t h e n u m b e r s xi, j(a, b), i-~j < n - - l , (i,j) # (19-- l, q - - l ) . I n order for the core of x E A v , q to exist, certain derivatives of x of lower order m u s t exist a n d be continuous; we say t h a t the l a t t e r derivatives are covered b y t h e core.

Thus in Ap, q t h e derivatives covered b y t h e core are precisely t h e following: All t h e derivatives in t h e core of Bp_l, q-l, except Xp_l,q_l(8 , t), a n d all t h e derivatives covered b y t h e core of Bv_~, q-1.

The n o r m of x in Ap, q, d e n o t e d b y [IX[lAp, q, is defined as t h e m a x i m u m of t h e absolute values of t h e derivatives of x in t h e core or covered b y t h e core, (s, t) e U 2.

(6:2) T a y l o r ' s f o r m u l a o n Ap, q. I f x ~ A p , q,

t

x(s, t) = ~ (s--a)i(t--b)Jxi, j(a, b)-~ ~ (s--a)i I (t--t)n~-~xi,,~_i_l(a, t)dt

i+j<n--1 i<p--1 b

(i,j) # (p-~, q--D

+ ~ (t--b)J 18 (s-- 8)n--J--2Xn_j_i, j ( i , b)di + T , (s, t) e U s ,

j<q--1 ~a

where

T =

8 t

t ( s - ~ ) p - 2 d i l (t--t)q-2xp-l,q-~(i, t)dt, 19 > 2, q ~ 2 ,

*)a *)b

i8 (s--g) P~-2XP -1 , o(i, t ) d i , 19 > 2, q = I , a

t

Ib(t--t)q-2Xo, q_l(s, t)dt, 19 = 1, q > 2 ,

Xo, o(S, t) , 1 9 = q = 1.

This f o r m u l a is established in t h e same w a y as Taylor's f o r m u l a (5:4) on By, q.

T a y l o r ' s f o r m u l a (6:2) expresses x in t e r m s of its complete core inAp, q. Applied 2 2 - 642138 Acta mathematica. 84

t o t h e d e r i v a t i v e s c o v e r e d b y t h e core, (6:2) a n d (4: 1) express these d e r i v a t i v e s also in t e r m s of t h e c o m p l e t e core. H e n c e I lxl lAy, q a n d t h e m a x i m u m of t h e a b s o l u t e values of t h e e l e m e n t s in t h e c o m p l e t e core of x are e q u i v a l e n t n o r m s in t h e sense t h a t each is a t m o s t a c o n s t a n t t i m e s t h e other.

F o r each x 9 q t h e r e is a c o m p l e t e core. Conversely, given constants: c i'j, i ~ - j < n - - l , (i,j) # ( p - - l , q - - l ) ; continuous functions

of

t: zi'n~-l(t), i < p - - l ; continuous functions of s : z n-j-i' J(s), j < q-- 1 ; and a continuous function of (s, t) : z p-I' q-l(s, t) ; there is a unique function x 9 Ap, q having these elements as its complete core. This is u n d e r s t o o d a n d established precisely as was t h e similar f a c t (5:9) for By, q.

T h u s t h e c o m p l e t e core of x 9 Ap, q m a y be t h o u g h t of as t h e i n d e p e n d e n t p a r t of x a n d as a n i n d e p e n d e n t variable.

(6: 3) L e m m a .

A functional linear on Av, q is afortiori linear on Ap, q.

Bp, q ~ Ap, q c Bp_I, q-l"

][Xt[Bp, q ~ [IX][Ap, q ~ [lX[IBp_l,q_l 9

is a fortiori linear on Bp, q; a functional linear on By_l, q-1

(6:4) L e m m a . Ap, q c Av_l,q, p > 2 .

IIxtli~,q > IlxltA~_l,a.

A functional linear on Ap_l, q is afortiori linear on Av, q.

Likewise for Av, q-l, q > 2.

(6: 5) M u s s t h e o r e m o n Ap, q. Suppose that R x is a functional linear on Ap, q and zero whenever x is a polynomial in (s, t) of degree n - - 2 that is free of the term 8P-lt q-1. Then there are functions o~ i' j e ~fl such that

(6:6) R x ---- ~ llx~, ~_i_l(a, t)do~"~-i-l(t)-~-11 tlxp_l, q_l(s, t)d~p--1,g-l(8,

t)

i < p - 1 "20 *)0 "10

~- ~ flxn_j_l.i(S, b)d~n-J-l'J(s) , x e A p , q.

j < q--1 vO

Conversely, given masses ~i,j 9 9 , (6:6) defines a functional which is linear on Ap, q and zero whenever x(s, t) is a polynomial of degree n - - 2 that is free of the term 8P--lt q-1"

The masses oJ "i may be defined as follows:

R e m a i n d e r s : F u n c t i o n s of S e v e r a l V a r i a b l e s . 335

(6:7)

t

al, n-i-l(t' ) --__ R ( s - - a ) i Ib(t--t)n-i-~Ot,(t)dt ,

is n-j-2Os'

o~n-j--l'J(8 ') ~-- R(t--b) j (s--g) (g)dg,

a

0J~-l,

q--l(.~',

t") =

i < p - - 1 , * ,

j < q - - l , * ,

R l'~( s - - P-~O~, ~) (~)d~ I t (t--t)q-2Ot,(t t , - -)d- p > 2, q > 2 *

a b

ROt,(t ) s--g) (g)dg, p > 2, q = 1, *

~ a

t

ROs'(S) l (t--t)q-~Ot'(t)dt ' p = 1, q > 2 , * b

RO~,(s) Ot,(t ) , p = q : 1, *

The proof is similar to t h a t of the mass theorem (5: 11) on Bp, q.

The definitions (6:7) are understood as definitions b y monotone limits (cf.

(5:15)).

If the functional R x is linear on Ap_~, q, p ~ 2, (hence on Ap, q), the first two relations (6:7) m a y he used directly, consistently with their interpretation as definitions b y monotone limits. Likewise for Ap, q--i, q ~ 2. If R x is linear on Av_~, q-l, p ~ 2, q ~ 2, all the relations (6:7) may, equivalently, be used directly. In these cases, the relations (6:7) may be transformed by (4: 11). In particular,

~%i, n--i--l(t, ) ~ R(s__a)i(t__t,)n-i-lOt,(t) , i < p - - 1 ; (6: 8)

o~-J-]'J(s ') : R(s~s')'~-J-l(t--b)JOs,(s), j < q - - 1 .

(6:9) A n e x t e n s i o n of R . Once the masses eci, j have been defined, we m a y extend the /t of theorem (6: 5) as follows. P u t

(6:10)

i<p-i

§ X I x n - J - l ' Y ( s ' b l d ~ - J - l ' J ( s ) ' x e A ; , q , j<q-1 U

where A +, q is the space of functions x for which Taylor's formula (6 : 2) holds with integrable derivatives and whose derivatives in (6: 10) are Lebesgue-Stieltjes inte- grable relative to their corresponding masses ~'~. Then Ap, q c A~, q and R x ~ .R+x for x e Ap, q. Furthermore, the relations (6:7), with R replaced b y R +, are valid with countable exceptions; the relations (6: 7) thus modified may be transformed by the use of (4:11).

(6: 11) K e r n e l t h e o r e m . Suppose that R x is a functional linear on Ap, q zero for degree n - - 1 . Then there are functions fi, j such that

(6: ~2)

and

Rx = i ~ fu lxi' n-i(a' t)fi'n-i(t)dt -~ I fu/t)' q(8, t)fP'q(8, t)dsdt

§ ~ " I x,~_j,j(s, b)ff-J'J(s)ds, x 9 B;:~q,

<q U

where B+,+q is the space of functions x whose Bp, q core derivatives are integrable and for which Taylor's formula (5: 4) holds. The kernels fi'J 9 ~ . The two particular kernels fp-1, q+l(t ) and fp+l, q-l(s ) vanish at 0 and 1 and are integrals of functions in o]).

Conversely, given a set of kernels fi, j with the properties listed in the preceding paragraph, (6: 12) defines a functional R x which can be extended so as to be linear on A p' q and zero for degree n - - 1 .

The kernels m a y be defined as follows:

(6 : 13)

f~, n-i(~) ~_ R + ( s _ a ) i ( t _ ~ ) n - i - l ~ b ' t(t), i < p, fn-j,j(~) : R+(s__~)n-j-i(t__b)J~a,s(~), j < q , fP' q(~, t)

-~ R + ( s - - ~ ) P - l ( t - - t ) q - l v 2 a , s(8)~Yb, t(~) ,

W, C, e.~

w.c.e.~

w.c.e.~

where R + is the extension (6:9) of R, and ~ is defined in (4:18).

Observe t h a t Bp, q c B~,+q c Ap, q; so t h a t , in particular, (6:12) holds for x 9 Bp, q. A c t u a l l y B~,+q is n o w the same as B~, q defined in (5: 18), b u t this fact is not used in the proof or the application of the present theorem.

The relations (6: 13) m a y be w r i t t e n in alternative forms, b y the use of (4: 18).

I n particular,

(6: 14)

fl,

n-i(~) : __R+(s__a)i(t__~)n-i-lo~(t ) = R+(s--a)i(t--t)n-i-l[1--O~(t)], i < p , w . c . e , with t h e dual relation for f~-J'i(~), j < q.

If R x is linear on Ap_l,q_l, p ~ 2, q __~ 2, the relations (6:13 a n d 14) hold with the q- and w. c. e. deleted.

Observe t h a t R x is surely linear on Ap, q if it is linear on Bp_l, q-l, b y l e m m a (6:3).

P r o o f of t h e o r e m . Since R x is linear on Av, q, we m a y write R x in the form (6: 6).

++ Taylor's formula (5:4), (4: 18), a n d t h e fact t h a t R x is Suppose t h a t x 9 Bp, q.

zero for degree n - - 1 i m p l y t h a t

Remainders: Functions of Several Variables. 337 (6: 15)

Rx = ~ R(s--a) i fu(t--t)n-i-~y~b,t(t)xi, n_i(a, t)dt

~- R f fu~ ( s - ~)v ~(t--t)q-~Y~' ~(S)~fb't(t)xp'

q('~'

t)d~dt

- - 8 ) n - 3 - 1 ~ a ' s(8)Xn_j,j(8,

b )d~

, j < q

x ~ Bp, q

§

The derivation of (6: 12) from (6: 15) a m o u n t s to t h e interchange of R a n d t h e integral operator in each t e r m of (6: 15). Consider one of t h e terms, for example:

The a r g u m e n t of R in W is an element of

A~), q

whose core in

Ap, q

is zero except for its derivative of index p - - l , q--1. Hence, by (6:6) a n d Fubini's theorem,

W ~ f10

fl0d~P-l'q-l(8, t)flud~dt~f~, ~(~)y~b,t(t)xp, q(~, t ) :

where

(6:16) f(8, t) = l IU~a,s(8)Y~b,t(t)do~P-l'q-l(8, t) .

I t follows from (6:16) t h a t

f(s,

t ) e ~ , since

~p-1,

q--1 ~ . (One m a y actually evaluate the integral (6:16), by using (4: 18)).

F u r t h e r m o r e ,

.fP'q

defined in (6: 13) equals

f

with countable exceptions. For, p u t

y ~-- y(s, t) -~ (s--~)P-~(t--t)q-~y~, ~(s)YJb, t(~) 9

As a function of s, Fa, 8(~) is continuous a n d constant except at s = ~, by (4: 18).

Also Fa, a(~) ~ 0. Hence the derivative yp_l,q_l(s, t) exists a n d equals ~a, 8(s)yJb, t(t) when s 4 ~, t ~= t; a n d t h e other Ap, q core derivatives of y exist a n d vanish. Now Fa,~(~)~b,t(t), s ~= ~, t =4= t, is integrable

daP-l'q-l(s, t)

except when ~ or t is a disconti- n u i t y of ~p-l,q-1. Also Taylor's formula (6:2) applies to

y,

since (s--5)P-l~fa, 8(~)

++ q l(s, t) has discontinuities at only countably m a n y is an element of B~)_a. As ap-1,

s a n d t [13], it follows t h a t y e A~,q with countable exceptions. Hence, by (6: 10),

~> = ~ = ..ii

~ a . ( ~ ) ~ , , ( ~ > ~ ~-~'~(',

'>

= i(~, ~), w

~ ~

f~, U

A similar t r e a t m e n t of t h e o t h e r t e r m s of (6: 15) establishes (6:12), (6: 13).

W e n o w show t h a t

fp-1, q+l(t )

vanishes a t 0 a n d 1 a n d is an i n t e g r a l of a f u n c t i o n in c,~. T h e kernel

fP+l'q-l(8)

is t r e a t e d similarly. T h u s ,

(6: 17) f p - 1 ,

q+l(~) __~ R(s_a)p-l(t_~)qWb ' t(~) ;

we write R i n s t e a d of R + as in (6: 13) because t h e a r g u m e n t of R in (6: 17) is an e l e m e n t of

Ap, q;

we h a v e s u p p r e s s e d t h e w. c. e., as we m a y . N o t e t h a t

fp-l,q+l(O

) =

R(s_a)P-ltq[Oo(b)_Oo(t)] = 0--0 = O,

since

tOo(t )

is zero f o r all t ~ 0. Also (4: 18), (6:6), a n d F u b i n i ' s t h e o r e m i m p l y t h a t

d~l i1

fp-l,q+l(~) = --R(s--a)p-l(t--~)qo-t(t) = -- ~ (t---t)Oi(t)do? -~'q-~(s, t) =

0 0

w h e r e

T h u s h e G];, since a ~G]~. Finally, f P - l ' q + l ( 1 ) = 0 b y (6: 17) since Wb, t ( 1 ) = 0.

I t r e m a i n s to p r o v e t h e converse. W e consider t h e t e r m s of (6:12) s e p a r a t e l y a n d i n t e g r a t e b y parts. F o r e x a m p l e , g i v e n t h a t fP' q E cp, consider t h e t e r m

U U

fP'q(1, 1)Xp_l,q_l(1 , 1)--fP'q(1, O)Xp_l,q_l(1 , O)--fP'q(o, 1)Xp_l,q_l(O,

1)

-F-fP'q(o,O)xp_l,q_l(O,O)--IiXp_l,q_l(1, t)dfP'q(1, t)-~-IiXp_Lq_l(O,t)dfP'q(o,t ) --flXp-l,q_l(s, 1)dfP'q(s, 1)~-flxp_l,q_l(s,O)dfP'q(s,O)

1 1

+ Io I0 "-~,,-~(''')':'''(''')' "~;,+.

T h e last m e m b e r of this equation is a linear functional on Ap, q, w h i c h reduces to W for x ~ Bp,+q a n d vanishes for degree n - - 1 .

Remainders: Functions of Several Variables. 339 T h e o t h e r t e r m s of (6 : 12) are t r e a t e d similarly b u t m o r e simply, e x c e p t for t h e t w o neighbors of W. Consider one of these. G i v e n t h a t f P - l ' q + l ( t ) = f~h(t)dt a n d

vanishes a t 1, w h e r e h e o]). T h e n ^{i . 0}

I uXp_I t)fP-1, q+t(t)dt ~- -- q(a, t)h(t)dt = -- (t) (a, t) -~

u

i

--h(1)xp_l, q_l(a, 1)-~h(0)Xp_l, q_l(a, O) -~- IoXp_l, q_l(a, t)dh(t) , x ~ B~,+q . B u t t h e last e x p r e s s i o n defines a f u n c t i o n a l linear on Ap, q a n d zero for degree n - - 1 .

(6: 18) R e m a r k . Suppose t h a t a f u n c t i o n a l R * x is linear on Bp, q or Ap, q b u t n o t necessarily zero for degree n - - 1 . As in (4:23), one m a y c o n s t r u c t a r e l a t e d f u n c t i o n a l R x which will be zero for degree n - - 1 :

R x = R * x - - ~ ci, jxi, j(a , b) ,

i ~ j < n

where ci, i ~ R * ( s - - a ) i ( t - - b ) j, i + j < n. T h e f u n c t i o n a l R x is linear on Bp, q if R * x is.

N o t e , however, t h a t g i v e n R * x linear on Ap, q, R x will be linear on Ap, q if a n d o n l y if cv_~, q -~ cv, q-1 ~ O,

7. I l l u s t r a t i o n . A simple i l l u s t r a t i o n of t h e k e r n e l t h e o r e m (6:11) is t h e following. T h e f u n d a m e n t a l r e c t a n g l e is - - 1 ~ s, t < 1;

R x = x(1, 1 ) - - x ( - - 1 , 1)-~x(1, - - 1 ) - - x ( - - 1 , - - 1 ) - - 4 x l , 0 ( 0 , 0) .

T h u s R x / 4 is, a m o n g o t h e r things, t h e r e m a i n d e r in t h e a p p r o x i m a t i o n of t h e deri- v a t i v e x~,0(0, 0) b y t h e i n d i c a t e d c o m b i n a t i o n of t h e c o r n e r values of x.

T h e f u n c t i o n a l R x vanishes for degree 2; R x is linear on B1, 0, h e n c e on A2, l, for all (a, b) ; R x is linear on B0,1, h e n c e on A1, 3, if a n d o n l y if b = 0.

T h e f o r m of R x on B ++~, 1 is .

(7:1) Rx = I Xo,~(a,t)f~ xl,~(a,t)fx'~(t)dt-4-II x~'l(s' t)ff'l(s' t)dsdt

[--l, 1] [--1, 1] --1 ~ 8, t ~ 1

I , B ++

Jr X3, o(S, b)ff'~ x E 2, 1 ^'

[--I, 13

where, for all a, b,

a n d

f~ = O, fl'z(t) = 2(I--It[), ff'~ = 2(1--[sl) 2 ,

I (1--1sl) s i g n u m

(st)

^if a = b = 0 ,

y'1(8,

t)

/

1 - - s - - 4 ~ if a = b = - - l ,

e q u a l i n g one if s < 0, t < 0 a n d zero otherwise.

T h e f o r m of

Rx

on B ++ ( a , b ) - - (0,0), is _1,2~

(7 : 2)

where

Rx = I xo, 3(O, t)g~ 47 I I x1,:(s, t)gi'2(s, t)dsdt

[--1, 1] --1 --< s, t _~ 1

-@ X2, 1( 8, 0 ) g 2' 1(8) d8 Jl- X3, 0(8, 0 ) g 3' ~ x 1, 2 ,

I-i, i] I - i , i]

g~ = g~'l(s) =

0;

g"~(s, t) =

1--]tl;

g3'~ =

2(1--1sl) 2.

T h e coefficients in f3, 0 a n d ga' 0 are 2 i n s t e a d of 1 because of o u r c o n v e n t i o n a b o u t e x p o n e n t s .

8. A p p r a i s a l s a n d b e s t a p p r o x i m a t i o n s . I f

Rx

is a r e m a i n d e r , we m a y be i n t e r e s t e d in its appraisal. W e m a y appraise t h e s e p a r a t e t e r m s in (5:12) or in (6: 12) in t h e c u s t o m a r y ways. Such s e p a r a t e appraisals are efficient, since t h e e l e m e n t s of the core of x in

Bp, q

are i n d e p e n d e n t .

A l t e r n a t i v e l y we m a y appraise (6:12) as follows b y a g e n e r a l i z a t i o n of HOlder's i n e q u a l i t y . L e t

K i'j

be t h e set on which t h e kernel

fi'Jis

n o t zero; let

[Ki'J{

be t h e m e a s u r e of K i'j in t h e

t, (s, t),

or s space, as is a p p r o p r i a t e ; a n d let K = ~Y'

IKi'Jl.

i+j=n

E x c l u d e t h e case K = 0, for t h e n

R x

= 0 a n d no appraisal is needed. T h e n

(s: 1)

where

[Rxl ~ K i ~ 2 7 ]3Ci, n - i ( a , t)l~dt-~

Ixp, q( s, t)l r

i < p

Ki, n--i KP, q

+ 2 2 b)l d8 i o , x

j < q Kn-], j

Ki, n-i KP, q

I 9 . , 11/e' 1 1

§

Ifn-J'J(s)l* ds , - § j

j < q e

Kn--], )

n l , e > l ,

l ~ e m a i n d e r s : F u n c t i o n s of S e v e r a l V a r i a b l e s . 341 a n d e x p o n e n t s are u n d e r s t o o d as powers in the o r d i n a r y sense. N o w M~ is i n d e p e n - d e n t of x, a n d t h e m u l t i p l i e r of M~ in (8:1) is a n a v e r a g e of t h e absolute v a l u e of t h e core d e r i v a t i v e s of x in

Bp, q.

F i x e a n d n. Suppose t h a t one has t h e choice of several f o r m u l a s (6: 12) (which m a y be d i f f e r e n t f o r m u l a s for t h e same r e m a i n d e r

Rx

or f o r m u l a s for different r e m a i n d e r s , all in t e r m s of n t h derivatives.) Suppose t h a t one i n t e n d s t o appraise b y (8: 1) a n d t h a t one has no r e a s o n to p r e f e r t h e a v e r a g e of t h e absolute v a l u e of t h e n t h d e r i v a t i v e s on one class of sets

K ~'j

to t h e a v e r a g e on a n o t h e r class of sets

K i'j.

T h e n it is r e a s o n a b l e t o say t h a t t h a t f o r m u l a (6:12) is

best

which minimizes M~ [12].

I n t h e usual w a y we m a y a d m i t t h e values e = 1 a n d e = oo in (8: 1). T h e appraisal (8:1) m a y be a d j u s t e d to assign different weights to t h e core derivatives.

I n practice, t h e calculation of M , is o f t e n decisively simpler t h a n t h a t of Mr, e 4 : 2 .

F o r t h e illustrations of section 7, M 2 has t h e following values: (176/5) ~/2 for (7:1) w i t h a = b = 0; (496/5) 1/~ for (7:1) w i t h a ~ b = - - 1 ; (52/5) 1/~ for (7:2).

I n a similar fashion, we m a y appraise (5:12) b y a g e n e r a l i z a t i o n of t h e t h e o r e m of t h e m e a n :

(8:2)

[Rxl

~ m a x e s s sup

[Ixl, n_i(a,t)l, [xp, q(s,t)l, Ix~_j,/(s,b)l]M, x ~ B ; , q , i<p,j<q (s,t)@ U~

w h e r e t h e essential s u p r e m u m of each d e r i v a t i v e is t a k e n r e l a t i v e to its corresponding mass

fli,j,

a n d

~/Z ~ ~ I Idfl i'n-i(t) l -]- I I [aft p'q(s, t), + ~ I ' d~n-j'j(s) l 9

i < p .) U 1 e) e) U 2 j < q ~ U 1

If t h e appraisal (8:2) is to be used on a class of f o r m u l a s (5: 12), it is r e a s o n a b l e to say t h a t t h a t f o r m u l a is

best

for which M is least.

T h e appraisals (8 : 1 a n d 2) are efficient in t h e sense t h a t a n y r e d u c t i o n of t h e i r second m e m b e r s w o u l d m a k e t h e inequalities false. This is t r u e of (8: 2) because, b y c o n s t r u c t i o n , t h e masses fl~'J are c o n t i n u o u s on t h e r i g h t for positive a r g u m e n t s . 9. F u n c t i o n s of m v a r i a b l e s . T h e results of t h e previous section e x t e n d t o f u n c t i o n s of several variables. F o r t h e m o s t p a r t t h e proofs are d i r e c t generalizations of the earlier proofs. H e r e we s t a t e t h e principle facts. F o r m ~ 1 or 2 t h e concepts a n d t h e o r e m s of t h e p r e s e n t section are t h o s e of t h e p r e c e d i n g sections.

L e t (a) : (a 1 .. . . . am) be a f i x e d p o i n t in Urn; a n d let (p) : (Pl . . . Pro) be f i x e d n o n - n e g a t i v e integers. P u t

n = p ] + . . . +Pm 9

W e d e f i n e t h e s p a c e B(p) --- B m ... p~ as t h e s p a c e of f u n c t i o n s x = x(s) - - x ( s 1 .. . . . sin) w h o s e d e r i v a t i v e s

(9: 1) xil . . . i r a ( a 1 , . . . , a m ) , i x @ ' ' . @ i m ~-- n ,

a l l e x i s t a n d a r e c o n t i n u o u s i n t h e i r v a r i a b l e s o n U m, w h e r e av, v = 1 . . . m , is e i t h e r t h e c o n s t a n t a v o r t h e v a r i a b l e s~, a c c o r d i n g t o t h e f o l l o w i n g i n d u c t i v e r u l e , w h i c h we call t h e r u l e of B m ... pra:

F o r m = 1, a l = sl 9

( 9 : 2 ) F o r m > 1, (al . . . am_z) --~ (a I . . . am_l) a n d a m = s m if i m > P m ; (a I . . . . am_l) is d e t e r m i n e d b y t h e r u l e of B m ... p~_~ if i m ~ Pm a n d a m = s m if i m = Pro, am ⁼ am if i m < P m 9

W e s h a l l u s e t h e r u l e ( 9 : 2 ) t o d e t e r m i n e (a) = (al . . . am), g i v e n i 1 .. . . , i m, i n cases i n w h i c h i l - k 9 9 9 q - i m => P l q - " 9 " -kPm. T h e o r d e r of d i f f e r e n t i a t i o n i n xjl ... j~, w h e n x is c o n s i d e r e d as a n e l e m e n t of B m ... p~, is r e s t r i c t e d as follows. P u t qv = m i n (Pv,Jv), v = 1 . . . m. T h e n xjl ... j~ is t o b e u n d e r s t o o d as a d e r i v a t i v e of xql ... qa; a l l o r d e r s of d i f f e r e n t i a t i o n s i n xq~ ... q,~ a r e a l l o w e d a n d a l l o r d e r s of t h e r e m a i n i n g d i f f e r e n t i a - t i o n s n e c e s s a r y t o c a r r y xq~ ... qra i n t o Xjl ... ]~ a r e a l l o w e d .

T h e core of a f u n c t i o n x i n B(p) is t h e s e t of d e r i v a t i v e s (9: 1); t h e c o m p l e t e core of x is t h e core t o g e t h e r w i t h t h e n u m b e r s Xjl ... jm (al . . . am), j l - k " 9 9 - k j m < n.

(Cf. I l l u s t r a t i o n (9 : 8) b e l o w . ) I n o r d e r f o r t h e core of x e B(p) t o e x i s t , c e r t a i n d e r i - v a t i v e s of x of l o w e r o r d e r m u s t e x i s t a n d b e c o n t i n u o u s ; we s a y t h a t t h e l a t t e r d e r i v a t i v e s a r e covered b y t h e core.

T h e n o r m of x i n B(p), d e n o t e d b y [IXl[B(p), ^{i s} d e f i n e d as t h e m a x i m u m of t h e a b s o l u t e v a l u e s of t h e d e r i v a t i v e s of x i n t h e core o r c o v e r e d b y t h e core, (s) e Um.

A b a r y c e n t r i c d i a g r a m of t h e d e r i v a t i v e s of o r d e r n is u s e f u l i n c o n s i d e r i n g t h e core of x i n B(p).

(9 : 3) T a y l o r ' s f o r m u l a o n B(p). I f x ~ B(p),

X ( 8 ) = s ( s l - - a l ) i l . . . ( 8 m - - a m ) i m x i l . . . ira(a) + ~ 1 1 1 2 . . . I m X i l ... ira(3) ,

i 1 + " 9 9 +irn < n i 1 + " " 9 +im : n

(s) e U m ,

w h e r e (3) = (31 . . . 3m) is d e f i n e d i n t e r m s o f (a), d e t e r m i n e d by ( 9 : 2 ) , as f o l l o w s : { sv i f a~ = s~ a n d i v >__ 1 ,

Ty ~--

a v o t h e r w i s e ; a n d the operators I v are d e f i n e d as f o l l o w s :