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On analytic extrapolation of scattering amplitudes
~
Nuclear Physics BI2 (1969) 110-118. North-Holland Publ. Comp., Amsterdam
ON ANALYTIC EXTRAPOLATION OF SCATTERING AMPLITUDES J. P I ~ 0 T and P. PRESNAJDER Katedra teoretickej fyziky , Comenius University, B~'atislava, Czechoslovakia Received 31 March 1969
Abstract: The method for extrapolation of amplitudes off the physical region lying interior to the domain of analyticity is proposed. It is essentially based on Cauchy theorem and subsequent approximations of the kernel (z - a ) - I by polynomial approximants. Particular attention is payed to instabilities and to the determination of the best degree of approximation. The method is compared with direct polynomial expansions proposed recently by Cutkosky and Deo and by Ciulli. i. INTRODUCTION
The analyticity of s c a t t e r i n g amplitudes and f o r m f a c t o r s enables to e x t r a c t by analytic extrapolation an i n f o r m a t i o n about amplitudes also in r e g i o n s not a c c e s s i b l e to d i r e c t m e a s u r e m e n t . In the p a s t ten y e a r s this method was applied to many p r o c e s s e s and a r e p r e s e n t a t i v e list of r e f e r e n c e s may be found in p a p e r s [1-5]. Most of the work on the subject used polynomial expansions of amplitudes in a suitably chosen variable. An i m p o r t a n t step in this d i r e c t i o n has r e c e n t l y been p e r f o r m e d by Cutkosky and Deo [1-3] and by Ciulli [4, 5] who found optimal v a r i a b l e s for polynomial expansions of this kind. Since the t r e a t m e n t of the extrapolation p r o b l e m given below p a r a l l e l s in c e r t a i n f e a t u r e s to those given in refs. [1-5] we find it convenient to s t a r t with a b r i e f d e s c r i p t i o n of the method developed there, Let us suppose that the s c a t t e r i n g amplitude f ( w ) is analytic in the cut w~plane with cuts _~o< w < -k and k < w < oo (k > 1) on the r e a l axis, see fig, !a. Let f u r t h e r the function f(w) may be m e a s u r e d in the p h y s i c a l r e gion -1 ~< w ~< 1, w real. Cutkosky and Deo[1-3] than map c o n f o r m a l l y the cut w - p l a n e onto the i n t e r i o r of an ellipse with loci in u = +1 in the u-plane. The physical region -1 ~
ANALYTIC EXTRAPOLATION
111
t Fig. 1. if(u) -
pn(u)] < MI[R/(o- 6)] n ,
(1)
holds u n i f o r m l y f o r any u i n t e r i o r and on the e l l i p s e with foe| at u = ,-1 and 1 + major semiaxis ~(R R - l ) , 1 < R< P. In eq. (1), M 1 is a constant depending
on R and 5 is an a r b i t r a r i l y s m a l l p o s i t i v e n u m b e r . W a l s h [6] al~o shows that t h e r e is no s e q u e n c e l~n(u) which s a t i s f i e s eq. (1) with P ehallged to > P. The s e q u e n c e ~Pn(U)} s a t i s f y i n g eq. (1) is called m a x i m a l l y c o n v e r g e n t in the W a l s h ~ense. Cutkosky and Deo then p r o c e e d to show that t h e i r m a p p i n g is o p t i m a l in the s e n s e of eq. (1). This m e a n s roughly that f o r any o t h e r m a p p i n g the r a t i o (R/p) in eq. (1) i n c r e a s e s . The equivalent o p t i m a l m a p p i n g was found by Ciulli [4, 51. Ciulli m a p s the cut w - p l a n e onto an a n n u l a r region with radii 1 and R in the v - p l a n e (see fig. lc). The i n t e r n a l c i r c l e of the r a d i u s 1 is the i m a g e of the p h y s i c a l r e g i o n (counted twice), and the e x t e r n a l c i r c l e of r a d i u s R is the i m a g e of the boundary G in the cut w-plane. The e q u i v a l e n c e of both m a p s is e a s i l y s e e n , s i n c e th~ t r a n s f o r m a t i o n u = ½ (v + v - 1) m a p s the a n n u l a r r e g i o n in fig. !c onto the e l l i p s e in fig. lb. Both a p p r o a c h e s by Ciulli and by Cutkosky and Deo a r e b a s e d on the WalshWs concept of m a x i m a l l y c o n v e r g e n t p o l y n o m i a l s and on the i m p o r t a n t f a c t [6] that the p o l y n o m i a l s obtained by i n t e r p o l a t i o n in the p h y s i c a l r e g i o n are m~ximally convergent. The i n t e r p o l a t i o n in a m a t h e m a t i c a l s e n s e when the exact value of the function is given is d i f f e r e n t f r o m the a c t u a l p h y s i c a l situation w h e r e the
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J. PIgleT and P. PRESNAJDER
v a l u e s (data) a r e a l w a y s given with s o m e i n a c c u r a c y . This point r e f l e c t s i t s e l f in the f a c t that the e x t r a p o l a t i o n in t e r m s of the s e q u e n c e of m a x i m a l ly c o n v e r g e n t p o l y n o m i a l s should a l w a y s be t r u n c a t e d on a f i n i t e n u m b e r of t e r m s . T h i s h a p p e n s not only b e c a u s e of p r a c t i c a l , s a y c o m p u t e r t i m e , l i m i t a t i o n s , but m o s t l y b e c a u s e of i n s t a b i l i t i e s which s t a r t g r o w i n g up f r o m s o m e n in p o l y n o m i a l e x p a n s i o n s . The p u r p o s e of the p r e s e n t p a p e r i s to d e s c r i b e an a p p r o a c h to a n a l y t i c e x t r a p o l a t i o n a l t e r n a t i v e to d i r e c t p o l y n o m i a l e x p a n s i o n s . The a p p r o a c h is b a s e d on the Cauchy t h e o r e m and s u b s e q u e n t a p p r o x i m a t i o n of the Cauchy k e r n e l ( z - a) -1 by p o l y n o m i a l s . T h e s e a p p r o x i m a t i o n s a r e intended to s u p p r e s s the c o n t r i b u t i o n s to f(a) (in the Cauchy f o r m u l a ) f r o m the unknown p a r t of the b o u n d a r y (e.g; the cut G in fig. l a ) r e l a t i v e l y to the c o n t r i b u t i o n of the p h y s i c a l r e g i o n The m e t h o d e n a b l e s to e s t i m a t e the i n s t a b i l i t i e s d e v e l o p i n g in h i g h e r a p p r o x i m a t i o n s and due to i n a c c u r a c i e s of the d a t a and c o n s e q u e n t l y it g i v e s an a p r i o r i e s t i m a t e of the b e s t o r d e r of a p p r o x i m a t i o n . The p a p e r i s o r g a n i z e d in the following way. In s e c t . 2 the p r i n c i p l e of the e x t r a p o l a t i o n m e t h o d i s d e s c r i b e d . The s e c t i o n i s s e p a r a t e d into two p a r t s , the f o r m e r d e a l s with the i d e a l c a s e , e.g. the e x p e r i m e n t a l e r r o r s a r e n e g l e c t e d , the l a t t e r t r e a t s m o r e r e a l i s t i c c a s e . The t h i r d s e c t i o n p r e s e n t s the way to d e t e r m i n e the b e s t d e g r e e of a p p r o x i m a t i o n and the b e s t mapping. The d i s c u s s i o n is given in s e c t . 4.
2. THE P R I N C I P L E O F THE E X T R A P O L A T I O N METHOD L e t us s u p p o s e that we know a p p r o x i m a t e o r e x a c t v a l u e s of the function
f(w) in the p h y s i c a l r e g i o n -1 ~< w ~< 1 in the cut w - p l a n e (see fig. l a ) . Our a i m i s to o b t a i n the a p p r o x i m a t e v a l u e of f(w) in a point wa off the p h y s i c a l r e g i o n and not lying on the b o u n d a r y G. It i s c o n v e n i e n t to s t a r t with m a p p i n g the r e g i o n i n t e r i o r to s o m e c l o s e d c u r v e GD (containing in i t s i n t e r i o r the p h y s i c a l r e g i o n and the point wa) in the cut w - p l a n e onto the a n n u l a r r e g i o n D in the z - p l a n e bounded by two c o n c e n t r i c c i r c l e s with r a d i i r (the i m a g e of the c u r v e GD in fig. l a ) and R (the i m a g e of the p h y s i c a l r e g i o n -1 ~< w < 1 in fig. l a ) . The i m a g e of GD i s d e n o t e d as CD, the i m a g e of the p h y s i c a l r e g i o n a s C e and that of the point w a a s a. The m a p i s shown in fig. 2. The a m p l i t u d e f(z) = f(z(w)) i s by c o n s t r u c t i o n a h o l o m o r p h i c function in the r e gion D. If q(z) i s any function h o l o m o r p h i c in D and continuous in £)= D U CD U C e we m a y w r i t e a s a c o n s e q u e n c e of the Cauchy t h e o r e m 1 f f ( z ) [ ( z - a ) -1 - q(z)]dz + ~-~ 1 Ce f f ( z ) [ ( z _ a ) - i _ q(z)]dz. f(a) =2-~-~ CD
(2)
L e t us w r i t e f ( z ) f o r z e C e in the f o r m * In a sense we may call our method a "passive one", as starting with suppressing our ignorance, in contradiction to the polynomial expansions which starts with making the best use of our knowledge.
ANALYTIC EXTRAPOLATION
/
/
113
\
/
Fig. 2.
/(z) =/e(Z) + ~(z),
(3)
w h e r e re(Z) r e p r e s e n t s t h e e x p e r i m e n t a l d a t a and ¢(z) i n a c c u r a c i e s d a t a . In t h i s w a y w e o b t a i n f r o m eq. (2)
f(a) - ~ 17
f
of t h e
re(z) Q(z)dz = E(a),
(4)
Ce where
E(a)
= ~-~i
Cef~(z)Q(z)dz+
2-~if., CD/(z)Q(z)dz,
Q(z) = ( z - a ) -1 - q(z).
(4,)
(4")
To f i n d a n e x t r a p o l a t i o n m e t h o d i s t h e n e q u i v a l e n t to f i n d i n g a f u n c t i o n
q(z) w h i c h m a k e s ]E(a) [ a s s m a l l a s p o s s i b l e . T h e m i n i m i z a t i o n c a n b e p e r f o r m e d o n l y if s o m e i n f o r m a t i o n on E(z) on C e a n d on f(z) e n C D i s
IE(a) I
a v a i l a b l e . E v e n if t h i s i s t h e c a s e t h e m i n i m i z a t i o n of may present a c o m p l i c a t e d m a t h e m a t i c a l p r o b l e m . In t h e f o l l o w i n g w e s h a l l p r o p o s e a s i m p l e m e t h o d b a s e d on m i n i m i z a t i o n of Q(z) on C D w i t h o u t c l a i m i n g t h a t no b e t t e r m e t h o d e x i s t s . M a k i n g a m i l d a s s u m p t i o n a b o u t t h e i n t e g r a b i l i t y of If(z) I on C D a n d [E(z)[ on C e w e m a y w r i t e
f
IE(z)l Idz[ -" 2~EM
Ce where E and M are constants.
and
f CD
If(z> l ]dz I = 2~M,
(5)
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J. PI~ffT and P. PRESNAJDER
Denoting further max z e Ce
]Q(z)l=pe
and
max IQ(z)I:pD, z e CD
(5')
we obtain f r o m eqs. (4) the bound on [E(a) I
IE(a) l
< M(PD+ ~Oe).
(6)
In the following it will a p p e a r c o n v e n i e n t to d i s c u s s s e p a r a t e l y two c a s e s (i) E = 0, which we shall call the " i d e a l c a s e " , and (ii) E ¢ 0, which will be r e f e r r e d to a s the " r e a l i s t i c c a s e " . 2.1. The extrapolation in the ideal case C o m b i n i n g eqs. (4)-(6) we e a s i l y o b t a i n f o r the i d e a l c a s e (¢ = 0) the following e s t i m a t e
1 f
If(a) - ~-~ Ce
f(z) Q(z) dz I < MOD.
(7)
O b s e r v i n g now that ( z - a) -1 is h o l o m o r p h i c in the i n t e r i o r of CD (and in f a c t in the r e g i o n [z[ < a) we can a l w a y s m a k e PD a r b i t r a r i l y s m a l l by c h o o s i n g a s q(z) the p o l y n o m i a l of the s u f f i c i e n t l y high d e g r e e w h i c h a p p r o x i m a t e s b e s t ( z - a ) - i in the u n i f o r m n o r m on C D. T h e s e p o l y n o m i a l s w e r e given by Rivlin [7] in an e x p l i c i t e f o r m . Making u s e of his r e s u l t we may write * for real a ( z - a) -1 = qn(Z) + Qn(z),
(8)
where
qn(Z) = - ( l / a ) [ 1 + (z/a) + (z/a) 2 + . . . + (z/a) n-1 + (z/a)n ~ 2 ] , i s the p o l y n o m i a l of the b e s t u n i f o r m a p p r o x i m a t i o n to ( z - a ) - i c l o s e d i n t e r i o r of C D ([z] = r ) and
in the
qn(z) = [ 1 / ( z - a)] (z/a)n[ (za - r2)/ (a2 - r 2 ) ] , PD = P(Dn ) =
]Q~(z)l =[r/(a2-r2)]
(r/a)n
for
izl
(9)
I n s e r t i n g this into eq. (7) we have ~f(z) (z/a)n[(za_r2)/(a2_r2)]dzl
if(a)_ ~1 f
~
(10)
e Due to the f a c t that r < a f o r any a ~ D we can m a k e the r i g h t - h a n d side of eq. (10) a r b i t r a r i l y s m a l l c h o o s i n g n high enough. * The formulae can easily begeneralized to any complex a, I al > r.
ANALYTIC EXTRAPOLATION
115
2.2. The extrapolation in the realistic case L e t us chose a g a i n q(z) and Q(z) as in eqs. (8) and (9) below and try to find the value n f o r which ]E(a) [ in eq. (6) t a k e s on the m i n i m a l value. The value of Pe is calculated in a s t r a i g h t f o r w a r d way to be
p(en) = (R/a)n(a 2 - r 2)-1[ (aR - r2)/ (R - a)]. The c o r r e s p o n d i n g e s t i m a t e of E(a) is then IE(a)[ < M(a 2 -r2)-l[r(r/a) n + e(R/a)n[(aR-r2)/(R-a)]].
(11)
!The eq. (11) shows in a c l e a r way that we can no longer p e r f o r m the l i m i t n ~ co, since the f i r s t t e r m in the b r a c k e t g o e s then to z e r o a s it should, but the second t e r m blows up. T h i s p a r t i c u l a r f a c t is n o t due to o u r choice of the function qn(Z) a p p r o x i m a t i n g ( z - a) -1. Any s e q u e n c e of p o l y n o m i a l s a p p r o x i m a t i n g ( z - a ) - I f o r ]z[ -< r < a m u s t blow up on the c i r cle [Z[ = R > a . The value of n which m i n i m a l i z e s the bound on E(a) as given by eq. (11) is
Jr(R-a)
og/~
n=
----g
log(a/r)l
k(aR - r ~) log(R/a) j log(R/r)
(12)
I n s e r t i n g the value of n given by eq. (12) into the eq. (12) we obtain [E(a) I ~< [M(a 2 - r2)] E1 - w ( a ) [ l o g ( R / a ) / l o g ( a / r ) ] w ( a )
x [(aR-r2)/(R-a)]l-w(a)[log(R/r/(a/r)],
(11')
where
w(z) = [log(R/lz])/log(R/r) ] is the h a r m o n i c m e a s u r e (see e.g. r e f s . [4, 9]) f o r annular region
r
fzf
R.
Note that the e s t i m a t e of IE(a) I in eq. (11) and the r e s u l t i n g e s t i m a t e of the b e s t d e g r e e of a p p r o x i m a t i o n in eq. (12) a r e b a s e d on the v e r y m i l d a s s u m p t i o n s about the b e h a v i o r of f(z) on C D and E(z) on C e. M o r e p r e c i s e e s t i m a t e s of ]E(a) ] and n in eq. (12) m a y be obtained if a m o r e detailed i n f o r m a t i o n about E(z) and f(z) on C D is available.
3. THE O P T I M A L M A P P I N G AND T H E D E G R E E O F APPROXIMATION The i m p o r t a n t concept of the o p t i m a l m a p p i n g introduced by Ciulli [4, 5] and Cutkosky and Deo [i, 2] (and a p p r o a c h e d e a r l i e r by Ciulli and F i s c h e r [8])
116
J. PI~IJT and P. PRESNAJDER
is applicable also to a p p r o x i m a t i o n s given above. Denoting the nth a p p r o x i mation in the ideal case as An(a) we have f r o m eq. (10)
An(a) = ~-~ 1 fc ~f ( z ) (z/a)n[(za -r2)/(a 2 - r 2 ) ] d z , e
(13)
If(a) - An(a) [ < M [ r / (a 2 - r2)](r/a) n.
(13')
The optimal mapping is defined as that one which for a given w a in the cut w-plane m a k e s the ratio (r/a) in eq. (13') minimal. It can be shown [4] that the ratio (r/a) is the s m a l l e s t if the mapping z = z(w) maps the whole cut w-plane onto the i n t e r i o r of the annular region r < lz I < R. We shall not give the p r o o f here since that would amount to an a l m o s t l i t e r a l r e p r o d u c tion of Ciulli's reasoning. The optimal mapping for our case differs then only by i n v e r s i o n and multiplication f r o m the one given by Ciulli [4] and we r e f e r the i n t e r e s t e d r e a d e r to the appendix to [4] for details of the mapping. Note, however, that due to the explicite s t r u c t u r e of the r i g h t - h a n d side of eq. (13) the optimal mapping is here applicable to any t e r m An(a) and not only to the whole sequence in the a s y m p t o t i c sense. Let us now turn to the related question of the rate of convergence. It is easy to see that non-polynomial a p p r o x i m a t i o n s An(a) given by eq. (13) a r e m a x i m a l l y c o n v e r g e n t in the s e n s e of eq. (1), if the optimal mapping z = z(w) is used. To see that let us p e r f o r m the mapping u = ½[(R/z) + ( z / R ) ] ,
(14)
the i n v e r s e being z = [R/(u + ~ u 2 - 1)] with the s q u a r e - r o o t positive and r e a l for u r e a l , u > 1. The annular region of fig. 2 is then mapped onto the i n t e r i o r of the ellipse in fig. lb. The physical region Izl = R is mapped onto the real s e g m e n t -1 --< u ~< 1, the boundary Izl = r onto the ellipse with foci in u = ±1 and the m a j o r s e m i a x i s ½[(R/r) + (r/R)]. The point a, r < a < R is m a p p e d into the point u a = ½[(R/a) + (a/R)]. A c c o r d i n g to eq. (1) the m a x i m a l l y c o n v e r g e n t sequence of p o l y n o m i a l s then obeys the relation
If(a) - Pn(a)[ < Ml(a)[(R/a)/(R/r)] n,!
(15)
which with the identification Ml(a) = (Mr)/(a2 - r 2 ) is the s a m e as eq. (13'). A p a r t f r o m the f a s t e s r a t e of c o n v e r g e n c e the advantage of the optimal mapping is that the constant M entering eq. (5) is d i r e c t l y related to the p r o p e r t i e s of the amplitude in the c r o s s e d channel. This may be p a r t i c u l a r l y useful when t h e r e a r e s o m e i n f o r m a t i o n s on the c r o s s e d channel available.
4. D I S C U S S I O N
In the p r e s e n t section we shall c o m p a r e the method p r o p o s e d above with polynomial expansions. We s t a r t with pointing out the d i f f e r e n c e s and then we shall c o n c e n t r a t e on c o m m o n t r o u b l e s , consisting mainly in instabilities developing in high d e g r e e a p p r o x i m a t i o n s .
ANALYTIC EXTRAPOLATION
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W i t h i n t h e p r e s e n t m e t h o d o n e d e t e r m i n e s f i r s t t h e b e s t v a l u e of t h e d e g r e e of a p p r o x i m a t i o n f r o m eq. (12) a n d e s t i m a t e s t h e c o r r e s p o n d i n g e r r o r by eq. (11). T h e n t h e a p p r o x i m a t e v a l u e o f f ( a ) i s c a l c u l a t e d by eq. (13). If w e r e w r i t e eq. (13) in t h e o b v i o u s n o t a t i o n a s
f(a) ~ An(a) = f
f(z)Kn(z,a)dz,
(16)
Ce w e c a n s e e t h a t w e n e e d a d i f f e r e n t k e r n e l Kn(z,a) f o r any p o i n t a. T h i s i s a b i t m o r e c o m p l i c a t e d t h a n p o l y n o m i a l e x p a n s i o n in w h i c h t h e a p p r o x i m a t e v a l u e of f(a) f o r a n y a i s c a l c u l a t e d in a m o r e s i m p l e w a y if c o e f f i c i e n t s in the e x p a n s i o n h a v e b e e n o n c e o b t a i n e d by f i t t i n g in the p h y s i c a l r e g i o n *. W e h o p e h o w e v e r t h a t t h i s t e c h n i c a l s h o r t c o m i n g i s b a l a n c e d by the p o s s i b i l i t y to o b t a i n t h e v a l u e of t h e m o s t s u i t a b l e d e g r e e of a p p r o x i m a t i o n . W e h a v e s e e n in s e c t . 2.2 t h a t i n s t a b i l i t i e s in a p p r o x i m a t i o n s of f(a) d e v e l o p e v e n in p o i n t s i n t e r i o r to t h e a n a l y t i c i t y d o m a i n of f(z) if we go to h i g h - o r d e r a p p r o x i m a t i o n s . O n e m i g h t s u s p e c t t h a t t h i s i s d u e to the p a r t i c u l a r m e t h o d u s e d a n d n o t to t h e n a t u r e of the p r o b l e m i t s e l f . In o r d e r to c l a r i f y t h e q u e s t i o n l e t u s s t a r t w i t h s h o w i n g how t h e s a m e e f f e c t a p p e a r s in p o l y n o m i a l e x p a n s i o n s . F o r the s a k e of d e f i n i t e n e s s w e s h a l l u s e the v a r i a b l e u ( s e e fig. l a ) . S u p p o s e t h a t f(u) i s a n a l y t i c in t h e e l l i p s e w i t h f o c i in u = +1 a n d t h e m a j o r s e m i a x i s e q u a l to ½(p+ p - l ) . S u p p o s e f u r t h e r t h a t we h a v e t h e d a t a in p o i n t s ui, u i ~ ( - 1 , 1). W e t h u s know t h a t f(u i) =fe(ui)+ ei, w h e r e f(ui) i s the (unknown) t r u e v a l u e , fe(Ui) i s the e x p e r i m e n t a l v a l u e and ¢i i s t h e t r u e e r r o r w h i c h i s o n l y k n o w n to b e b o u n d e d by s o m e E > 0. To s i m p l i f y the a r g u m e n t l e t u s c o n s t r u c t f i r s t the p o l y n o m i a l Pn(U) w h i c h i n t e r p o l e s to e x p e r i m e n t a l d a t a Pn(Ui) = fe(Ui) f o r i = 1 , 2 , . . . . n + 1. T h e p o l y n o m i a l Pn(U) i s now t h e d i f f e r e n c e of two (unknown) p o l y n o m i a l s Pn(U) and 7rn(U); t h e f o r m e r i n t e r p o l a t e s to f(u i) e . g . to an a n a l y t i c f u n c t i o n and the l a t t e r to t r u e e r r o r s nn(Ui) = ei" L e t a b e any p o i n t i n t e r i o r to the e l l i p s e in the u - p l a n e w i t h l o c i a t +1 a n d t h e m a j o r s e m i a x i s e q u a l to ½ ( R + R - 1 ) , 1 < R < P. T h e n w e m a y w r i t e
If(a)-Pn(a)l < IfCa)-pn(a)l +
I n(a>l,
(1~)
a n d e s t i m a t e t h e f i r s t t e r m on t h e r i g h t - h a n d s i d e by eq. (1). T h e s e c o n d t e r m c a n n o t b e e s t i m a t e d in a s i m i l a r w a y s i n c e ~n(U) i s n o t a n a p p r o x i m a t i o n to a n a n a l y t i c f u n c t i o n . W e o n l y know t h a t ~n(u) i s b o u n d e d by e f o r -1 < p < 1. A l l w h a t w e m a y g e t f r o m t h i s f a c t i s g i v e n by B e r n s t e i n l e m m a ( s e e e . g . r e f . [6], § 5.2). M a k i n g u s e of eq. (1) and B e r n s t e i n l e m m a w e o b t a i n f r o m eq. (18) t h e e s t i m a t e If(a) -
Pn(a) 1 --
(19)
w h e r e M 1 a n d L a r e s o m e c o n s t a n t s . S i n c e R > 1 t h e s e c o n d t e r m b l o w s up w i t h n ~ ~ in t h e s i m i l a r w a y a s d o e s the c o r r e s p o n d i n g t e r m ** in eq. (11). * One is frequently i n t e r e s t e d only in the extrapolation to one fixed point a. In this c a s e both approaches r e q u i r e about the s a m e labour. ** Consequently the polynomial expansion has to be truncated at some finite n. This point was d i s c u s s e d in a slightly different way by Ciulli [4].
118
J. PI~I~T and P. P R E ~ N A J D E R
Note h o w e v e r that c o n s t a n t s M 1 and L a r e unknown in eq. (19) what f o r b i d s to e x t r a c t the b e s t value of n f r o m eq. (195 in c o n t r a d i s t i n c t i o n to eqs. (115,
(125. The c o m m o n o r i g i n of i n s t a b i l i t i e s d e v e l o p i n g f o r h i g h e r a p p r o x i m a t i o n s i s e a s i l y t r a c e d . In o u r a p p r o a c h it i s due to the f a c t that s u c c e s s i v e p o l y n o m i a l a p p r o x i m a t i o n to (z - a ) - I on C D blow up on C e and in the p o l y n o m i a l e x p a n s i o n it i s due to the f a c t that p o l y n o m i a l s Pn(Z) a r e not o b l i g e d to be u n i f o r m l y bounded in the r e g i o n of a n a l y t i c i t y of f(zS. The a r g u m e n t b e c o m e s c l e a r if we r e c a l l s t a b i l i t y a r g u m e n t s b a s e d on the N e v a n l i n n a p r i n c i p l e , s e e e.g. r e f s . [4, 9]. In a c t u a l e x t r a p o l a t i o n s the d a t a is fitted and not i n t e r p o l a t e d . T h i s , howe v e r , d o e s not i n f l u e n c e the c o n c l u s i o n s i n c e any r e s u l t of the fit to e x p e r i m e n t a l d a t a with e r r o r s m a y a p o s t e r i o r i be thought of as a fit of the above m e n t i o n e d type. Note f i n a l l y a p a r t i c u l a r c a s e when the m e t h o d p r o p o s e d above is p r e f e r a b l e to p o l y n o m i a l e x p a n s i o n . Suppose that c o n s t a n t s M 1 and L in eq. (19) a r e known and that the v a l u e of n, n = n o which m i n i m i z e s the r i g h t - h a n d s i d e of eq. (19) i s found. A p o l y n o m i a l of d e g r e e n o m a y h o w e v e r be i n c a p a b l e of fitting the d a t a w e l l (for i n s t a n c e n o = 2 but the d a t a r e q u i r e a p o l y n o m i a l of the s i x t h d e g r e e ) . T h i s t r o u b l e cannot o c c u r in t h e m e t h o d given a b o v e s i n c e the function fe(Z) m a y h e r e be c h o s e n in a m o r e f l e x i b l e way. T h e d e g r e e of a p p r o x i m a t i o n in our m e t h o d i s e s s e n t i a l l y given by the m a g nitude of e r r o r s a n d is not r e l a t e d to the a n a l y t i c s h a p e of the data. We have d i s c u s s e d a b o v e only the s i m p l e s t c a s e of the e x t r a p o l a t i o n p r o b l e m . The e x t r a p o l a t i o n to the b o u n d a r y is a much m o r e d e l i c a t e t a s k and it i s c l e a r t h a t the p r e s e n t m e t h o d without m o d i f i c a t i o n s would not w o r k in that case. The a u t h o r s ' i n t e r e s t in the e x t r a p o l a t i o n p r o b l e m g r e w out of d i s c u s s i o n s with Dr. M. Roos (CERN) on the p i o n - p i o n d a t a e x t r a c t i o n f r o m pion p r o d u c t i o n . U s e f u l c o r r e s p o n d e n c e with Dr. Roos i s g r a t e f u l l y a c k n o w l e d g e d . P a r t of the p a p e r w a s w r i t t e n while one of the a u t h o r s ( J . P . ) was at the I n s t i t u t e of T h e o r e t i c a l P h y s i c s at the U n i v e r s i t y of H e l s i n k i . The h o s p i t a l i t y extended to him by D r . A. G r e e n and P r o f e s s o r P. T a r j a n n e i s g r a t e f u l l y a c knowledged. We a r e a l s o i n d e b t e d to Dr. C. C r o n s t r 6 m and Dr. G. T i k t o p o u l o s ( H e l sinki) who c o n t r i b u t e d m o r e than s i g n i f i c a n t l y to the c l a r i f i c a t i o n of the a p p r o a c h used. L a s t but no l e a s t we would l i k e to thank Dr. J. F i s c h e r (Prague) for interesting discussions. [ 1] [2] [3] [4] [5] [6]
R.E. Cutkosky and B. B. Dee, Phys. Rev. Letters 22 (1968) 1272. R.E. Cutkosky and B.B. Deo, Phys. Rev. 174 (1968) 1859. R.E. Cutkosky, preprint DAMTP 68/'30, Cambridge (December 1968). S. Ciulli, CERN preprint, TH937, 1968; and Nuovo Cimento, to be published. S. Ciulli, CERNpreprint, TH987, 1968. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Coll. Publ. Vol. XX (Amer. Soc., 1956) ch. IV to VII. [7] T . J . Rivlin, Bull. Am. Math. Soc. 73 (1967) 467. [8] S. Ciulli and J. Fischer, Nucl. Phys. 24 (1961) 465. [9] M. Bertero and G. Viano, Nuovo Cimento 38 (1965) 1915.
Nuclear Physics BI2 (1969) 110-118. North-Holland Publ. Comp., Amsterdam
ON ANALYTIC EXTRAPOLATION OF SCATTERING AMPLITUDES J. P I ~ 0 T and P. PRESNAJDER Katedra teoretickej fyziky , Comenius University, B~'atislava, Czechoslovakia Received 31 March 1969
Abstract: The method for extrapolation of amplitudes off the physical region lying interior to the domain of analyticity is proposed. It is essentially based on Cauchy theorem and subsequent approximations of the kernel (z - a ) - I by polynomial approximants. Particular attention is payed to instabilities and to the determination of the best degree of approximation. The method is compared with direct polynomial expansions proposed recently by Cutkosky and Deo and by Ciulli. i. INTRODUCTION
The analyticity of s c a t t e r i n g amplitudes and f o r m f a c t o r s enables to e x t r a c t by analytic extrapolation an i n f o r m a t i o n about amplitudes also in r e g i o n s not a c c e s s i b l e to d i r e c t m e a s u r e m e n t . In the p a s t ten y e a r s this method was applied to many p r o c e s s e s and a r e p r e s e n t a t i v e list of r e f e r e n c e s may be found in p a p e r s [1-5]. Most of the work on the subject used polynomial expansions of amplitudes in a suitably chosen variable. An i m p o r t a n t step in this d i r e c t i o n has r e c e n t l y been p e r f o r m e d by Cutkosky and Deo [1-3] and by Ciulli [4, 5] who found optimal v a r i a b l e s for polynomial expansions of this kind. Since the t r e a t m e n t of the extrapolation p r o b l e m given below p a r a l l e l s in c e r t a i n f e a t u r e s to those given in refs. [1-5] we find it convenient to s t a r t with a b r i e f d e s c r i p t i o n of the method developed there, Let us suppose that the s c a t t e r i n g amplitude f ( w ) is analytic in the cut w~plane with cuts _~o< w < -k and k < w < oo (k > 1) on the r e a l axis, see fig, !a. Let f u r t h e r the function f(w) may be m e a s u r e d in the p h y s i c a l r e gion -1 ~< w ~< 1, w real. Cutkosky and Deo[1-3] than map c o n f o r m a l l y the cut w - p l a n e onto the i n t e r i o r of an ellipse with loci in u = +1 in the u-plane. The physical region -1 ~
ANALYTIC EXTRAPOLATION
111
t Fig. 1. if(u) -
pn(u)] < MI[R/(o- 6)] n ,
(1)
holds u n i f o r m l y f o r any u i n t e r i o r and on the e l l i p s e with foe| at u = ,-1 and 1 + major semiaxis ~(R R - l ) , 1 < R< P. In eq. (1), M 1 is a constant depending
on R and 5 is an a r b i t r a r i l y s m a l l p o s i t i v e n u m b e r . W a l s h [6] al~o shows that t h e r e is no s e q u e n c e l~n(u) which s a t i s f i e s eq. (1) with P ehallged to > P. The s e q u e n c e ~Pn(U)} s a t i s f y i n g eq. (1) is called m a x i m a l l y c o n v e r g e n t in the W a l s h ~ense. Cutkosky and Deo then p r o c e e d to show that t h e i r m a p p i n g is o p t i m a l in the s e n s e of eq. (1). This m e a n s roughly that f o r any o t h e r m a p p i n g the r a t i o (R/p) in eq. (1) i n c r e a s e s . The equivalent o p t i m a l m a p p i n g was found by Ciulli [4, 51. Ciulli m a p s the cut w - p l a n e onto an a n n u l a r region with radii 1 and R in the v - p l a n e (see fig. lc). The i n t e r n a l c i r c l e of the r a d i u s 1 is the i m a g e of the p h y s i c a l r e g i o n (counted twice), and the e x t e r n a l c i r c l e of r a d i u s R is the i m a g e of the boundary G in the cut w-plane. The e q u i v a l e n c e of both m a p s is e a s i l y s e e n , s i n c e th~ t r a n s f o r m a t i o n u = ½ (v + v - 1) m a p s the a n n u l a r r e g i o n in fig. !c onto the e l l i p s e in fig. lb. Both a p p r o a c h e s by Ciulli and by Cutkosky and Deo a r e b a s e d on the WalshWs concept of m a x i m a l l y c o n v e r g e n t p o l y n o m i a l s and on the i m p o r t a n t f a c t [6] that the p o l y n o m i a l s obtained by i n t e r p o l a t i o n in the p h y s i c a l r e g i o n are m~ximally convergent. The i n t e r p o l a t i o n in a m a t h e m a t i c a l s e n s e when the exact value of the function is given is d i f f e r e n t f r o m the a c t u a l p h y s i c a l situation w h e r e the
112
J. PIgleT and P. PRESNAJDER
v a l u e s (data) a r e a l w a y s given with s o m e i n a c c u r a c y . This point r e f l e c t s i t s e l f in the f a c t that the e x t r a p o l a t i o n in t e r m s of the s e q u e n c e of m a x i m a l ly c o n v e r g e n t p o l y n o m i a l s should a l w a y s be t r u n c a t e d on a f i n i t e n u m b e r of t e r m s . T h i s h a p p e n s not only b e c a u s e of p r a c t i c a l , s a y c o m p u t e r t i m e , l i m i t a t i o n s , but m o s t l y b e c a u s e of i n s t a b i l i t i e s which s t a r t g r o w i n g up f r o m s o m e n in p o l y n o m i a l e x p a n s i o n s . The p u r p o s e of the p r e s e n t p a p e r i s to d e s c r i b e an a p p r o a c h to a n a l y t i c e x t r a p o l a t i o n a l t e r n a t i v e to d i r e c t p o l y n o m i a l e x p a n s i o n s . The a p p r o a c h is b a s e d on the Cauchy t h e o r e m and s u b s e q u e n t a p p r o x i m a t i o n of the Cauchy k e r n e l ( z - a) -1 by p o l y n o m i a l s . T h e s e a p p r o x i m a t i o n s a r e intended to s u p p r e s s the c o n t r i b u t i o n s to f(a) (in the Cauchy f o r m u l a ) f r o m the unknown p a r t of the b o u n d a r y (e.g; the cut G in fig. l a ) r e l a t i v e l y to the c o n t r i b u t i o n of the p h y s i c a l r e g i o n The m e t h o d e n a b l e s to e s t i m a t e the i n s t a b i l i t i e s d e v e l o p i n g in h i g h e r a p p r o x i m a t i o n s and due to i n a c c u r a c i e s of the d a t a and c o n s e q u e n t l y it g i v e s an a p r i o r i e s t i m a t e of the b e s t o r d e r of a p p r o x i m a t i o n . The p a p e r i s o r g a n i z e d in the following way. In s e c t . 2 the p r i n c i p l e of the e x t r a p o l a t i o n m e t h o d i s d e s c r i b e d . The s e c t i o n i s s e p a r a t e d into two p a r t s , the f o r m e r d e a l s with the i d e a l c a s e , e.g. the e x p e r i m e n t a l e r r o r s a r e n e g l e c t e d , the l a t t e r t r e a t s m o r e r e a l i s t i c c a s e . The t h i r d s e c t i o n p r e s e n t s the way to d e t e r m i n e the b e s t d e g r e e of a p p r o x i m a t i o n and the b e s t mapping. The d i s c u s s i o n is given in s e c t . 4.
2. THE P R I N C I P L E O F THE E X T R A P O L A T I O N METHOD L e t us s u p p o s e that we know a p p r o x i m a t e o r e x a c t v a l u e s of the function
f(w) in the p h y s i c a l r e g i o n -1 ~< w ~< 1 in the cut w - p l a n e (see fig. l a ) . Our a i m i s to o b t a i n the a p p r o x i m a t e v a l u e of f(w) in a point wa off the p h y s i c a l r e g i o n and not lying on the b o u n d a r y G. It i s c o n v e n i e n t to s t a r t with m a p p i n g the r e g i o n i n t e r i o r to s o m e c l o s e d c u r v e GD (containing in i t s i n t e r i o r the p h y s i c a l r e g i o n and the point wa) in the cut w - p l a n e onto the a n n u l a r r e g i o n D in the z - p l a n e bounded by two c o n c e n t r i c c i r c l e s with r a d i i r (the i m a g e of the c u r v e GD in fig. l a ) and R (the i m a g e of the p h y s i c a l r e g i o n -1 ~< w < 1 in fig. l a ) . The i m a g e of GD i s d e n o t e d as CD, the i m a g e of the p h y s i c a l r e g i o n a s C e and that of the point w a a s a. The m a p i s shown in fig. 2. The a m p l i t u d e f(z) = f(z(w)) i s by c o n s t r u c t i o n a h o l o m o r p h i c function in the r e gion D. If q(z) i s any function h o l o m o r p h i c in D and continuous in £)= D U CD U C e we m a y w r i t e a s a c o n s e q u e n c e of the Cauchy t h e o r e m 1 f f ( z ) [ ( z - a ) -1 - q(z)]dz + ~-~ 1 Ce f f ( z ) [ ( z _ a ) - i _ q(z)]dz. f(a) =2-~-~ CD
(2)
L e t us w r i t e f ( z ) f o r z e C e in the f o r m * In a sense we may call our method a "passive one", as starting with suppressing our ignorance, in contradiction to the polynomial expansions which starts with making the best use of our knowledge.
ANALYTIC EXTRAPOLATION
/
/
113
\
/
Fig. 2.
/(z) =/e(Z) + ~(z),
(3)
w h e r e re(Z) r e p r e s e n t s t h e e x p e r i m e n t a l d a t a and ¢(z) i n a c c u r a c i e s d a t a . In t h i s w a y w e o b t a i n f r o m eq. (2)
f(a) - ~ 17
f
of t h e
re(z) Q(z)dz = E(a),
(4)
Ce where
E(a)
= ~-~i
Cef~(z)Q(z)dz+
2-~if., CD/(z)Q(z)dz,
Q(z) = ( z - a ) -1 - q(z).
(4,)
(4")
To f i n d a n e x t r a p o l a t i o n m e t h o d i s t h e n e q u i v a l e n t to f i n d i n g a f u n c t i o n
q(z) w h i c h m a k e s ]E(a) [ a s s m a l l a s p o s s i b l e . T h e m i n i m i z a t i o n c a n b e p e r f o r m e d o n l y if s o m e i n f o r m a t i o n on E(z) on C e a n d on f(z) e n C D i s
IE(a) I
a v a i l a b l e . E v e n if t h i s i s t h e c a s e t h e m i n i m i z a t i o n of may present a c o m p l i c a t e d m a t h e m a t i c a l p r o b l e m . In t h e f o l l o w i n g w e s h a l l p r o p o s e a s i m p l e m e t h o d b a s e d on m i n i m i z a t i o n of Q(z) on C D w i t h o u t c l a i m i n g t h a t no b e t t e r m e t h o d e x i s t s . M a k i n g a m i l d a s s u m p t i o n a b o u t t h e i n t e g r a b i l i t y of If(z) I on C D a n d [E(z)[ on C e w e m a y w r i t e
f
IE(z)l Idz[ -" 2~EM
Ce where E and M are constants.
and
f CD
If(z> l ]dz I = 2~M,
(5)
114
J. PI~ffT and P. PRESNAJDER
Denoting further max z e Ce
]Q(z)l=pe
and
max IQ(z)I:pD, z e CD
(5')
we obtain f r o m eqs. (4) the bound on [E(a) I
IE(a) l
< M(PD+ ~Oe).
(6)
In the following it will a p p e a r c o n v e n i e n t to d i s c u s s s e p a r a t e l y two c a s e s (i) E = 0, which we shall call the " i d e a l c a s e " , and (ii) E ¢ 0, which will be r e f e r r e d to a s the " r e a l i s t i c c a s e " . 2.1. The extrapolation in the ideal case C o m b i n i n g eqs. (4)-(6) we e a s i l y o b t a i n f o r the i d e a l c a s e (¢ = 0) the following e s t i m a t e
1 f
If(a) - ~-~ Ce
f(z) Q(z) dz I < MOD.
(7)
O b s e r v i n g now that ( z - a) -1 is h o l o m o r p h i c in the i n t e r i o r of CD (and in f a c t in the r e g i o n [z[ < a) we can a l w a y s m a k e PD a r b i t r a r i l y s m a l l by c h o o s i n g a s q(z) the p o l y n o m i a l of the s u f f i c i e n t l y high d e g r e e w h i c h a p p r o x i m a t e s b e s t ( z - a ) - i in the u n i f o r m n o r m on C D. T h e s e p o l y n o m i a l s w e r e given by Rivlin [7] in an e x p l i c i t e f o r m . Making u s e of his r e s u l t we may write * for real a ( z - a) -1 = qn(Z) + Qn(z),
(8)
where
qn(Z) = - ( l / a ) [ 1 + (z/a) + (z/a) 2 + . . . + (z/a) n-1 + (z/a)n ~ 2 ] , i s the p o l y n o m i a l of the b e s t u n i f o r m a p p r o x i m a t i o n to ( z - a ) - i c l o s e d i n t e r i o r of C D ([z] = r ) and
in the
qn(z) = [ 1 / ( z - a)] (z/a)n[ (za - r2)/ (a2 - r 2 ) ] , PD = P(Dn ) =
]Q~(z)l =[r/(a2-r2)]
(r/a)n
for
izl
(9)
I n s e r t i n g this into eq. (7) we have ~f(z) (z/a)n[(za_r2)/(a2_r2)]dzl
if(a)_ ~1 f
~
(10)
e Due to the f a c t that r < a f o r any a ~ D we can m a k e the r i g h t - h a n d side of eq. (10) a r b i t r a r i l y s m a l l c h o o s i n g n high enough. * The formulae can easily begeneralized to any complex a, I al > r.
ANALYTIC EXTRAPOLATION
115
2.2. The extrapolation in the realistic case L e t us chose a g a i n q(z) and Q(z) as in eqs. (8) and (9) below and try to find the value n f o r which ]E(a) [ in eq. (6) t a k e s on the m i n i m a l value. The value of Pe is calculated in a s t r a i g h t f o r w a r d way to be
p(en) = (R/a)n(a 2 - r 2)-1[ (aR - r2)/ (R - a)]. The c o r r e s p o n d i n g e s t i m a t e of E(a) is then IE(a)[ < M(a 2 -r2)-l[r(r/a) n + e(R/a)n[(aR-r2)/(R-a)]].
(11)
!The eq. (11) shows in a c l e a r way that we can no longer p e r f o r m the l i m i t n ~ co, since the f i r s t t e r m in the b r a c k e t g o e s then to z e r o a s it should, but the second t e r m blows up. T h i s p a r t i c u l a r f a c t is n o t due to o u r choice of the function qn(Z) a p p r o x i m a t i n g ( z - a) -1. Any s e q u e n c e of p o l y n o m i a l s a p p r o x i m a t i n g ( z - a ) - I f o r ]z[ -< r < a m u s t blow up on the c i r cle [Z[ = R > a . The value of n which m i n i m a l i z e s the bound on E(a) as given by eq. (11) is
Jr(R-a)
og/~
n=
----g
log(a/r)l
k(aR - r ~) log(R/a) j log(R/r)
(12)
I n s e r t i n g the value of n given by eq. (12) into the eq. (12) we obtain [E(a) I ~< [M(a 2 - r2)] E1 - w ( a ) [ l o g ( R / a ) / l o g ( a / r ) ] w ( a )
x [(aR-r2)/(R-a)]l-w(a)[log(R/r/(a/r)],
(11')
where
w(z) = [log(R/lz])/log(R/r) ] is the h a r m o n i c m e a s u r e (see e.g. r e f s . [4, 9]) f o r annular region
r
fzf
R.
Note that the e s t i m a t e of IE(a) I in eq. (11) and the r e s u l t i n g e s t i m a t e of the b e s t d e g r e e of a p p r o x i m a t i o n in eq. (12) a r e b a s e d on the v e r y m i l d a s s u m p t i o n s about the b e h a v i o r of f(z) on C D and E(z) on C e. M o r e p r e c i s e e s t i m a t e s of ]E(a) ] and n in eq. (12) m a y be obtained if a m o r e detailed i n f o r m a t i o n about E(z) and f(z) on C D is available.
3. THE O P T I M A L M A P P I N G AND T H E D E G R E E O F APPROXIMATION The i m p o r t a n t concept of the o p t i m a l m a p p i n g introduced by Ciulli [4, 5] and Cutkosky and Deo [i, 2] (and a p p r o a c h e d e a r l i e r by Ciulli and F i s c h e r [8])
116
J. PI~IJT and P. PRESNAJDER
is applicable also to a p p r o x i m a t i o n s given above. Denoting the nth a p p r o x i mation in the ideal case as An(a) we have f r o m eq. (10)
An(a) = ~-~ 1 fc ~f ( z ) (z/a)n[(za -r2)/(a 2 - r 2 ) ] d z , e
(13)
If(a) - An(a) [ < M [ r / (a 2 - r2)](r/a) n.
(13')
The optimal mapping is defined as that one which for a given w a in the cut w-plane m a k e s the ratio (r/a) in eq. (13') minimal. It can be shown [4] that the ratio (r/a) is the s m a l l e s t if the mapping z = z(w) maps the whole cut w-plane onto the i n t e r i o r of the annular region r < lz I < R. We shall not give the p r o o f here since that would amount to an a l m o s t l i t e r a l r e p r o d u c tion of Ciulli's reasoning. The optimal mapping for our case differs then only by i n v e r s i o n and multiplication f r o m the one given by Ciulli [4] and we r e f e r the i n t e r e s t e d r e a d e r to the appendix to [4] for details of the mapping. Note, however, that due to the explicite s t r u c t u r e of the r i g h t - h a n d side of eq. (13) the optimal mapping is here applicable to any t e r m An(a) and not only to the whole sequence in the a s y m p t o t i c sense. Let us now turn to the related question of the rate of convergence. It is easy to see that non-polynomial a p p r o x i m a t i o n s An(a) given by eq. (13) a r e m a x i m a l l y c o n v e r g e n t in the s e n s e of eq. (1), if the optimal mapping z = z(w) is used. To see that let us p e r f o r m the mapping u = ½[(R/z) + ( z / R ) ] ,
(14)
the i n v e r s e being z = [R/(u + ~ u 2 - 1)] with the s q u a r e - r o o t positive and r e a l for u r e a l , u > 1. The annular region of fig. 2 is then mapped onto the i n t e r i o r of the ellipse in fig. lb. The physical region Izl = R is mapped onto the real s e g m e n t -1 --< u ~< 1, the boundary Izl = r onto the ellipse with foci in u = ±1 and the m a j o r s e m i a x i s ½[(R/r) + (r/R)]. The point a, r < a < R is m a p p e d into the point u a = ½[(R/a) + (a/R)]. A c c o r d i n g to eq. (1) the m a x i m a l l y c o n v e r g e n t sequence of p o l y n o m i a l s then obeys the relation
If(a) - Pn(a)[ < Ml(a)[(R/a)/(R/r)] n,!
(15)
which with the identification Ml(a) = (Mr)/(a2 - r 2 ) is the s a m e as eq. (13'). A p a r t f r o m the f a s t e s r a t e of c o n v e r g e n c e the advantage of the optimal mapping is that the constant M entering eq. (5) is d i r e c t l y related to the p r o p e r t i e s of the amplitude in the c r o s s e d channel. This may be p a r t i c u l a r l y useful when t h e r e a r e s o m e i n f o r m a t i o n s on the c r o s s e d channel available.
4. D I S C U S S I O N
In the p r e s e n t section we shall c o m p a r e the method p r o p o s e d above with polynomial expansions. We s t a r t with pointing out the d i f f e r e n c e s and then we shall c o n c e n t r a t e on c o m m o n t r o u b l e s , consisting mainly in instabilities developing in high d e g r e e a p p r o x i m a t i o n s .
ANALYTIC EXTRAPOLATION
117
W i t h i n t h e p r e s e n t m e t h o d o n e d e t e r m i n e s f i r s t t h e b e s t v a l u e of t h e d e g r e e of a p p r o x i m a t i o n f r o m eq. (12) a n d e s t i m a t e s t h e c o r r e s p o n d i n g e r r o r by eq. (11). T h e n t h e a p p r o x i m a t e v a l u e o f f ( a ) i s c a l c u l a t e d by eq. (13). If w e r e w r i t e eq. (13) in t h e o b v i o u s n o t a t i o n a s
f(a) ~ An(a) = f
f(z)Kn(z,a)dz,
(16)
Ce w e c a n s e e t h a t w e n e e d a d i f f e r e n t k e r n e l Kn(z,a) f o r any p o i n t a. T h i s i s a b i t m o r e c o m p l i c a t e d t h a n p o l y n o m i a l e x p a n s i o n in w h i c h t h e a p p r o x i m a t e v a l u e of f(a) f o r a n y a i s c a l c u l a t e d in a m o r e s i m p l e w a y if c o e f f i c i e n t s in the e x p a n s i o n h a v e b e e n o n c e o b t a i n e d by f i t t i n g in the p h y s i c a l r e g i o n *. W e h o p e h o w e v e r t h a t t h i s t e c h n i c a l s h o r t c o m i n g i s b a l a n c e d by the p o s s i b i l i t y to o b t a i n t h e v a l u e of t h e m o s t s u i t a b l e d e g r e e of a p p r o x i m a t i o n . W e h a v e s e e n in s e c t . 2.2 t h a t i n s t a b i l i t i e s in a p p r o x i m a t i o n s of f(a) d e v e l o p e v e n in p o i n t s i n t e r i o r to t h e a n a l y t i c i t y d o m a i n of f(z) if we go to h i g h - o r d e r a p p r o x i m a t i o n s . O n e m i g h t s u s p e c t t h a t t h i s i s d u e to the p a r t i c u l a r m e t h o d u s e d a n d n o t to t h e n a t u r e of the p r o b l e m i t s e l f . In o r d e r to c l a r i f y t h e q u e s t i o n l e t u s s t a r t w i t h s h o w i n g how t h e s a m e e f f e c t a p p e a r s in p o l y n o m i a l e x p a n s i o n s . F o r the s a k e of d e f i n i t e n e s s w e s h a l l u s e the v a r i a b l e u ( s e e fig. l a ) . S u p p o s e t h a t f(u) i s a n a l y t i c in t h e e l l i p s e w i t h f o c i in u = +1 a n d t h e m a j o r s e m i a x i s e q u a l to ½(p+ p - l ) . S u p p o s e f u r t h e r t h a t we h a v e t h e d a t a in p o i n t s ui, u i ~ ( - 1 , 1). W e t h u s know t h a t f(u i) =fe(ui)+ ei, w h e r e f(ui) i s the (unknown) t r u e v a l u e , fe(Ui) i s the e x p e r i m e n t a l v a l u e and ¢i i s t h e t r u e e r r o r w h i c h i s o n l y k n o w n to b e b o u n d e d by s o m e E > 0. To s i m p l i f y the a r g u m e n t l e t u s c o n s t r u c t f i r s t the p o l y n o m i a l Pn(U) w h i c h i n t e r p o l e s to e x p e r i m e n t a l d a t a Pn(Ui) = fe(Ui) f o r i = 1 , 2 , . . . . n + 1. T h e p o l y n o m i a l Pn(U) i s now t h e d i f f e r e n c e of two (unknown) p o l y n o m i a l s Pn(U) and 7rn(U); t h e f o r m e r i n t e r p o l a t e s to f(u i) e . g . to an a n a l y t i c f u n c t i o n and the l a t t e r to t r u e e r r o r s nn(Ui) = ei" L e t a b e any p o i n t i n t e r i o r to the e l l i p s e in the u - p l a n e w i t h l o c i a t +1 a n d t h e m a j o r s e m i a x i s e q u a l to ½ ( R + R - 1 ) , 1 < R < P. T h e n w e m a y w r i t e
If(a)-Pn(a)l < IfCa)-pn(a)l +
I n(a>l,
(1~)
a n d e s t i m a t e t h e f i r s t t e r m on t h e r i g h t - h a n d s i d e by eq. (1). T h e s e c o n d t e r m c a n n o t b e e s t i m a t e d in a s i m i l a r w a y s i n c e ~n(U) i s n o t a n a p p r o x i m a t i o n to a n a n a l y t i c f u n c t i o n . W e o n l y know t h a t ~n(u) i s b o u n d e d by e f o r -1 < p < 1. A l l w h a t w e m a y g e t f r o m t h i s f a c t i s g i v e n by B e r n s t e i n l e m m a ( s e e e . g . r e f . [6], § 5.2). M a k i n g u s e of eq. (1) and B e r n s t e i n l e m m a w e o b t a i n f r o m eq. (18) t h e e s t i m a t e If(a) -
Pn(a) 1 --
(19)
w h e r e M 1 a n d L a r e s o m e c o n s t a n t s . S i n c e R > 1 t h e s e c o n d t e r m b l o w s up w i t h n ~ ~ in t h e s i m i l a r w a y a s d o e s the c o r r e s p o n d i n g t e r m ** in eq. (11). * One is frequently i n t e r e s t e d only in the extrapolation to one fixed point a. In this c a s e both approaches r e q u i r e about the s a m e labour. ** Consequently the polynomial expansion has to be truncated at some finite n. This point was d i s c u s s e d in a slightly different way by Ciulli [4].
118
J. PI~I~T and P. P R E ~ N A J D E R
Note h o w e v e r that c o n s t a n t s M 1 and L a r e unknown in eq. (19) what f o r b i d s to e x t r a c t the b e s t value of n f r o m eq. (195 in c o n t r a d i s t i n c t i o n to eqs. (115,
(125. The c o m m o n o r i g i n of i n s t a b i l i t i e s d e v e l o p i n g f o r h i g h e r a p p r o x i m a t i o n s i s e a s i l y t r a c e d . In o u r a p p r o a c h it i s due to the f a c t that s u c c e s s i v e p o l y n o m i a l a p p r o x i m a t i o n to (z - a ) - I on C D blow up on C e and in the p o l y n o m i a l e x p a n s i o n it i s due to the f a c t that p o l y n o m i a l s Pn(Z) a r e not o b l i g e d to be u n i f o r m l y bounded in the r e g i o n of a n a l y t i c i t y of f(zS. The a r g u m e n t b e c o m e s c l e a r if we r e c a l l s t a b i l i t y a r g u m e n t s b a s e d on the N e v a n l i n n a p r i n c i p l e , s e e e.g. r e f s . [4, 9]. In a c t u a l e x t r a p o l a t i o n s the d a t a is fitted and not i n t e r p o l a t e d . T h i s , howe v e r , d o e s not i n f l u e n c e the c o n c l u s i o n s i n c e any r e s u l t of the fit to e x p e r i m e n t a l d a t a with e r r o r s m a y a p o s t e r i o r i be thought of as a fit of the above m e n t i o n e d type. Note f i n a l l y a p a r t i c u l a r c a s e when the m e t h o d p r o p o s e d above is p r e f e r a b l e to p o l y n o m i a l e x p a n s i o n . Suppose that c o n s t a n t s M 1 and L in eq. (19) a r e known and that the v a l u e of n, n = n o which m i n i m i z e s the r i g h t - h a n d s i d e of eq. (19) i s found. A p o l y n o m i a l of d e g r e e n o m a y h o w e v e r be i n c a p a b l e of fitting the d a t a w e l l (for i n s t a n c e n o = 2 but the d a t a r e q u i r e a p o l y n o m i a l of the s i x t h d e g r e e ) . T h i s t r o u b l e cannot o c c u r in t h e m e t h o d given a b o v e s i n c e the function fe(Z) m a y h e r e be c h o s e n in a m o r e f l e x i b l e way. T h e d e g r e e of a p p r o x i m a t i o n in our m e t h o d i s e s s e n t i a l l y given by the m a g nitude of e r r o r s a n d is not r e l a t e d to the a n a l y t i c s h a p e of the data. We have d i s c u s s e d a b o v e only the s i m p l e s t c a s e of the e x t r a p o l a t i o n p r o b l e m . The e x t r a p o l a t i o n to the b o u n d a r y is a much m o r e d e l i c a t e t a s k and it i s c l e a r t h a t the p r e s e n t m e t h o d without m o d i f i c a t i o n s would not w o r k in that case. The a u t h o r s ' i n t e r e s t in the e x t r a p o l a t i o n p r o b l e m g r e w out of d i s c u s s i o n s with Dr. M. Roos (CERN) on the p i o n - p i o n d a t a e x t r a c t i o n f r o m pion p r o d u c t i o n . U s e f u l c o r r e s p o n d e n c e with Dr. Roos i s g r a t e f u l l y a c k n o w l e d g e d . P a r t of the p a p e r w a s w r i t t e n while one of the a u t h o r s ( J . P . ) was at the I n s t i t u t e of T h e o r e t i c a l P h y s i c s at the U n i v e r s i t y of H e l s i n k i . The h o s p i t a l i t y extended to him by D r . A. G r e e n and P r o f e s s o r P. T a r j a n n e i s g r a t e f u l l y a c knowledged. We a r e a l s o i n d e b t e d to Dr. C. C r o n s t r 6 m and Dr. G. T i k t o p o u l o s ( H e l sinki) who c o n t r i b u t e d m o r e than s i g n i f i c a n t l y to the c l a r i f i c a t i o n of the a p p r o a c h used. L a s t but no l e a s t we would l i k e to thank Dr. J. F i s c h e r (Prague) for interesting discussions. [ 1] [2] [3] [4] [5] [6]
R.E. Cutkosky and B. B. Dee, Phys. Rev. Letters 22 (1968) 1272. R.E. Cutkosky and B.B. Deo, Phys. Rev. 174 (1968) 1859. R.E. Cutkosky, preprint DAMTP 68/'30, Cambridge (December 1968). S. Ciulli, CERN preprint, TH937, 1968; and Nuovo Cimento, to be published. S. Ciulli, CERNpreprint, TH987, 1968. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Coll. Publ. Vol. XX (Amer. Soc., 1956) ch. IV to VII. [7] T . J . Rivlin, Bull. Am. Math. Soc. 73 (1967) 467. [8] S. Ciulli and J. Fischer, Nucl. Phys. 24 (1961) 465. [9] M. Bertero and G. Viano, Nuovo Cimento 38 (1965) 1915.