A guide to analytic extrapolations

Computer Physics Communications 18 (1979) 215—244 © North-Holland Publishing Company

A GUIDE TO ANALYTIC EXTRAPOLATIONS Part I: A program for optimal extrapolation to interior points M. CIULLI and S. CIULLI Th. Div., CERN, Geneva, Switzerland Received 27 January 1978; in revised form 8 June 1979

PROGRAM SUMMARY Title ofprogram: ANALYT

Key-words: nuclear physics, scattering amplitude, analytic continuation, inverse problems, stability, dispersion relations, functional methods, zeros, poles

Catalogue number: AAUT Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

Nature of the physical problem In the scattering theory of elementary particles as well as in any other branch of physics in which the analyticity of some function of physical interest has been established, one is often faced with the practical question of continuing data given on parts of the cuts, in an analytic way towards some interior points of the analyticity domain. Since this problem is “ill posed” in the Hadamard sense, in order to get sensible predictions, one has to “stabilize” the output by means, for instance, of a boundedness condition, given on those parts of the cuts where the actual data is lacking (e.g. a Froissart bound). The present program processes the input data and the stabilizing condition in an optimal function-analytic way, yielding estimates of analytic continuations in any desired interior point, as well as the value of some important nonlinear functionals [1] which “measure” the “analyticity” of the input. These numbers might then be used to correlate in an analytic way data spread on different energy regions, to get informations about the asymptotics, or to find zeros and poles of the scattering amplitude.

Computer: IBM 370/3033, Kernforschungszentrum, Karlsruhe, tested also on UNIVAC 1108 and on CDC 6600 and CDC 7600 Operating system: OS/VS 2, Compiler Fortran H opt. (2) Programming language used: FORTRAN IV High speed storage required (in GO step): 212 k

*

No. of bits in k-word: 8 bits in a byte, 1024 bytes in 1 k Overlay structure: none

No. of magnetic tapes required: none Other peripherals used: card reader, line printer

Method of solution Calling once the subroutine ANALYT, all the pertinent information (the complex-valued data and the points where it is given, the form of the error corridor and the function-bounds), is loaded. Then, according to the name of the extrapolation function one calls, one gets either the PoissoN weighted EXtrapolation [2], PNEX(s), which optimizes dispersion relations versus the uniform (L~)norm, the CauchY weighted 2 norm,EXtraof polation, CYEX(s), (optimization versus the L Cutkosky’s modified x2-test, see ref. [3]),or the Central Analytic EXtrapolation, CAEX(s), which gives at any point s, the center of gravity of the (complex) values taken there by all the analytic functions compatible with the initial data

No. of cards in combined program and test decks: 2500 (No. of cards of the ANALYT deck: 1800)

Card punching code: EBCDIC

*

The memory requirements may be considerably lowered if the dimension statements are changed accordingly (see outlook of ref. [1]). 215

216

M. (‘jul11, S. ciulli /Analytic extrapolations!

and boundedness condition [41.The values of the other four external functions described in part 1, may be computed by calling the corresponding entry points. Further, calling EMZERO/or EPZERO/ (see part 2 [1]), one computes a number (~the width of the smallest bound, or smallest error corridor, still compatible with the initial data and with the analyticity), which might be used then to recognize “good” (i.e. “analytical”) data from “wrong” ones. A further facility permits to see whether the data and the analyticity condition is compatible or not with the existence of a (or a pair of) zero(s) or pole(s), placed at some given point Szero. Restrictions on the complexity of the problem None, provided sufficient data points and suitable dimensions were taken (in the test run these numbers were 181 and 501, respectively).

Typical running times The (‘auchy weighted or the Poisson weighted extrapolations need only some 0.2 s per point, but the central analytic or the other special extrapolations are more time consuming (approx. 1 mm, see remark 3.1.0). Their subsequent calls need again only 0.2 s per extrapolated point. Some 10 s are necessary for EMZERO/EPZERO. Further details might be found in the two test programs descriptions at the end of ref. [1]. References [1] I. Caprini, M. Ciulli, S. Ciulli, C. Pomponiu, M. Sararu and IS. Stefanescu, Comput. Phys. Commun. 18 (1979) [2] S. Ciulli and J. Fischer, Nuci. Phys. B24 (1970) 537. ]3] RE. Cutkosky, Ann. Phys. 54 (1969) 110. [4] S. Ciulli and G. Nenciu, J. Math. Phys. 14 (1973) 1675; S. Cmli, C. Pomponiu and IS. Stefanescu, Phys. Rept. 17C (1975) 133.

LONG WRITE-UP 1. Introduction In present-day strong interaction physics, where no universal algorithm is yet available, the analyticity (Martin [1]) of the scattering amplitude provides a much-exploited binding material for theories otherwise so different in nature, such as current algebra, Regge poles, eikonal and so on. This binding ability of the analyticity is related to the well-known theorem which states that once an analytic function is given along an arbitrarily small continuum (henceforth called the “data region”), it is uniquely determined everywhere. The analyticity of the scattering amplitude has therefore been seen as an ideal information conveyer from regions (in the energy- or momentumtransfer complex plane) with denser experimental or theoretical information, to less tangible ones. Unfortunately, the enormous literature which flourished around these lines has almost completely disregarded the explosive instabilities of these continuation methods. Indeed, the smallest uncertainties in the initial data could lead to results differing by arbitrary amounts, even in regions very close to the initial one (see appendix A of ref. [2]). Hence, since the experimental or theoretical information is always affected by uncertainties, analyticity by itself has no predictive properties at all, if no supplementary conditions a~eadded. To find precisely what qualities this supplementary stabilizing information should have in order to control the instabilities, we shall visualize the information flow in the process of analytic continuation in the form of an infinite dimensional (and infinitely divergent!) flow inside some suitable functions’ Banach space. A ready example is provided by the analytic continuation between two concentric circles Fi and F0 of radii z~= r1 and ~zj = (r1
21rd(o)(0l\

f

d~1~(0)1 ~

~

jO

2ir

)r 0 e r1~e —r1e

do’

f

do’ K(0’, 0)d~°~(0’)fKd~°~(O).

An alternative way of dealing with this problem functions u~(O)of the integral operator l~ ~Un(0)

f

is to expand both

do’ K(O’, 0) u~(O’)= X~u~(O).

(1)

d~°~(0) and d~’~(0) in terms of the eigen-

(2)

M. Ciulli, S. Ciulli / Analytic extrapolations!

Since

~n

satisfies the Cauchy formula as well,

Zn

=

217

dz’ Z~n/(ZP—

~

we find that the eigenfunctions of 1K are simply un(0)=etT~O,

(3)

while its eigenvalues are T’~ with r E ln(r = e

0/rj) >0.

(4)

Hence the expansion coefficients of

0~u~(0)

(5)

~i)d~1

d~°~(0) = and

d(1~(0)

~II~ d~n~) Um(O)

(6)

in our case are nothing but the Fourier coefficients d~ =

dO u~(0)d(k)(0).

(7)

Introducing the expansions (5) and (6) into eq. (1), [i.e. into d~”(0) Kd~°)(0)], we get d~1~ = ~

(8)

so that, if the data function d~°~(8) is known [and hence also its expansion coefficients d~°~] ,we can compute d~”(0),or, what is quite the same, the infinite component vector dW~,in a simple and direct way. Since the X~ occurring in (8) tend rapidly to zero [see eq. (4)], only a few coefficients d~°~ would be relevant to our continuation problem, and we say that our flow * has contracting properties. Reversing the direction of “time”, we get a dilatating (diverging) flow. Indeed, if the function d~’~(O), from the smaller circle, were known instead of d(°~(0), we could, in principle, compute the latter by reversing eq. (8), d~°)=d~’~/X~,

*

(9)

The analytic continuation from the circle of radius r 0 to the circles of radii r1

exp(—r) r0, r2

=

exp(—2r) r0, and so on (where

r is some fixed positive quantity), is nothing but an instance of 2dt2T(O) discrete “time” = ... =(tK~h~d~(O), = r, so kr,that ...) the contracting interpretation flow problem. of our

problemdkT(O) Writing as a flowdQ’)(e), is obvious. we have Continuity indeed can dt(O) be =restored Kdt~(O) by =varying K r; the operator K, as defined by eq. (1) depends, of course, on T in r 0/r1. This flow has manifest contracting properties, since after some “time” r, the projection, on the nth dimension, of the distance between two lines of current, falls down (see eqs. (8) and (4)] by a factor exp(—nt/r); when t grows to infinity (this corresponds to the analytic continuation to the circle of radius zero), the projections on all n * 0 dimensions of this distance vanish, which is nothing but the theorem that the value of an analytic function at the origin is the average of its values on the circle Z I r0 (and hence is not affected by the changes of its higher Fourier components). It should be noticed that even for finite “times”, this (projection of the) distance tends to zero, when n goes to infinity. Reversing the “time” direction (reversing the sign of r), we get a divergent flow: namely an inifinite rapid diverging flow in the remotest dimensions of the Banach space (if n is not bounded in some way).

M. Ciulli, S. Ciulli / A nalytic extrapolations I

218

but, in this case, even the smallest uncertainties in d~’~(O) [and hence in its coefficients d~,’~] would be enhanced by the factor l/Xn up to infinity (when n goes to infinity) and so the result becomes meaningless. In the more general case of a continuation between two arbitrary curves, one surrounding the other, the simple formulae (3) and (4) are, of course, no longer valid, but the eqs. (5) to (9) as well as the general features, namely that lim 1/X~=oo

(10)

remain unchanged. We have just seen that in continuing from inner to outer curves, the stream lines of the information flow diverge among them with increasing strength in the remotest dimensions of the functions’ Banach space. It is therefore clear that, if we had been provided with some supplementary piece of information, telling us that our result should remain confined inside some “object”, even infinite dimensional but “thinner and thinner” (infinitely flattened) in the higher dimensions, we would have been able to control the “spread off’ of our initial information flow. A compact set, for instance, has the above property. On the other hand, compactness has been intensively studied by mathematicians and a whole variety of conditions leading to compactness are known. For instance, if for a scattering amplitude “measured” [approximated by d(s)] in the data region F1,

If(s)

—

d(s)1r1
(11)

we could add, on the remainder ~‘2 of the cuts F, a boundedness condition of the type f(s)I~2
(12)

the problem of analytic continuation of the data d(s) to any interior point of the cut s-plane would become stable. Fortunately, in the last decade, much attention [31was devoted by the theorists to the problem of obtaining bounds for the amplitude: owing to the extreme importance of this subject for practical problems, the reader is kindly invited to consult the papers [31on this subject before starting any actual computations. In fact, as we have just emphasized, it is the shape of the “stabilizing object” [determined by inequality (12)] which is the “leading entity” in the process of analytic continuation, so that the latter inequality is of greatest importance for the figures we get. Since, in many practical situations, the right-hand side of inequality (12) is known only up to a constant (e.g. Froissart bounds, etc.), we have split off its functional dependence p(s), so that the computation could be performed for different values of the constant M, if this is found to be necessary. There is [41a smallest value forM (we shall call it M0), M0

=

inf(sup f(s)I/p(s)),

f

sEI~

(13)

for which there is still a holomorphic function f(s) satisfying both formula (11) and formula (12), and below which the set of “admissible” functions becomes null. This is an important quantity (functional) for problems correlating “in an analytic way” low- and high-energy data, asymptotics, and so on (it is important also for the poles and zeros search [5]), to which the second part of this paper is devoted (see problems Bi and B2 of part 2). In the first part of this paper, once given M (greater than M0), we shall: Problem Al: find the whole set and its “centre of mass” f(s) (~ the optimal extrapolation) of the values of all holomorphic functionsf(s) satisfying (11) and (12) at every interior points of the complex cut plane [6]; Problem A2: for every interior point s, find [7] the value d(s) yielded by the optimal weighted dispersion relation. Since in problem A2 the optimization is carried out only among the class of the variously weighted Cauchy integrals (which would be tautologic if the data were actually error free), d(s) has, of course, less predictive value than f(s). Nevertheless, since this program is less time-consuming than that connected with problem Al, it can be used every time a quick result is wanted, or when the size of the errors or uncertainties about M or p(s), would make the solution of problem Al too pedantic. A final remark: the error condition (11), as it stands, is a sharp one; one could nevertheless run the programs

M. Ciulli, S. Ciulli / Analytic extrapolations 1

219

for some (say, three) values of e, to find results with different confidence levels. This is why, as in (12), we have split in formula (11) the functional dependence of the error corridor in the form of the function a(s). For practical reasons, the functions a(s) and p(s) are normalized to 1 at the boundary point 5~between F’1 and F2.

2. Mathematical statement of the problems Al and A2 We shall suppose that the scattering amplitude is holomorphic in the complex s-plane cut along (_oo, s1] and 00), and that the data function d(s), i.e. both the real and imaginary parts of the (approximately) measured amplitude, are given along the line segment r1 [s2, 5~].More complicated analyticity domains, such as Martin’s axiomatic one [1,8] for the partial waves, present no special difficulties: only the form of the conformal mapping f(s) (see below) will be affected. [S2,

The best analytic extrapolation Problem Al: Given the complex function d(s) and the real function a(s) both defined for ~2 5~)and the constants and M, find for every s of the cut plane, the centre f(s) and the radius fl(s) of the values taken by the functions of the set 9 of holomorphic functions satisfying formulae (11) and (12): If(s)—d(s)I~1
If(s)

p2


As is apparent from appendix 1, this set of values fills densely a circle whose position and size depends on s. In order to solve the problem Al (see appendix 1), the conditions (11) and (12) have to be brought into a more symmetric form, using the exterior * weight functions C0(s) and C~(s)defined by the moduli condition ICo(s)I=(~/M I ~ w()~=

~:

1/u(s) onF1, ~1/p(s) onF2.

(14)

( 15

Then, with the definition d(s)= (Co(5)Cw(S)d(s). 0,

on F1, onF2,

(16)

f(s) = C0(s) C~(s)f(s), everywhere in the cut plane,conditions (11) and (12) merge into the single one: Ij~s)—~(s)I~1+p2
(bothonF~and F2).

(17)

As shown in appendix 1, every f~s)satisfying (17) can be written via a chain of homographic transformations by means of a holomorphic function i,(INf subjected only to the condition (18) and otherwise arbitrary. (For more details the reader isrefered to appendix 1 or to refs. [6,9]). The numberNf of these transformations depends on the degree of accuracy with which wet’Nf(S)satisfying would like to represent the every “non-analytic (18) fill, for s, a part”, d2, of d’, in eq. (A.2). Since the set of the values of all possible ‘~ circle of centre ‘YNf = 0 and radius flNf = 1, and since the homographic transformations map circles onto circles, the centrefof the values of the f’s can be found by recurrence (see ref. [6] or [9]) in terms of the Taylor coeffi*

An exterior (or outer) function

means an analytic function without zeros.

M. Ciulli, S. Ciulli /Analytic extrapolations!

220

cients i,LikJ of the various functions

i,l./k(Z)

defined in appendix 1:

1’

YNf~°~ ~7Nf=

~

Yk-1

77k—t

— —

~l

~k—1.0 s-

TIk SI

ii

[(1— I~k~1,0I)(l IWk—1,0

i~i

2~lr1

)RI’ +

+ ~k_1,o~*~)]/(I1 5_*

*

Vk—i,OS Yk

2

e ~ç~.rio

—(Ni) f(~)=diG~)+d 2 (~)~~Yo(~); rio -~

—

÷~k-1,0~YkI

-,

Irik~k~1,0~l)},

i 5-2 71k~’k—1,0S

(19)

.

Here

s’

=

~(si +s2

—

2so)s

+ (Si

+s2)s0

—

2s1s2~/{(s—

5O)(52

(20)

Si)~

is the mapping which transforms the s-cut plane onto the unit disk [the (upper lip of the) data region ~2
=

Optimal dispersion relations As was emphasized in the introduction, a less ambitious (and much simpler program) is provided by the optimization (only) inside the class of continuation methods using weighted dispersion integrals. The epistemologic value of the result is of course lower, but it can nevertheless be successfully used when the errors c are great, or/and the knowledge of the stabilizing condition [Mp(s)] is poor. In any case this program is much less time-consuming than that connected with problem Al. To give a mathematical formulation to this second problem, let us note that if the boundary valuesf(s’) of the holomorphic functionf(s) had been known on the cuts with absolute accuracy, we could have computed f(s) at every interior point, either by the conventional Cauchy integral (23)

ds’f(s’)/(s’-s),

2iri + “2

or by any weighted integral f(s)

ds’ f(s’)g5(s’)/(s’

=

2irzg5(s)

-

s),

(24)

where g~(s’)is any holomorphic function (not identical to zero) in the variable s’, with arbitrary dependence of the second argument s. If, however, in (23) and (24) we use, instead of f(s’), some approximant d(s’) of it (which in general is not the exact boundary value of some holomorphic function), the symmetry (the tautology) between the various formulations (24) of the Cauchy integral is broken. Indeed, we shall get different results for different g’s:

d[g] (s)

2il()

J

ds’ ~

‘‘

d~’1(s)

2h1()

,J~ds’ ~~s)

(withg’ *g).

(25)

M. Ciulli, S. Ciulli /Analytic extrapolations!

221

Hence a meaningful mathematical problem arises, namely: Problem A2: Find among all tautological formulae (24) whose symmetry is broken when instead of the exact f(s’) some approximation d(s’) satisfying (11) is introduced in the integrand, that one [namely that weight function g5(s’)] which leads for every s to the smallest deviation from the exact function f(s). As shown in appendix 2, the solution of problem A2 is yielded [7]by the formula

f ds’ ~(s, s’) C0(s’; e/M)C~(s’)d(s’),

cl(s) = ~

(26)

where P(s, s’) is Poisson kernel of the domain and C~(s’) and C0 (s’, elM) are the weights defined by the boundary conditions (14) and (15). Since both the Poisson kernel and the weights C0 and C~have simple forms in a circular domain, it is again convenient to use the mapping ~(s) defined in eq. (20). Indeed, the Poisson kernel of the unit disk has the wellknown form [here ~ = r exp(iO), ~‘ = exp(iO’)]: 2)/[1 2r cos(O 0’) + r2], (27) 2ir ?(~, eiO’) = 2ir Re[(eiO’ + ~)/(ei6’ ~)] 2ir 1(1 r

I

—

I

—

—

—

while the C weights can readily be computed by means of the Schwarz—Villat formula. Indeed, the function C(~), having by definition no zeros in the disk I~i~ 1, in C(~T)is also holomorphic there, so that its real part Re ln C(~) ln C(~)Iis harmonic and hence may be computed from its boundary values [seeeqs. (14) and (15)] by means of the Poisson kernel (27): Re in C(~)=-~— 2ir

f

do’ Re[(ei0’

+ ~)/(ei0’

—

~)]in

C(e~°)I,

(28)

}.

(29)

so that C(~)= exp

2ir

do’ [(ei0’ + ~)/(ei8

—

~)]in

IC(eiO )I

For instance, in the case of Co(~,M/e),replacing in eq. (29) in IC 0I by [seeeq. (14)] In 1 In(e/M)

0 on F1 and by

on F2, we get

Co(~e/M) = exP(_ln(M/e)[0.5

+

~-

ln(1

+

i~)/(l it)] ~-

(30)

.

Eq. (29) can be successfully written in terms of u(O) and p(O) also for the weight C~(~), defined in eq. (15). In order to save computer time, an alternative method is nevertheless preferred: ln(C~(~)), being holomorphic in the unit ~-disk, can be expanded there in a Taylor series. The coefficients of the latter can then be computed, once and for all, during the first call of the subprogram EXFUN. The function C~(~) can then be quickly computed for every value of ~, by summing and exponentiating the above series. Outputs of the analytic extrapolation subroutine package Besides the Central Analytic EXtrapolation (CAEX) [f(s), defined in eq. (21)] and the PoissoN-weighted EXtrapolation (PNEX) [seeeq. (26)] , this subroutine package produces also the values of the CauchY-weighted EXtrapolation (CYEX):

J

21TiC0(~e/M)C~(~) d~’[C0(~’; e/M) C~(~’)d(f)]/(~’~), together with (four) other extremal functions to be described below.

-

(31)

Al. Ciulli, S. Ciulli /Analytic extrapolations!

222

The Cauchy extrapolation (31) has the merit of optimizing the Cutkosky L2-norm problem [10,11] (see also ref. [9]), i.e. the problem we get when replacing the conditions (11) and (12) by those written for the L2-norm: dO f(elU)_d(eth)12/u2(0)
(32)

dO f~&°)I2/p2(0)
(33)

It should be emphasized that the Cauchy-weighted extrapolation J~s)is nothing but what we would get if the negative frequency part ~ 2(exp(i0)) [see eq. (A.2)] of the weighted data J(exp(iO)) of eq. (16) were neglected in f(s). Indeed, the Cauchy integral acts on a(exp(iO)) = CoC~d(exp(i0))= d~(exp(iO)) + ~i2(exp(i0)) [see eq. (A.2) from appendix 1], like a projection operator ~ d~’d(~’)/(~’ ~) = ~i (~). = e’~, (34) -

~‘

2rn. since, for ~I< 1,

±~ d~’d2(~/(~ ~) =0. -

(35)

2iri

In other words J(~)~d1(~)/~C0(~ e/M) C~(~)}

(36)

and hence CYEX is nothing but one of the terms of CAEX, or of the other extremal functions EXEX, EXEO, EXMO or EXEOO. We shall come back to this subject at the beginning of the next section (see, for instance, the remark 3.1.1 c) where we shall discuss the properties of the different extrapolation functions. For some theoretical problems it might be important to construct analytic functions with constant or given modulus on some part (F2) of the cuts. For this kind of problems one might resort to the (holomorphic) EXtremal EXtrapolation EXEX, which saturates, by construction, the conditions (11) and (12). Finally, the other three extremal functions which may also be computed by means of this analytic extrapolation package are related to the situation when, decreasing c, M, or both of them, the set ~ of admissible holomorpluc functions satisfying (11) and (12) shrinks down to a unique, extremal function. For instance, if the right. hand side of the weighted problem (17) decreases down to the value of the Hermitian norm ~ of the Fourier coefficients matrix defined in eq. (A.7), the function f0 (s) corresponding to the unique weighted function .i~ (s) still satisfying eq. (17) (modified in this way) is called EXEO (the EXtremal function corresponding to eo). It is merely of academic interest, but the next two extremal functions might also be of practical interest. The first of them, EXMO, is nothing but the unique function fM0 (s) which survives when Mis replaced in eqs. (11) and (12) by its smallest permitted value M0 [see eq. (13)]. Hence, among all the holomorphic functions which pass through the error corridor (11), E)(M0 is precisely that one which has the least modulus on F2. The second function will be denoted by EXEOO [~ ~ (s)], and represents the best approximation of the data cl(s) on F1 when Mis given. It corresponds to the extremal problem (11), (12) in which was replaced by ~ A more detailed discussion of the properties and uses of these two important numbers, ~ and M0 (actually they are nonlinear functionals of the data function) might be found in the second part of this paper, where the subprograms EMZERO/EPZERO are described. 3. Description of the analytic extrapolation package 3.1. Purposes and input quantities The analytic extrapolation package was designed to compute, at every interior point of the complex cut s-plane,

223

M. Ciulli, S. Ciulli /Analytic extrapolations I

the value of each of the seven functions (entries) described in section 2. But before any of these entries are called, the input data have to be introduced via the subroutine ANALYT(S0, Sl, S2, NPOINT, SEXP, DATA, ERROR, FBOUND, NOFBD, EPS, BOUND). ANALYT has a preparatory mission and therefore has to be called only once, at the beginning of the program: upon calling the specific subprograms, it performs the conformal mapping, directs the weighting of the input data, and loads the COMMON/PWDR/ with the relevant information [mainly the conformal angles and the C~(~)-weighted data] to be used in the remaining part of the program. Remark 3.1.0 CYEX and PNEX are both quick programs. But for the other entries big matrices of coefficients have to be computed, while for EXMO and EXEOO also some time consuming extremal problems have first to be solved. Therefore, if one should like to save time, it is preferable to call at once the same entry point for all points s of interest. Hence (see section 3.3), these large matrices are computed only once for each of these functions. The difference between the time needed for the first call of these entries and the subsequent ones (for the different s’s, but for the same EPS and BOUND) might be as large as 200 to 1. 3.1.1. CYEX The complex function CYEX(S, EPS, BOUND)

dV(s), defined in eq. (31) represents the optimal weighted

2-norm problem (32), (33) is used instead of the sharp

Cauchy extrapolation (Cutkosky [10,111) when anL

error channel (11) and bound condition (12). Therefore: Remark 3.1. la CYEX could be valuable in problems in which the errors are not due to some theoretical prejudice (wrong parametrization of the data, wrong branch of the phase shifts, etc.), but are merely statistical. Remark 3.l.lb Although the present program is designed for internal point extrapolations, CYEX, as well as the other holomorphic * extrapolations EXEX, EXEO, EXMO and EXEOO, can be continued up to the boundaries. As has been proved elsewhere [12] (see also section 2.4 of ref. [9]), if the scattering amplitude is subjected to a smoothness condition of the Holder type, the Cauchy-weighted extrapolation converges (when 0) to the latter, also for boundary points. —~

Remark 3.1.lc As has already been emphasized at the end of the last section [cf. eqs. (36), (21) and (19)], CYEX enters as a first term each of the more elaborate functions CAEX, EXEX, EXEO, EXMO and EXEOO. 3.1.2. PNEX The complex function PNEX(S, EPS, BOUND) (~d(s)) [see (26)] represents [7] the best estimate we can find using weighted dispersion relations with a general data function d(s) satisfying the conditions (11) and (12) (it is not necessary that the actual errors should be symmetric or random around the actual amplitude!). Remark 3.1.2a As has been emphasized in appendix 2, PNEX, like CAEX, is not holomorphic: they both coincide by construction at each interior point s, with some (optimal!) analytic function, but since these optimal functions are differ-

ent for different s, they are themselves not analytic. On the boundary, for instance, d(s) reproduces faithfully the

*

f)

The central analytic extrapolation CAEX (~ and the Poisson-weighted extrapolation (PNEX

functions.

d)do not represent holomorphic

M. Clulli, S. Ciulli /Analytic extrapolations I

224

input data (and hence it is ~0 on F’2!): this is a direct consequence of the Poisson kernels, namely that of producing harmonic functions with prescribed [see eq. (A.18)] boundary values. Therefore: Remark 3. l.2b

PNEX should not be used for points which are too close to F’2. In general, PNEX can be used along with CYEX when quick evaluations are wanted, for big errors, or when the knowledge ofM and p(O) is poor (see below, remark 3.l.3a), especially when theoretical biases are suspected; in this latter case PNEX has neat advantages over CYEX, which works in a satisfactory way (see remark 3.1 .la) only when the errors are distributed statistically. (Remember that on F2, “the errors” are never distributed statistically!) 3.1.3. CAEX The complex function CAEX(S, EPS, BOUND) (nsf(s)) [see eq. (21)] gives by far the best analytic extrapola-

tion one could get for a given d(s) satisfying (11) and (12). Nevertheless, Remark 3.1.3a Since CAEX is a powerful extrapolation procedure, and hence is very sensitive to “wrong” inputs [toinexact error function eu(s) or wrong function bounds Mp(s)], we have to be cautious about this point. If we are in doubt about the value of M or e, we should first check that the value of BOUND (= M) is not below the least permitted value of M, namelyM0. We can check this point by watching the printed output or using the real function EMZERO (EPS, SZERO, 0) described in part 2. If little is known about a and p, it is preferable to use PNEX. It should also be borne in mind that in contradistinction to CAEX and to the other three extremal extrapolations, PNEX and CYEX depend on EPS and BOUND (e and M) only via the ratio EPS/BOUND, and are thus less vulnerable to a bad choice. Remark 3. l.3b Like PNEX, CAEX not being a holomorphic function, it should not be used for points too close to F2. Of course, CAEX being by definition the best estimate one could get combining analyticity with the conditions (11) and (12), none of the other entries will give better results in the framework of this information. Nevertheless, if some supplementary smoothness information is available (or assumed), CYEX (see remark 3.l.lb) or one of the other holomorphic extrapolations could be used also (caution!) on F’2. 3.1.4. EXEX The complex function EXEX(S, EPS, BOUND) constructs the extremal holomorphic function saturating both the error-channel and the bound condition: EXEX(s)— d(s)I ~ea(s),on F’1,

EXEX(s) =Mp(s),

on F2.

Remark 3.1.4 The function EXEX (together with EXEOO, see remark 3.l.7b, below) is especially useful in problems in which, beside the approximate data on F1, also the modulus off(s) on F2 is given. In these cases (iff(s) is also HOlder continuous on F2), EXEX provides a stable extrapolation even on the cuts. 3.1.5. EXEO(S, EPS, BOUND) This function is just the function f0(s) which survives if one replaces in the weighted problem (17), by the norm e~of coefficients’ matrix of eq. (A.7). Remark 3.l.5a The modulus of

[EXEO(s) d(s)]/C~(s)is —

constant

all along F1, and IEXE0(s)I/IC~(s)= f0(s)I/p(s) is con-

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stant along F

2. This property of the extremal function EXEO (valid also for EXEX, EXMO and EXEOO) may be useful in some problems. 3.1.6. EXMO(S, EPS, BOUND) This function returns the function fM0(s) which, among all holomorphic functions satisfying (11), has on F2 the least modulus [but divided by p(s)!]. Remark 3.1. 6a Owing to the property (3.l.5a), IfM0(s)I/p(s)Mo,

for ailsE F2.

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Remark 3.1. 6b In the data region (on F1) the approximation is not so good since the bound (11) has to be saturated identically (see remark 3.1.5a) for every s E F1: IfM0(s)

—

d(s)I/u(s) = e,

for all s E F1.

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Remark 3.1. 6c Owing to its definition EXMO does not depend on its third argument, i.e. on BOUND. 3.1.7. EXEOO(S, EPS, BOUND) This function returns that holomorphic function f~00(s),satisfying (12), which gives the best analytic approximation to the data d(s), given on F1. Remark 3.1. 7a —

d(s)!/u(s)

e,~,

all over F1.

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Remark 3.1. 7b Owing to a property complementary to remark 3.1.6b, EXEOO would have the largest permitted value on F2 and hence represents a bad approximation there. This property of EXEOO (see also 3.1.4) of saturating the modulus condition (12)

If~00(s)I/p(s)= M,

all over F2,

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may nevertheless be very useful in some theoretical problems [2] in which the actual value ofM is known, as, for instance in (A) phase-shift analysis or (B) in the construction (without N/D equations!) of unitary amplitudes on the right-hand cut F2 which approximate in a best way some complex data function (and/or some bound conditions *) given on the left-hand cut F1. Remark 3.1. 7c In analogy with 3.1.6c, EXEOO does not depend on its second argument, EPS. 3.1.8. ETA The maximal error of PNEX or CAEX at point s may be computed by calling CABS(ETA(s)) immediately *

A bound condition is indeed very similar to a data condition: if, in problem 3.1.7b, the partial wave data d’(s) is given only along the region P’1 (of the left-hand cut 1’i), and on the remainder, r’j, only a function bound is known, one would merge both conditions into a single one.

226

Al. Ciulli. S. Ciulli / Analytic extrapolations!

after the above two function names: ETA(s) should not be called after CYEX, EXEX, EXMO or EXEOO: The error of CYEX depends on the smoothness condition assumed on the boundary, while that of EXEX, EXMO and EXEOO equals that of CAEX, multiplied by a factor which varies between one and two. Never call ETA after other extrapolation function but PNEX and CAEX. 3. 1.9. The quantities to be provided by the user are the following 3.1.9a. Input arguments ofANALYT SO real, the end point of the data region [can be left undetermined, since it is redefined by ANALYT (SO = SEXP(NPOINT))]. Sl real, the rightmost point of the left-hand cut; if there is only a right hand cut, take SI positive, = the rightmost end of the cut (1010 if the cut extends up to infinity) S2 real, the leftmost point of the right-hand cut (Si
All the variables described above represent the input of the first subroutine of the package ANALYT, to be described in section 3.2. Once ANALYT has been called (only once, at the beginning of the program), the analytic extrapolation may be performed using the seven entries, just described, of the function subprogram EXEX, called for the different points of interest s, and for different over-all error channel widths EPS( e) or bound constants BOUND (= M). Namely, these two constants are first returned by ANALYT, being by definition the factors which multiply the normalized error width a(s) and the normalized function bound p(s) [both are normalized to one at the end point of the experimental region: O(5~)= p(s~)= 1]. Nevertheless, before calling the extrapolation functions, these scale factors can be changed at will, as often as we wish. (The user ought nevertheless to observe remark 3.l.3a.) On the other hand, in order to save computer time, remark 3.1.0 should be observed strictly. Further, if EPS is set negative, a signal is transmitted to the subprogram (FF) performing the recurrence (19) permitting to write the last analytic function ~Nf as a Blaschke product, whose number and parameters might then be given by the user via COMMON/EXEXFF/NZK,ZK(1O). We are grateful to G. Atkin for pointing us the interest of this problem *~ The negative sign of EPS is further disregarded. With the exceptions 3.1 .6c and 3.1 .7c, the relevant arguments for the extrapolation functions are hence: S *

complex value of the point from the cut s-plane where the extrapolation of the data is meant. If, G. Atkin, private communication, see also ref.

[131.

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227

exceptionally, we would like to have s on the cuts, we should note that the actual values on the cuts are assimilated with those infinitesimally above them (they are mapped on the upper halfof the unit semicircle): EPS real, equal to [seeeq. (11)]; if set negative, some supplementary parameters might be introduced, as explained above. BOUND real, equal toM [seeeq. (12)]. Further, with the exceptiondiscussed above (when EPS is set negative) no COMMON variables have to be provided by the user, since the latter are meant rather as a link among the various subroutines of the present package. 3.1.10. Usage As already emphasized, in a normal run only the subroutine ANALYT (called only once, at the beginning) and the different entry points (see 3.1.1 to 3.1.7) of the function subprogram EXEX have to be called by the user’s subprogram. All other subprograms of the package listed below are automatically called by ANALYT and EXEX.

Remark 3.i.iOa The subprograms marked with (e), see below, are also contained in the EMZERO package, to be discussed in part 2. The analytic ANALYT EXEX ZETA I THETA 1 STH1 EXFUN CYWGHT FF PSITGR PSI FBOUND

extrapolation package contains 14 subprograms subroutine (e) complex function; entry points: CYEX, PNEX, CAEX, EXEX, EXEO, EXMO, EXEOO, ETA complex function (e) real function (e) real function (e) complex function (e); entry points: EXFUN, CW, CW1 complex function; entry points: CYWGHT, PNWGHT complex function; entry points: FF, FF0, CRHO subroutine subroutine real function (e), has to be provided by the user if a non-constant bound is assumed (if NOFBD = .FALSE.) MOEOO subroutine (e) COEF subroutine (e) NORM subroutine (e) The description of the above subprograms (except the last three) will be given below. The description of MOEOO, COEF, and NORM will be deferred to part 2. — — — — — — — — — — —

— — —

3.2. The subroutine ANAL YT (SO, Si, S2, NPOINT, SEXP, DATA, ERROR, FBOUND, NOFBD, EPS, BOUND) 3.2.1. This subroutine has the preparatory functions already announced (see section 3.1) and has to be called (only once) before the entry points of the extrapolation subprograms EXEX. The meaning of its arguments, except the last two, was explained in section 3.1.9; EPS = e and BOUND = M are output variables and are the constant factors in front of the normalized error corridor u(s) [eq. (11)] and the normalized function bound p(s) [eq. (12)]. Indeed, ANALYT normalizes the input error array ERROR(I) [corresponding to, see 3.1 .9a, eu (s = SEXP(I))] to its last component value [EPS ERROR(NPOINT)] such that u (SO SEXP(NPOINT)) should be equal to I. If a non-constant function bound FBOUND(X) is supposed to exist (NOFBD = .FALSE.), the latter will also be normalized to its value at s = s0: BOUND = FBOUND(S0). If no function bound is assumed (NOFBD = .TRUE.),

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228

the value of BOUND is set equal to 1. If the internal control parameter LWRITE is set greater or equal to zero, the values of EPS and BOUND are printed before returning to the principal programs.

3.2.2. The COMMON blocks The analytic extrapolation package uses COMMON blocks which are meant as internal links among the different subprograms. Therefore, as was already stated in 3.1 .9b, with the unique exception when EPS is set negative, none of the common arguments has to be determined by the user’s program. The first common block COMMON/ANLYT/SSO, SS1, SS2, NFOUR, KFLAG transmits to EXEX the relevant parameters (SO, Si, S2) of the conformal mapping ~(s),the number of the nega-

tive Fourier coefficients, as well as an internal flag, set zero when ANALYT is called. COMMON/PWDR/LWRITE, GRAD, NPT, THETA(501), CARL(5Ol), ERR(501), N, KSWICH, ER is divided in three parts,/PWDR1/, /PWDR2/ and /PWDR3/, and represents the main link between ANALYT and the other subprograms of the package. Its description will be deferred to section 3.7 where EXFUN is discussed since, indeed, the latter suprogram uses most of its arguments. We shall nevertheless come back to this subject in

the next subsection (3.2.3). COMMON/PSS/EPSS, PS(50,50) represents a link between EXEX and the subprograms involving the computation and the use of i~1~jcoefficients enteringjifl [seeeqs. (19), (A.13), etc.],while COMMON/EXEXFF/NZK, ZK(1O) allows the user (if EPS is set negative) to transmit the parameters of the Blaschke factors entering the last unimodular but otherwise arbitrary analytic function ~ Otherwise (if EPS is set positive) it is disregarded and is set equal to 1. 3.2.3. One of the first tasks of ANALYT is that of computing the angles THETA(I) = 01, which corresponds by exp(iO~)= ~(SEXP(i))to the points of the F1 part of cut, where the data are given. This is done by means of the THETA 1(RS, SO, Sl, S2) function subprogram, for RS = SEXP(i). Owing to the property of the ~-mapping,all the 0’s corresponding to the upper lip of the F1 cut, are between 0 and ir/2. The THETA(I) and the corresponding data values CARL(I) = DATA(I) (I = 1, NPT; NPT NPOINT) are then loaded in COMMON/PWDR/. Once the normalization factors EPS and BOUND (see 3.2.1) are known, ANALYT loads the array ERR(I) of COMMON/PWDR/ [aliasthe array EPS(I) appearing in EXFUN, see section 3.7] with the values of the normalized error corridor ERR(I) a (SEXP(I)) ERROR(I)/EPS,(I = 1, NET). Now, if a non-constant function bound is assumed to exist (setting NOFBD = .FALSE.), a number of equidistant points THETA(I), (I = NPT + 1, ..., N), is produced by ANALYT, corresponding to “new data” on F’2. The corresponding values of ERR(I) p (s(I)) FBOUND (S(I))/BOUND are computed via STH1 (the inverse function of THETA 1) and stored into the last (1 = NPT + 1 N) locations of ERR(I) [alias EPS(I) in EXFUN]. Hence: if NOFBD = .TRUE., ERR(I) has N = NPT NPOINT components, containing the values of a (SEXP(I)). if NOFBD = FALSE., the first NPT NPOINT components of ERR(I) correspond to u(s), the last ones [NPT+ 1, ..., N(> NPOINT)] correspond to p(s). [Inview of eq. (16), the corresponding CARL(J = NPT + 1, N) are all set equal to zero.] At this point, ANALYT calls EXFUN. Indeed, in its actual form, ERR(1) contains [see eq. (15)] the reciprocal boundary values C~(e’°)I, which are the main ingredients to be used by EXFUN in the computation of the weight function C~(~). As was explained in section 2, the Taylor coefficients of ln(C~(~)) are computed and then stored in the internal memory of EXFUN, permitting quick subsequent calculations of C~(~) or C4.,’~).As a result of the fact that the control parameter KSWICH was set equal to 2 (for details, see section 3.7), EXFUN returns in CARL the partially weighted data, — —

d~(e’°i)=

C~(e’°f)d(e’°!)

if f~NPT, (O<0.
0,

if NPT
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229

while the array ERR(J) is reloaded with the “weighted” u(J) and p(J) (i.e. iC~(exp(i01))lERR(J)). Since the latter should all be equal to 1 (apart from computational errors), this enables the subroutine ANALYT to check the effi-

ciency of EXFUN; if we set the internal control parameter LWRITE > 4, both the input data and the conformal angles, as well as the deviations from 1 of the “weighted errors” are printed. Summing up, at the moment in which ANALYT is left: the commons /PWDR/ contain the conformal angles [in the real array THETA(J)] and the corresponding d~(exp(i01))[in the complex array CARL(J)] as well as the “weighted” a(J) and p(J). EXFUN contains the Taylor coefficients of ln C~(~), which, as mentioned, permits quick calculation of C~(fl or ~ every time this is needed. —

—

3.3. The complex function EXEX(S, EPS, BOUND) 3.3.1. Entry points CYEX the “Cauchy extrapolation”, d(s), PNEX the “Poisson extrapolation”, cl(s), CAEX the “Central analytic extrapolation”,f(s), EXEX the “Extremal extrapolation” saturating both the error channel and the bound condition, EXEO the “Extremal extrapolation corresponding to en”, i.e. to the function fo(s) of the weighted problem (17), EXMO the “Extremal extrapolation corresponding toM0”, EfM0, EXEOO the “Extremal extrapolation corresponding to ~ ~f~00, ETA the real part of ETA contains the error bound of PNEX or CAEX, if called immediately after these latter. —

—

—

—

—

—

— —

3.3.2. Arguments S complex value, the point of the cut s-plane where the extrapolation is performed, EPS (as e), the over-all error channel width [the scale factor multiplying the normalized error channel u(s)]; if set negative (this sign is further disregarded), the last ~j,function might be written by the user as a Blaschke product, as explained at 3.1.9b, BOUND (asM), the scale factor multiplying the normalized function bound p(s). — —

—

3.3.3. Procedure Once the input data have been prepared and stored by ANALYT in the commons /PWDR/, the entries of EXEX may be called. This subprogram computes first, upon calling ZETA!, the ~ corresponding to the point s, and then directs and returns the result of all the calculations needed in the process of analytic extrapolation. The meaning of each of the entry points was already discussed in section 3.1. As explained there, the scale factors EPS and BOUND may be changed at will before calling the EXEX entries. In this way, the flexibility of the program is large enough, but for the sake of time saving we should nevertheless observe remark 3.1.0. Indeed, the program, as it stands, recognizes the situations in which the old values of EPS and BOUND are equal to the new ones. In these cases it avoids much computation no new Fourier coefficients are computed and no new extremalM0 or 600 problems are solved. Further, the old i,Li coefficients are used. Of course, all these remarks are pertinent only to the “total approach” entries (CAEX, EXEX, EXEO, EXMO, EXEOO), but not to CYEX and PNEX, whose task is only to call the corresponding entries (CYWGHT and PNWGHT) of the subroutine CYWGHT, performing the weighted Cauchy/Poisson integrals and dividing the result by the value of C~(~T), produced by EXFUN (entry CW). As explained in section 3.1, CYEX and PNEX do not depend on the absolute value of the parameters BOUND and EPS, but only on their ratio R elM. On the other hand, EXMO and EXEOO, by their very definition (see remarks 3.1.6c and 3.1.7c), depend only on EPS and on BOUND, respectively. Beside LWRITE, which controls —

M. Ciulli, S. Ciulli / Analy tic extrapolations!

230

* facilities, an important (internal) knob to be changed (if this is felt necessary by the user) is the number Nf as NFOUR of negative frequencies Fourier coefficients. NFOUR is taken into account by the subroutine MOEOO, which directs the solution of the M0 or ~ problems, and by the PSI or PSITGR subroutines,

the writing

which compute the ~

k coefficients. Of course, the bigger NFOUR, the better the accuracy of the final result. But the amount of work increases with the square of NFOUR in MOEOO [since e~is the norm of the matrix (A.7)] and with a much more complicated law in the PSI subroutines. The PSITGR subroutine computes the l,hIjk coefficients entering the recurrence (19) in an algebraic way, by means of eq. (A.l3). But the number of combinations of integers satisfying (A.14) increases dramatically withNf (this was a classical problem, treated by Gauss). Hence forNf greater than 35 ‘l”I’, the work usually done by PSITGR is committed to a parallel subroutine PSI, which computes the 1l1j,k coefficients by means of Cauchy-type integrals (see section 3.1 1). If EXMO or EXEOO is called, the program needs the value ofM0 or that of 00, respectively. Indeed, these values enter [via the weights C0(~, e/M0) and C0(~,e00/M~)]the Fourier coefficients of a’2(exp(iO)) and hence also the 1~1~j coefficients. M0 or may be calculated by the (time-consuming) MOEOO subroutine; but, as has already been said, in order to avoid wasting time, the old iii’s together with the old values ofM0 or Ø~are memorized. Indeed, a new call of MOEOO, COEF, and PSI is performed only if at least one of the following three propositions are false:

KEY.EQ.KEYOLD,

BOUND.EQ.OLDB,

EPS.EQ .OLDE.

Here, OLDB = “old bound”; OLDE = “old epsilon”. The ~ depend not only on the various ratios (Ri = elM and so on) which parametrize the weighted Fourier coefficients, but also on the absolute value of the “epsilon” used, transmitted to PSITGR or PSI via “EPSS” of the COMMON/PSS/. To summarize, the ~1i’s are computed with: Ri Ri Rl Ri

= = = =

e/M0 e00/M e/M e/M

and and and and

EPSS EPSS EPSS EPSS

=

e,

= ~~1’ = =

e~, e,

for EXMO; for EXEOO; for EXEO; for CAEX and EXEX. ‘~“~‘

(43)

Once the values of l,Llkj are stored in COMMON/PSS/ the function FF (or FF0, if EXEX, EXEO, EXMO or EXEOO were entered) cojnputes the “negative Fourier contribution” —x d~Nf)(~) (~/~.Nf)70(~.) to the weighted extrapolation function of eq. (19), (Nf as NFOUR, c= EPSS). The positive frequency part a’i (~)of the weighted data function d(~)is just the value of CYEX (see remark 3.1.1 c) for the corresponding ratio R1 of eq. (38). The computation of the weighted function (19)J(~)= d1 (~)+ d~Nf)(ç~ (~ftNf)70(~)is hence completed by calling CYWGHT(Z, Ri). Dividing by C0(Z, Ri) and CW(Z) (a subsequent entry of EXFUN), we finally get JI3~)defined

7

by eq. (21). 3.3.4. Entry ETA We could compute the value of the error bound of the central analytic extrapolation f(s) or of the Poissonweighted dispersion relation d(s) at some given s upon calling CABS(ETA(S)) immediately after the corresponding calls of CAEX(S, EPS, BOUND) or of PNEX(S, EPS, BOUND). The value of s may be altered (in this case, a

slightly longer computation will result for the CAEX error bound), but of course not the values of EPS and BOUND. Since the last ~I’f,k~stored in the memory are used, ETA(S) should not be called after EXEX, EXEO, EXMO, or *

** ~‘~‘~‘

The “LWRITE steps” are 0, 1 and 3: for LWRITE ~ 3 the name of the entry used as well as the returned value displayed; for LWRITE ~ 1 some information concerning the entries 2 to 5 is given, if the input e provided to the subprogram is less than the computed e00 (and LWRITE ~ 0), a warning message is displayed. The threshold may nevertheless be changed by the user, if this is felt to be necessary. Indeed, the extremal function for the weighted problem [i.e. the unique function which survives when we replace e in eq. (17) by e0] corresponds to the initial value of the ratio R = elM [nevertheless, in eq. (A.l0) we have to replace e by rol. In contradistinction to what happens with EXMO and EXEOO, e~is here the norm e0[d, elM] of the matrix (A.7) of the Fourier coefficients computed for the initial ratio e/M.

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231

EXEOO. According to the theory, the error bounds corresponding to these latter three functions lie between the value of the error bound of CAEX and its double (eCAEX ~ EEXEx, 6EXM0~6EXEOO ~ 2CAEX). Neither should ETA(S) be called after CYEX, since, on the one hand, the Cauchy-weighted dispersion relation J(s) may not satisfy the conditions (11) and (12), and on the other hand the approximation power of d(s) depends mainly on smoothness conditions, which are outside the scope of the present paper. Hence ETA should be called only after CAEX or PNEX. 3.4. Complex function ZETA] (S, SO, Si, S2) 3.4.1. Purpose Performs the conformal mapping from the cut s-plane to the unit disk, such that the point s = Sl and S2 are mapped into —1 and +1, respectively, while s = SO + ie are mapped into ±i. 3.4.2. Arguments S = the complex point to be transformed. SO = real, the end of the known (experimental) region F Si S2

1. real, the threshold of the left-hand cut. = real, the threshold of the right-hand cut. Usually Si
[—°°,

also in problems having a unique cut running between S2 and Si. In this case, S2 < SO
cut extends to infinity, take Si

=

1010

or so.

3.4.3. Procedure The mapping f(s) = (~/i~? \/ii7)/(\/i~7 + \/f’7), (44) where s’ = [(s1 +52 2so)s + s2)— 2s1s2]/[(s s0)(s2 Si)], is performed. If the points fallsjust on the cuts (i.e. if Im s as 0, while Re s lies between and Si or between S2 and +oo), the value of ~ returned lies on the upper half unit circle [e.g.s = SO is mapped into ~(S~) = +i]. —

—

5~(5j

+

—

—

—°°

3.5. Function THETA1(RS, SO, Si, S2) 3.5.1. Purpose Performs the transformation from the boundary of the cut s-plane to the boundary of the unit disk. 3.5.2. Arguments RS = the current value of Re(s) (s being on the upper lip of the cuts). SO = the real upper end of the experimental (known) region. Si = the real threshold of the left-hand cut. S2 = the real threshold of the right-hand cut. 3.5.3. Procedure The mapping performed is 0

i~’ln~(s)

for sEF1 +I’2,

where v(s) is defined by the function ZETA! [see eq. (44)]. The values of angles 0 returned all lie between 0 and ir.

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232

3.6. Function STH1(TH, SO, Si, S2) 3.6.1. Purpose Performs the inverse transformation of THETA1 (from the boundary of the unit disk) to the cuts of the complex s-plane. 3.6.2. Arguments TH = the current 0 value (real, between 0 and is). SO = the real end of the known (experimental) region. Sl, S2 = the real threshold of the left- and right-hand cuts, respectively. 3.6.3. Procedure By inverting the transformation: exp(iO)

=

3.7. The complex function EXFUN(Z) 3. 7.1. Purpose Produces the exterior (outer) weight function CW(Z) and also its derivative CW1(Z), both on the unit disk. 3. 7.2. Theoretical considerations It was shown that one can bring the general variable error-channel and bound conditions to a “canonical” form [eq. (17)] by introducing a suitable weight external function, defined by: l/u(0),

(46)

=

l/p(O), Since, by definition, C~(~) has no zeros in the unit disk, ln C~(~) is also holomorphic, and so it can be expanded

in a Taylor series:

1nC~(~)~I~ ck~.

=

(47)

Here the Ck’5 are real quantities because C~(~) is a real-type function. On the boundary of the unit circle, exp(iO), (47) takes the form of a Fourier series which can be further split into its real and imaginary parts: —lna(0),

Re[ln C~(e’6)]as ln Cw(e’°)j=

ck cos kO

is

is

——<0 <—, 2 2 is 3is —lnp(0), -~<0<~-

=

=

k=0

(48)

The coefficients Ck are completely determined by eq. (48). Standard formulae tell us that c 0

=

f dO ln IC~(e’°)! =

—lr

1

7/2

0

dO ln a(0)

—

f

dO ln p(O), (49)

ir/2

ck_fdOln~Cw(e16)tcoskO~IdOlna(0)coskO__fdOlnp(O)cosk0,

k=1,2

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233

where, on the right-hand side we have used the fact that the error functions u(0) and p(0) are symmetrical with respect to 0 —0. The values of u(0) and p(O) are transmitted to EXFUN in the form of an N-component real array EPS(J) corresponding to the angles 0~= T(J), both contained in the COMMON/PWDR/; so: -+

EPS(J) = o(O~) for 0 EPS(J)

=

s~o~

p(O~) for~<0 ~ is.

(50)

If p(O) as ~, the last °~=N[contained in the array T(J)J should be ir/2 rather than is; in this way the integrations in (49) (which are performed by means of approximating parabolas) stop at ir/2, in complete agreement with the vanishing of the ln p(0) for 0 between ir/2 and is! Every time a new c1 coefficient is computed, a test is performed 2jC~(e’°)I dO

171

—

IT TEST=—f ln

~/ d
(5i)

.~

J can go up to NF (the maximal number of Fourier coefficients, fixed at the beginning of the subprogram), if TEST remains larger than a given error level, ERROR. As soon as TEST becomes smaller than ERROR, NF is set to the last J value and the already computed coefficients (c 1,/ = i, NF) are stored, for future use, in the array C(J). If the value of TEST becomes negative * at some step J, the computation of the Fourier coefficients stops, and only the previous (J i) coefficients are retained (though the corresponding TEST is, of course, larger than ERROR). If LWRITE ~ 2 (a key parameter transmitted by the COMMON/PWDR/), a message concerning the number NF, the TEST and the ERROR is printed. —

Sometimes, better results can be achieved by using a modified form of representation, the so-called Fejer sums,

defined as arithmetic means of the corresponding Fourier sums. To this end, starting from an internally adjustable rank NFJ = FEJER*NF fixed at the beginning of this subprogram, the Fourier coefficients CNFJ+K are weighted by a factor (1 (K NFJ)/(NF NFJ + i)), i.e.: —

—

—

NF+ i_NFJ)’

cNFJ+K ~cNFJ+K(i

K

1,2

NF+ i —NFJ.

(52)

The number of the weighted coefficients is fixed by the user giving a fractional value to FEJER: 0 ~ FEJER ~ In order to enable a check on the accuracy, we compute the relative discrepancy U(i) between the G~’F(exp(i0j))Igiven by the truncated series C~(exp(iO~)) = exp(~~o Ck exp(i01)) and the exact IC~(exp(i0,))~ defined by eq. (46). Since, as already stated, the values of u(0’) and p(0’) are transmitted to EXFUN in the form of an array EPS(i), the discrepancy U(i) can be written as U(i) = Ii

—

ICt’JF(e1°i)jEPS(i)~,

(53)

and if LWRITE ~ 1, the sup U(i) (~the uniform error) is printed out. *

For the mathematically minded reader, we add that TEST should indeed tend to zero through non-negative values (provided no machine errors enter

the game) N

271

-i--

J’

271o

~f(O)I~do > ~ j0

due to Bessel’s inequality

4,

and to the Parceval relation, valid for complete orthonormal systems: 2 do

i’— f tfto)1

=

~ c/.

M. Ciulli, S. (‘iulli / Anal tic extrapolations I

234

Lastly, the computed coefficients allow C~Fto be represented at interior points ~ = r exp(iO) of the unit disk as well. Dropping the indexNF, we shall write: C~(exp~

ck~}.

(54)

This is done by the entry CW(Z). For some purposes, the derivative C~(~) of C~(~) may also be important. It can be computed using the entry CW1(Z) (of course, CW1 returns the derivative with respect to the conformal variable ZETA1, not s!). 3.7.3. Usage ofthe subprogram Upon calling EXFUN(Z) once, the expansion coefficients of ln(C~(~)) are stored, so that a quick computation of the outer function or of its derivative could be easily performed calling CW(Z) or CW1(Z). 3. 7.4. Input values Z complex variable whose modulus has to be less than or equal to 1. Further, some quantities in the commons /PWDR1/, /PWDR2/ and /PWDR3/ (see section 3.i) should be assigned: LWRITE integer governing the amount of printed output; if a) LWRITE = 0, no output is printed. b) LWRITE 1, the maximal relative deviation U(I) [seeeq. (53)] is printed; also the first entry in EXFUN is reported. c) LWRITE 2; the number of the computed Fourier coefficients NE, the TEST [see eq. (51)] and the ERROR values are also printed. d) LWRITE ~ 5, the headings EXIT EXFUN and EXIT FROM EXFUIN1 are also printed, every time the entries CW or CW1 are used (otherwise nothing is printed when the entries CW or CW1 are used). GRAD real parameter which does not concern EXFUN; it controls the fineness of the intervals of integration in some routines of the package, where special integration methods have to be used (CYWGHT and COEF). NPOINT integer; does not concern EXFUN, = the number of °k ~ ir/2, corresponding by the conformal mapping to the experimental points SEXP(K). T(SO1) real array, holds the angles O~,where the input C~(exp(iO1))I~’ is given. ~‘

~‘

CARL(5Oi) complex, output array; containing, if KSWICH = 0 (see below), the values of CW(exp(iOJ)) at the points 01as T(J). If KSWICH ‘ 0, the values already existing in the memory are not erased, but simply weighted with C~(exp(iO1)).Hence, if, initially, CARUJ) holds the data d(exp(iO/)), it will return d~(exp(iO1)). EPS(5O1) real array, containing the input values of C~(exp(iO1))[’.If KSWICH (see below) equals 2, these values are finally multiplied by the (approximate) modulus of the weight function produced by EXFUN. Since the values EPS(J)*ICw(exp(IO1))l should all equal to i, this enables ANALYT to perform a check of the activities of EXFUN. N integer, the number of angles T(J) where the input ICW(exp(iO/))~ ~ are given. KSWICH integer, control parameter. As already explained, if KSWICH = 0, the values C~(exp(iO1))are stored in CARL(J). If KSWICH ‘ 0, these values are multiplied by those already existing in the memory. If KSWICH = 2, the EPS(J) (see above) are multiplied by C~(exp(i01))j. 2 norm, of lnIC~(~)~. ERROR real value, the desired degree of accuracy in the L 3.8. Complex function CYWGHT(Z, R) 3.8.1. Entry points CYWGHT(Z, R) PNWGHT(Z, R).

M. Ciulli, S. Ciulli / Analytic extrapolations!

235

3.8.2. Purpose CYWGHT (and PNWGHT) computes at every point z of the unit disk (lzj ~ 1) the values of C0(Z, R)-weighted Cauchy (and Poisson) integrals [see eq. (30), for the definition of C0(Z, R)]. It is supposed that the data function presented in COMMON/PWDR/ is of real type, i.e. d(exp(iO)) = d*(exp(_iO)), and that it vanishes identically on the left-hand semicircle F2: d(e’°)”‘O

(55)

for~
If we want to compute the Cauchy- or Poisson-weighted integrals, for data not vanishing on F2 , we should call CYWGHT twice, as explained below [seeeq. (61) from the remark 3.8.2a]. The distinction between the functions J(s) or d(s) [seeeqs. (3i) and (26)] and the CYWGHT or PNWGHT integrals consists in the fact that the latter contain only the discontinuous weight C0(Z, R) but not the smooth one C~(z). Thus, if we are interested in d or d, the data function which CYWGHT or PNWGHT are presented with in COMMON/PWDR/ should already be weighted by C~(exp(iO)).Further, the result of the integration should be divided by C~(z).In the present form of the analytic extrapolation package, the weighting of the data contained in COMMON/PWDR/ is performed during the first call of EXFUN. Then, C~(z)can then be quickly calculated using its subsequent entry point CW(Z). Remark 3.8.2a If d(exp(iO)) or d~(exp(iO))does not vanish identically on F2, we should add to the integral between —sr/2 and ir/2 done by CYWGHT (or/and) PWGHT, a second one on F2. To this end, advantage could be taken of the functional relation: 1(—~e/M) ~ C C0(~e/M)’~ C~ 0(—~M/e). (56) Using (56)we get (0”” 0’ —is): 6’”37r/2

2isCo(~R)6=71/2 1 2isCo(~R)

=

2isC

—

0 Co(_eiO; R) d(ei(0”471))/[e10’

f

=

dO’ ei@ Co(etO ; R) d(&O)/(eiO

—

dO” e’

0(—~i/R)

O~71/2

do” eiO Co(eiO ; i/R)d(ei(0”+n))/[eiO”

—

(—~)].

(57)

The right-hand side of eq. (57) is just equal to CYWGHT(—Z, l/R), if the latter is now fed (via COMMON! PWDR/) with d(exp(i0” + is))) rather than with d(exp(iO”)). Further, if d(exp(iO7)) is known, let us say, atN’ points 0~(j =N, N +i, ...,N +JV’ 1) between ir/2 and is, the d(exp(iis + iO’)), 0< 0’
ir—O~asO;_ir,

*

O”=O’—is

*

(58)

The mapping THETA1(S) identifies the points of the ~‘2cut with those slightly above it, and transforms them into the quadrant [71/2,71]. On the other hand, the values d(exp(i8~~ + ir)), (0
M. Ciulli, S. Ciulli / A nalytic extrapolations I

236

one has d(e~°~)= d*(&o).

(59)

From eq. (58) it follows that the 0 and the 0” are supplementary. Now, since the angles 0”, between 0 and is/2, contained in the COMMON/PWDR/, have to be ordered according to their magnitude, we shall label the o;’ ‘s as follows: 0N+N’—l (= 0, since 01v+N’—! is). 0~= is °N+N’~-2 is —

=

—

=

—

O~

(

is/2, since

0N

(60)

is/2).

Putting all this together,

2isCo(~R)

I

dO e’6 Co(etO, R) d~eiO)/(ei0

—

(61)

0 =

CYWGHT(~,R)[{O 1}, {d1~,j= 1

N]

+

CYWGHT(—~,l/R)[~is 0N+N’/},{d~+N’~J},j = 1

N’],

—

where the quantities in the square brackets represent the angles and the data arrays transmitted to the respective subprograms via the COMMON/PWDR/. A similar formula could be written for the Poisson-weighted integral PNWGHT. Remark 3.8.2b The usual (not weighted) Cauchy or Poisson integrals, with data vanishing or not [in this latter case eq. (61) should be used], may be performed by putting R equal to 1 in the above formulae. Indeed, C0(Z, 1) as 1. 3.8.3. Description of the arguments a) Function arguments Z complex variable, with ZI < 1. R real parameter, equals the ratio e/M which enters the definition (30) of Co(Z, e/M). b) Arguments from the COMMON/PWDR1/ and COMMON/PWDR2/ LWRITE here dummy integer. GRAD real argument (in degrees) controlling the magnitude of the integration intervals (see section 3.8.4). NPT number of points. THETA real array containing NPT angles between 0 and is/2. D complex array containing the corresponding d~(exp(iO1))= d(exp(i01)) C~(exp(iO1)). 3.8.4. Procedure The Cauchy—Riemann equations for in C0(z), written in natural coordinates on F, 3Im(ln C0) ~s

=

~3 lnICoI

(dsas do, dn~as—dr),

(62)

an1

tell us that in the vicinity of the discontinuity points 0 = ±ir/2of the modulus of C0 on F, the phase of C0(Z, R) quickly. Therefore, the conventional numerical integration techniques cannot be used here. We shall nevertheless take advantage of the facts that: I) the rapidly varying part of C0(exp(iO), R) around x as is/2 0 0 can be factorized in a simple way:

varies extremely

—

Co(e’°,R) Svc(X,R)exP(finRlnX~

(63)

M. Ciulli, S. Ciulli / Analytic extrapolations I

237

[here Svc(X, R) is the “slowly varying part of Cs], and ii) the integrals of exp[(i/2ir) ln R ln X] as x~’~ R/2ir) multiplied by the different powers of X, have simple, close forms [putting r = (1 /2ir) ln R]: xj+1

f

dX XirXk

=

{X~~~rx7~’~~}/(k + I + ii).

(64)

—

x! Hence, we shall split off the slowly varying part of C0, multiply it with the weighted data d~(exp(iO))and with the Cauchy kernel CY(Z, X) as eiO/(eiO

—

Z)

1/(1

=

.f.

iZei~~),

(65)

and approximate this product, in each interval [X1,X1~1],by second-degree curves: CY(Z, X) d~(e’°) Svc(X, R)

bkXk,

X1 ~ X ~ X1+1.

(66)

For Poisson integrals, the CY of eq. (65) should be replaced by PN(Z, X) as Re[(e~°+ Z)/(eiO

—

Z)]

=

Re[(l— iZeiX)/(i

+ iZeiX)] .

(67)

Combining (63) with (66), the integrals have then, in every interval [X1,X1+1], expressions of the form ~~akbk. Now, since the CY(Z, X), PN(Z, X), and Svc(X, R) occurring in (66) have all close analytical expressions, the overall accuracy will be enhanced if the initial intervals are subdivided into finer ones (especially if some of them are too large). Indeed in this case only the values of d~(exp(iO1))have to be found by interpolation. All these operations are automatically performed in CYWGHT, if 01+1 O~,for some j, exceeds the value of GRAD (in degrees). Finally, it should be mentioned that if IZI approaches 1 (if FAR = .FALSE.), an automatic subtraction is performed in order that the singular part of the Cauchy or Poisson integrals should be computed in a suitably proper way. Thus CYWGHT and PNWGHT may be used up to the boundaries. —

3.9. Complex function FF(Z, IV) 3.9.1. Entry points FF FF0 FFFF CRHO 3.9.2. Purpose FF(Z, N) returns, for every z inside the unit disk, the value of —x1(z) [see eq. (A.6)], i.e. the contribution of the negative Fourier coefficients to the optimal (weighted) extrapolation 7[see eq. (19)]: FF(~,N) as —xi (~)= a’~”~(~)

—

~~-)

(68)

Here, N is the number of negative Fourier coefficients taken into account via ~ coefficients with which FF is fed via COMMON/PSS/. FF0 equals the right-hand side of eq. (68) in the case in which in the eq. (17) e is replaced by the norm e0 of the matrix (A.7), where only a unique analytic function —x1(z) survives. Further, the real part of the CRHO holds the value of the error bound of FF(Z, N), [see again eq. (i9)], at the point Z = ~(s) Re(CRHO(Z, N))

=

hoG) = (e/~)flo.

(69)

MI. Ciulli, S. Ciulli / Analytic extrapolations I

238

If EPS (transmitted to FF from EXEX via corn mon/PSS/) is set negative in the calling program, a do-loop is activated, which permits the last ~,1ifunction (~PN~ defined in eq. (18); see also the footnote of appendix 1)to be written as a Blaschke product: ~ZK

~N~(~([Z~k(k)1/L1

~Z~(k)ZI).

This might provide the user with some supplementary parameters [introduced via COMMON/EXEXFF/NZK, ZK (10)] which might be precious [13] in some extremal problems using EXEX(s) or EXEOQ(s). 3.9.3. Arguments a) Function arguments

Z complex variable, with IZI < I. N integer, the number of negative Fourier coefficients taken into account. b) Arguments in the COMMON/PSS/EPS, PS(51, 51) and COMMON/EXEXFF/NZK, ZK(10) EPS real, the value of e(or e~,if we wish to use FF0). PS real matrix, containing: K: K = 1 N) of the ~‘K_l(Z) functions, defined 1) the Taylor coefficients PS(K, J) = 1~K—1,J—1(J = 1 in appendix 1: 2) the elements PS(K, N + 2 K) on the second diagonal coincidence with the ‘y~computed (also by the PSI and PSITGR subroutines) for Z 0. ZK complex array, containing the parameters (zeros) of the NZK-Blaschke factors of IJAf. —

3.9.4. Procedure

—x(z) is computed mainly by means of the recurrence (19). For small IzI, an alternative method is used in order manipulation of negative powers of z; this branch which is automatically used if IzI is small, might be forced calling FFFF instead of FF. Once —~(z)hasbeen computed, to get the value offof eq. (21), we need also the Cauchy-weighted integral CYWGHT: to avoid

=

CYWGHT(~,R)/CW(~)+ FF(~,N)/{CW(~)C (70)

0(~,R)}, 3.10. Subroutine PSITGR (N, COEF) 3.10.1. Purpose To compute the Taylor coefficients PS(K + 1, J (A.Il)(KO,...,N—l;J0,...,N_1 —K).

+

1) of the recurrent unit module functions

~IK(Z),

defined in

3.10.2. Arguments a) Subroutine arguments

N number of negative Fourier coefficients. COEF real array, holding the Fourier coefficients c1, c2 c_N of the weighted data function. b)Arguments contained in COMMON/PSS/EPS, PS(51, 51) and COMMON/P WDRJ/L WRITE, GRAD EPS real, input value of e PS real matrix, containing the output ~,1i’s. LWRITE, as usual controls the printout: if LWRITE 2~’2 a table of the ~1i~coefficients is printed which permits the user to verify in a direct way if their moduli are less than 1, as they should be. Further, if LWRITE ~ 6, the Gauss integers k~defined in appendix 1 [see eq. (A.i4)] are also printed.

239

M. Ciulli, S. Ciulli / Analytic extrapolations I

3.10.3. Procedure The first PS(l, J) = ,tio,j_~coefficients are computed from the c_1’s according to the definition (A.I0). The remaining i4’jj’s are then computed by PSITGR in a purely algebraic manner, by means of eq. (A. 13). 3.11. Subroutine PSI(N, L, ERROR, COEF) 3.11.1. Purpose To compute the Taylor coefficients 4’(K + 1, J + I) of the recurrent unit module functions ijJ~(Z) defined in (A.l 1) (K = 0, ..., N 1; J = 0, ..., N 1 K). In fact, PSI is an alternative to PSITGR, if N is large. If the num~ ber of the Fourier coefficients will never exceed 35, this subroutine might be discarded. —

—

—

3.11.2. Arguments a) Subroutine arguments N integer, number of negative Fourier coefficients. L integer, controls the number of integration intervals (~N*2L).L increases if the required accuracy is not obtained. ERROR real, controls the accuracy of the integration. COEF real array, holding the input Fourier coefficients. b) COMMON/PSS/EPS, PS(5i,51) and COMMON/P WDR1/L WRITE, GRAD EPS real, input value of e. PS real matrix, holding the output ~,1i(K + I, J + 1)’s. LWRITE commands as usual, the output. 3.11.3. Procedure In contradistinction to PSITGR, which is purely algebraic, PSI computes the ~(i’sin an analytic manner. Since, as is proven by the very existence of formula (A.13), the first/Taylor coefficients of —

~1K_1,oI/[l

—

(71)

iP~_1,otPK_1(z)] ,

depend only on thej + 1 Taylor coefficients of 1IIK_1(Z), we could replace in eq. (71) the function ,1I~_1(z)by its truncated Taylor series, without altering the results. Then 1, J = 0,1,..., N —1 K, (72) 2iri ~dz’ ~4~(z’)/z’~~ PS(K + 1, J + l)as ~K,J =~~T where ,t4~(z)iscomputed from the truncated ,/,t~r(Z)[from (c_N/c) + ... + (c... 1/e) zN_l] via the recurrence (71). The accuracy of the integration is checked only for the PS(I, J) with I IPROOF (IPROOF is set equal to 2, in the actual form of the program). This check is performed as usual, by making a comparison between the integrals using N * 2L intervals and N 2L1 intervals. If this difference is larger than the prescribed ERROR, L is increased by one and the test is repeated. —

“~

*

3.12. Subroutine MOEOO (EOO, MO, KEY, NFOUR) This subroutine, as well as the following two, COEF and NORM, will be given a detailed description in part II, where theM0 and e~ problem is extensively treated. Only the list of the arguments will be given here. a) Arguments E00 real, input (e) if KEY = 1, output (e~~) if KEY = 2 MO real, output (M0) if KEY = 1, input (M) if KEY = 2 KEY integer, control key. NFOUR number of negative Fourier coefficients.

M. Ciulli, S. Ciulli /Analytic extrapolations I

240

b) COMMON/PWDRJ/L WRITE, GRAD LWRITE integer, controls the written output GRAD real, controls the size of the integration intervals of COEF. 3.13. Subroutine COEF (EML, NC, SUM) a) EML NC SUM

Arguments real, EML= —(1/2is) ln(e/M) integer. real, the output c_NC Fourier coefficient of the C0(Z, e/M)-weighted data. b) COMMON/PWDRJ/L WRITE, GRAD and COMMON/PWDR2/NPT, THETA (501), DATA (501) The same arguments with the same meaning as those described in section 3.8.

3.14. Subroutine NORM (NN, C, ERROR, EPSIL) a) Arguments NN integer, number of coefficients. C real array, holding the C_K coefficients which form the matrix (A.7). ERROR prescribed relative error (real). EPSIL real, output variable, equals the norm (e0) of the matrix (A.7). b) COMMON/PWDR/L WRITE LWRITE as usual, controls the written output. 3.15. Real function FBOUND(X) Finally, a real function has to be provided by the user as explained in section 3.1, if a non-constant function bound p(s) is assumed to exist on F2. Here, Xis real (X ~ F2).

Appendix 1 1(exp(iO)) on the unit circumference [see eq. (16)], we shall construct here all the functionsf(~), Given analytic in the unit disk and satisfying (17): —

~(e’°)I

~

c

,

0<0< 2ir

(A.1)

.

The method (see ref. [6], or section 4 of ref. [9]) is not based on usual dispersion relations (Cauchy-type integral formulae), but merely on the Fourier decomposition of the data function on the unit ~(s)-circle: d(e’°)

~ c~e”0 +

Cn

em0 as~ 1(e’°) + ~2(eiO) ,

(A.2)

with C_k = ~

f~(e10) eikO

dO.

(A.3)

Here, both the positive frequency (d1) and the negative frequency (~2) parts are present, this being necessarily required by the vanishing of~’on F2. While ~1(exp(iO)) can be continued analytically inside the unit disk in a trivial way,

~)

as~ ~

(A.4)

241

M. Ciulli, S. Ciulli / Analytic extrapolations I

some problems arise with the “non-analytical part”, ~2(exp(iO))(in practice we shall approximate ~2(exp(iO)) by its first Nf terms, so that d2(exp(iO)) ~d~.IVf)(exp(iO))). In fact, writing a Cauchy integral means nothing but taking into account only d (~),while totally neglecting d2(~).Thus, if we write: (A.5) the problem (A.l) is reduced to the problem of finding a holomorphic function x1(~)such that ~~f~(e’°)l
+

(A.6)

.

This problem can be solved (see appendix E of ref. [9]) only if e is greater than the modulus (0) of the largest eigenvalue of the matrix c_~

c_1

c_Nf

0 (A.7)

o

c_NfO

If this should happen, let us define the holomorphic function Nf

111o(~)= — (d~t’)(~) + x(~)).

(A.8)

According to (A.6), ~o(~)is subjected to the condition I

~oG)I

1

‘~

(A.9)

,

while its first N1 Taylor coefficients are fixed by those of d~1~),via eq. (A.8): = C_Nf/C,

~D0,,= C_Nf+1/

,

...‘

~PO,Nf—1= c_1/e .

(A.l0)

As is known, conditions of the type (A.9) are not altered by Blaschke transformations. We can hence define recurrently the iterated analytic functions *

~k(~)

r’[l,lIk_l(c)

—

~k_1,0]/[1

_1,o~k_1(~)1,

—

6 (and hence (holomorphy!) also for

fl

(All) < 1), we shall get the same condition

so that kL’k_1(~)I< 1 for ~ = e’ also forif~Pk(~) l~k(c)I
(A.12)

,

—

k Taylor coefficients are completely determined by the Nf ~

~k,n = (k 1,k2

kn+i)ki!k2!

~ ‘~‘fJ ...

II * ‘.Wk—1

kn+i! (1

—

k + 1 ones of

~k_1(fl:

0)

I~k_l,oI2)E”+t~~k~1,1

Here the sum is taken over all the sets of Gauss non-negative integers (k1, k2, k, +2k2 +3k3 +...+(n+l)k0+, “‘n+l

—

.

...

...,

.

~

kn

+

i)

(A.l3)

which satisfy: (A.14)

Hence the number of predefined Taylor coefficients diminishes with k, such that the function ~P,~N1(~),apart from a boundedness condition of the type (A. 12), is any arbitrary analytic function in the unit disk. Reversing eq. (A.ll), we can then compute ~1i~and hence x1(fl and)~),starting with such an arbitrary ,l’N1(~)(bounded *

Since the pole at ~-= 0 is annihilated by the zero of the numerator of eq. (A.i 1).

M. Ciulli, S. (‘iulli /Analytic extrapolations I

242

by 1). With some arithmetic [6], we can find in this way also the position of the centre ills), as well as the radius fl(s), of the set of values of all possible holomorphic functions .1(s) satisfying eqs. (11) and (12). These formulae are displayed in section 2 [see(19), (21) and (22)]. Since the function-transforms (A.1 1) have the property that, for ~ = exp(iO), they map the boundary of the 1~’k~_l-unit disk onto the boundary of the /1k-unit disk, then, if for some ~ = exp(i0) the arbitrary function is chosen to saturate (A.l2)so will all the ~,1k computed from 1,lINç via the reverse of eq. (A.l 1). This way (choosing for instance ~‘N~ ±1),one is able to construct the extremal functions which saturate the error-channel and the bound conditions. These functions may be useful in some theoretical problems, in which also the modulus off(s) is given. m,LINf(~’)

~

Appendix 2 Among all weighted dispersion integrals written for a general data function d(s) satisfying [f(s) —d(s)I~

1
I.t(s)IF2
,

(A.15)

we shall find here, following ref. [7], that one which leads to the smallest error bound, at every interior point s of the complex cut plane (problem A2). As already emphasized, all weighted integrals

~irig5(s) I

F1+F2

ds’g5(s’)f(s’)/(s’

-

s),

(A.l6)

[here the g5(s’) are holomorphic with respect to the variable s’ (in the cut s-plane) but otherwise arbitrary] yield exactly the same result, as long asf(s’) are the 100% exact boundary values of some holomorphic function. Once f(s’) is replaced by some approximant d(s’), and/or the integration is performed on only a part F1 of the cut F [this amounts to taking d(s) 0 on F2 = F\F1], the tautology of the integrals (A.16) is broken. Problem A2, then, amounts to finding that special g50(s) which corresponds to the smallest error bound of the result [for a general d(s) satisfying (A.15)]. For convenience, a) we shall use instead of (A.l5), the equivalent “weighted” problem [i~s)d(s)p1+p2 —

(A.l7)

<.

For the definition of~i(s)and of the weight factors C0(s) and C,,,,,(s), see eqs. (16) and (29). Further, (b) instead of the Cauchy kernel we shall use the Poisson one. Indeed, by its very definition, (1 /2is)f Ids’ I~(s,s’) a(s’) represents an harmonic function in D, whose boundary values coincide with that of a(s’), on F. Thus, for a quite general class of functions a(s’), defined on F, writing a(s)~f

Ids’I ~(s,s’)a(s’),

(A.l8)

1~’1+I~’2=F

we have 2a 0 and lini a(s) a(s’) s-÷sE F V/ If (and this time, only if!)j(s’ E F) are boundary values of some holomorphic functionj(s) in D, the real and imaginary parts of a holomorphic function being harmonic, one may write: ~s) 1J~ds’I~(s, s’)~s’), (A.19) 2ir~ =

*

=

A more general choice would be some Blaschke product.

M. Ciulli, S. Ciulli / Analytic extrapolations!

243

and along with (A.l9) also 1 fds’I~(s,s’)g5(s’)Jts’). 2irg5(s)

(A.20)

~.

The equality of the integrals (A.19) and (A.20) would have been violated iff(s’) were not the boundary values of an analytic function. [Notice also that (A.I9) and (A.20) can, of course, be rewritten in the form (A.l6), using suitable g-functions.] Stated in these terms, problem A2 amounts to finding the optimal ge(s), such that the “extrapolated” function I fIth’I~(s s’)g5(s’)~!(s’), 2irg~(s)1,

(A.21)

leads to the least error bound. We shall now prove that the best weights [with (A.21)] are g~(s’)= 1. Proof (see ref / 7]) From (A.20) and (A.21) and from the fact that the Poisson kernel ~(s,s’) is a real and positive quantity, we have for every interior points (since IJ~s’) d(s’)I~
V(s) —~[gj(5)I
(A.22)

g5~s, ~ r

The best error bound E1g1 one could find with the weight g5(s’) and with a general data function d~(s)satisfying (A.17) is then defined by V(s) —J111(s)I
2ir

=

(A.23)

e.

Indeed, the reader could easily imagine a data function ~ satisfying (A. 17), and which, for a given g~(s’),saturates the bound E[g~. But on the other hand, since g~0(s)is itself analytic in s, we have g50(s) —~--Jjds’I~?(s, s’)g~0(s’), 2ir

(A.24)

and hence, for every possible ~ and s, we get Ig50(s)I

,~-J~ds’I ~(s,s’)1g50(s)I,

i.e. E[g) ~E111 as

~.

(A.25)

Equality in (A.25) is possible only if g~0(s’)has a constant phase along F: g~0(s’)necessarily being analytic in s’, this means that g~(s’)should be a constant with respect to s’, and hence drops out of the definition of d[g~. Hence, the bound E111 = e is obtained only with d111(s)

-~--

J’Ids’I ~(s,s’) ~(s’),

(since ~(s E F2) = 0),

which, coming back to the unweighted problem, proves the optimality of “the Poisson-weighted dispersion relation” (26): 3(s)

f

2irCo(s) C~(s) Ids’I ~ (s, s’) C0(s’) C~(s’)d(s’).

(A.26)

The function cl(s), as it stands, is not holomorphic in (D). Nevertheless, it can be shown [7] that at every interior

Mi Ciulli, S. Ciulli/Analytic extrapolationsi

244

point s

0, cl(s0) coincides with some “optimal weighted Cauchy integral”: =

f

2isiC0(~0(s))C~(~o(s)) d~C0(~)C~(~) d(~)/[~

-

~

(A.27)

namely, with that one which is written in terms of the mapping ~~(S) which transforms the s-cut plane into the

unit ~odisk and brings the point ~ into the origin: I~~(s’ E F)I

=

I

,

~

=

0 ;

(A.28)

(the Cauchy weighted integrals are, indeed, dependent on the mapping ~ Of course the d(~(s))are holomorphic ins, but at each points = ~ cl(s) coincides with a different ‘(us)), and hence cl(s) is not analytic.

References [1]

Martin, Scattering theory, unitary, analyticity and crossing, Lecture Notes in Physics, vol. 3 (Springer Verlag, Berlin, 1969); A. Martin and F. Cheung, Analyticity properties and bounds of the scattering amplitude (Gordon and Breach, New York, 1970). [2] S. Ciulli, Nuovo Cimento 19A (1974) 621. [3] A. Martin, Stanford University preprint ITP-134 (1964); L. Lukaszuk and A. Martin, Nuovo Cimento 52A (1967) 122; A.

L. Lukaszuk, Nuovo Cimento 51A (1966) 67; B. Bonnier and R. Vinh-Mau, Phys. Rev. 165 (1968) 1923;

J.B. Healy, Phys. Rev. D8 (1973) 1904; G. Auberson, L. Epele, G. Mahoux and F.R.A. Simã~o,Nucl. Phys. B73 (1974) 314; Ann. Inst. Henri Poincaré 22(1975) 317; NucI. Phys. B94 (1975) 311; G. Mahoux, Bounds on scattering amplitudes and form factors, Lecture given at Les Houches Summer School (1975) P. 363, to appear; B. Bonnier, C. Lopez and G. Mennessier, Phys. Lett. 60B (1975) 63; C. Lopez and G. Mennessier, Bounds on the ir°lr°amplitude, CERN preprint TH 2235 (1976) Nucl. Phys., to be published; F. Schwarz, A new method for deriving absolute bounds on the ir°lr°scattering amplitude, Kaiserslautern preprint (1976).

[4] S. Ciulli and G. Nenciu, Nuovo Cimento 8 (1972) 735. [5]I. Caprini, S. Ciulli, C. Pomponiu and I. Sabba-StefI’nescu, Phys. Rev. D5 (1972) 1658. [6] S. Ciulli and G. Nenciu, J. Math. Phys. 14 (1973) 1675. [7] S. Ciulli and J. Fischer, Nucl. Phys. B24 (1970) 537. [8] G. Auberson and N.N. Khuri, Phys. Rev. D6 (1972) 2953; G. Mahoux, S.M. Roy and G. Wanders, Nuci. Phys. 70B (1974) 297. G. Auberson and L. Epele, Nuovo Cimento 25A (1975) 453.

[9] S. Ciulli, G. Pomponiu and!. Sabba-Stelinescu, Phys. Rep. 17C (1975) 133. [10] [11] [12] [13]

R.E. Cutkosky, Ann. Phys. 54(1969)110. R.E. Cutkosky, J. Math. Phys. 14(1973)1231. S. Ciulli and G. Nenciu, Commun. Math. Phys. 26 (1972) 237. G. Atkin, Ph.D. Thesis, University of Birmingham (1979).