Rational interpolation to functions of Stieltjes' type

*

3

... . ' ™ "• ' ' - * 'S

Rational Interpolation to Functions

of Stieltjes^ Type by

Jan Gelfgren

No 6 1978

UNIVERSITY OF UMEÅ

DEPARTMENT OF MATHEMATICS

S-901 87 UMEÅ

SWEDEN

of Stieltjes^ Type by

Jan Gelfgren

No 6 1978

L J

0. Introduction . 0.

1. Historical review 1.

2. Definition and properties of

the classes M(a,b) and M*(a,b) 8.

3. The method of Baker 14.

4. The rational interpolant belongs to M(a,b) 26.

5. Convergence results. 51.

6. Proofs of the lemmas from section 4. 60.

7. References 85.

0. Introduction.

In this paper we study rational interpolation to functions of ^

Stieltjes' type, i.e. functions that can be written f(z) - J ^(t) , Z"~L

where a,b are extended real numbers and a(t) is a bounded, non-decreasing a function.

Two polynomials Q^(z), P^n-^(^z) ^are defined by the relation (when the interpolation points are real and finite)

f(z)Qn(z) - = A(z) (z-x^ (z-x2) ... (z-x2n>, where A(z) is bounded at x^, x„»...,X2 £ |R [a,b],

n Pn-1^

In section 4 we show that the rational function .. . Q (z) .— can be written

Pn-l(z) — = / b dß(t) z_t— for some bounded, non-decreasing function 3(t).

ti a

This is also true when we interpolate at infinity and at complex conjugated points.

Pn-l(z)

The fact that _ Qn(z) ^{/ N} is of the same type as f(z) is used in section 5

to show some convergence theorems. The idea behind the proof of Theorem 1 of section 4 is revealed in section 3.

Acknowledgement : I want to thank Professor Hans Wallin for his guidance and encouragement during my work with this paper.

1. Historical review.

In his memoirs from 1894, "Recherches sur les fractions continues", Stieltjes [23] studied continued fractions of the form

1 ( la)

a^z + 1

a2 + 1

a3Z + 1

a4 + 1 a^z + . where a. > 0, i = 1,2,...,

He proved that if 00 E a. = ⁰⁰ then (la) converges to a function i=l 1

da(t) z+t

A more convenient way of writing (la) is

f(z) = / —ti— where a(t) is a bounded non-decreasing function.

0

|a¹z [ a² |a³z | a⁴ |a -5Z

The truncated fraction 1—^-1 + 1—+ 1—I + ... + r-^—1 (if n even) I a^xz I a² I a³z |aⁿ

is called the n:th convergent of (la). When putting zw = 1 and b = —, o a b. = 1 a.a. _{1 l+l} , (la) can be written

00 00

which converges to w / > when E a^ diverges.

With the identity

bl ^bl^b2

z + T - z + b. - ,

1 + —2 2

X

the continued fraction (la) can be transformed into (Ic) b

z + ^bi - ^bl^b2

(Ic)

Z + (b

2

V ~

3

z + (b^+b^) - .

A o (Id)

Z + \

Xo z + a2 - 2

X<*

z + " £_

z + a. - 4

where A = b , A. = b.b. - and an = b- , a. = b. + b. « o o'- i i î+l 1 1 J J J+l Now, zt = 1 gives

A t (le)

o o

1 + a-1 - 1 o

1 • a„t - V 1 + a^t

In 1903, van Vleck [28], started to extend the Stieltjes theory to continued fractions of the form (Id), in which the A are arbitrary positive numbers and the arbitrary real numbers. He was able to connect in certain cases these continued fractions with integrals of the form / . A complete extension of the Stieltj es theory to

—00 Z + t

these continued fractions was first obtained by Hamburger CL2 ] in 1920. The reader can find these results in the booksby Wall [27] and Perron [18]•

In his "Recherches ..." Stieltjes also proposes and solves the following problem which he calls "Problems of Moments":

Find a bounded non-decreasing function a(t) in the interval [0, such that its "moments" ^t da(t), n = 0,1oo ,2,..., have a pre­

scribed set of values

00

j tnda(t) = y , n = 0,1,2... (1.1)

0 n

Stieltjes makes the solution of the Moment-Problem dependent upon the nature of the continued fraction "corresponding" to the integral

7 ^{d a}^ A ^y° ^ n ^ ' = J ^ i 2 ⁺ 3 " ••• ⁽¹*²⁾

O Z Z

* 1J + _JJ + t-LJ + p-Ll + ...

I axz I a2 I a3z | a4 t

He shows that all a^ are positive and also that this condition is sufficient for the existence of a solution of the Problem of Moments.

The solution is unique when the series oo E a. diverges, thàt is when o 1

the continued fraction (1.2) converges; we speak of a determined Moment- Problem.

The continued fraction (1.2) may converge for certain z (to the value 4

oo # # ^

I(z,a)), while the series E (-1) y.z diverges for all z.

o 1

The Moment Problem is connected with the theory of Padé approximation (Padé [17] ).

Définition:The rational functionP^(z)/Q^(z) is called the [n/m] Padë approximant to the formal power series f(z) _if pn(z) and Qm(z) are polynomials of degree at most n and m, respectively,

t 0 and the formal product f(z)Q^m(z) - Pⁿ(^z) only contains terms of degree greater than n+m. In the sequel the [n/m]

Padé approximant to f(z) is denoted f[n/m](z)

Putting wz = 1 and using (lb) we can transform (1.2) into oo

I*(w,a) = w J ^ w(u - y w + p,w2 + ...) 0

wb i b.w i b0w i

^ + l_l + , ² I +

If f(z) = E y (*-w)n, where y = / tnda(t) and a(t) has infinitely

o ^{n n} 0

many values on [0,°°), then f(z) is called a series of Stieltjes, and it is known that the 2 n:th convergent of the continued fractions expansion of f(z) is just f[n-l/n](z).

Now the Moment-Problem gives us a condition assuring that f[n-l/n](z) -> g(z)^: oo

= / excePt on the negative real axis.

• • • • ri Many authors since Stieltjes have discussed conditions on ^iv

O

which will make f[n-l/n](z) -> g(z).

Carleman [5] in 1926 proved that

00 ^ l/2n

E ( ) = 0 0 i s s u f f i c i e n t . 1 yn

The reader interested in the Moment-Problem should consult the books by Achiezer [l] and Shohat, Tamarkin T203.

In a paper from 1968, Gragg [11] estimates the rate of convergence for Padé approximants and proves the

i/^{R m}

Theorem: If f(z) = f . and z £ (-°°,-R], then 0

|f(z) - f[n-l/n](z)I < C(z)

2n 1 - /1+z/R 1 + /1+z/R

where C(z) is bounded in any compact set disjoint from (-⁰⁰,-

In his proof he uses the convergents to a certain type of continued fractions.

Karlsson, von Sydow, in a paper from 1976 [13], use orthogonal polynomial to present convergence results on Pade approximation to series of

Stieltjes. In this paper they also show that the geometric degree of convergence in general cannot be improved.

Now, f[n-l/n](z) is the n:th order Gaussian quadrature approximation i/^R

to the function f(z) = J ^ (Perron [18, p. 200-201]).

0

Using this, von Sydow, in the paper [24 ] from 1977, proves the following

Theorem; Let f(z) = / 77^7—, z £ (-°°,-l] and cp(z) = 1/1+2 1

0 u z t — « s » i

Then |f(z) - f[n-l/n](z)| < ^ I—I —

- |/ïïî|-u- I l«z) I ) 1

where | dot | = / da(t) 0

In the paper [2] from 1969 Baker constructs a rational

function which interpolates to f(z) = jl/R at finite real 0 ^1+zt

points. He uses the relation f(x^o)

f(z) = l+(z-x^o)g(z) ' ^(1,3)

and shows that g(z) is a series of Stieltjes if f(z) is. This relation has its origin in the continued fraction 1(b),

He then iterates this procedure getting f(X )

f (z) = 2— (1.4)

1 + (z-xo)g1(x1) 1 + (z-x1)g2(x2)

1 + <z"xk)Sktl(z)

1/R da.(t)

where g. (z) = / —^—, j= 1,2 k+1.

J ~ 1+zt

Putting E 0 (1*4) gives a rational functions which inter­

polates to f(z) at 9x^ 9 • • • i f t w o p o i n t s a r e e q u a l t h i s m e a n s

that both the value and the derivative of the rational function coincide with those of f(z). He says that these rational interpolants converge to a series of Stieltjes.

Barnsley [4] in 1973 uses (1.4) to find bounding properties of the rational interpolants used by Baker. He says that the n:th convergent

n 1

can be written Aⁿ+ E where >0,X, > 0 and a. G (~°°,- —) ,

U k^z ^+ak ^{u k} k R

ak i a., k ^ j.

1/R dß(t) /VlitLCll J

0 bounded, non-decreasing function ß(t)

This means that the convergent can be written ^J -——- for some 0 zt

f[mk/nk](z) + f(z) = J , when lim ^- < 1.

a k-x» k

fr d ( t ) In this paper he also considers functions of the form f(z) = ^{J ,}

a where a(t) changes sign a finite number of times.

Nuttall and Singh, in the paper [*16J from 197 7, study Padé approximants to functions of the form

£(z) , ; «Ü. ,

S (t-z) /x+(t)

where the set S consists of a number of analytic jordan arcs, ending at the given distinct finite points d^ = 1,2,...,2&.

21

a(t) is a complex, non-vanishing function defined on S, x(t) = n (t-d.) i=l 1

-1/2 ~l/2

and x+(t) denotes the value of x (t) on a particular side of S.

von Sydow studies matrix-valued series of Stieltjes in the paper [25]

from 1977.

In the paper [22] from 1977 Stahl investigates functions of the type

f(z) = J ^ , where E c [R is compact and y is a complex-valued E t~Z

measure.

Goncar [9 ], in 1975, and Rachmanov [19] in 1977, have shown that f[n/n](z) -> f(z) = J z-t^ + w^en certain conditions on the

a

rational function r(z) and on a(t) are satisfied.

Goncar, in the paper [8] from 1978, using Gauss^ quadrature formula^

shows that the [n-l/n] rational interpolant to the function f(z) = f ^

J z-t

n

x

can be written £ where A > 0 and a, € (a,b), a.,

k=1 z a^k R R R j

if k 4- j . The numbers a,b are finite. When the interpolation points belong to a fixed set E cz (p \ [a,b], he shows that the sequence

n Ak °°

(f(z) - E —-— } ! z ak n-1 is uniformly bounded in compact sets in C v [a,b]

Goncar and Lopez, in the paper [10] from 1978, proves

Theorem 1: Let f(z) = f , where S cz |R is compact, and let (a,b)

— — s z_t

denote the smallest interval that covers S. Let a = {a ,,a

n n.| n>2

an 2n^ 9 n € N, be the interpolation points which belong to (t ^ [a,b] and are symmetrical with respect to (R. Let af = lim a

n-*»

belong to t ^ [a,b]. Let ^and be polynomials of

degree n-1 and n, respectively, which satisfy f(z)Qn(z) - =

A(z)«(z-a -)-(z-a n, 1 n,z ⁿ) ...(z-a n,zn ⁰ ), where A(z) is bounded at

an,l^,an,2^,##-,an,2n*

Pn-l^(z)

Then , .— converges uniformly towards f(z) in (C ^ [a,b],

\z)

vn

and for each K(ê t ^ [a,b] we have lim n-x»

P n-1 Qn

1/2n K

< e , -H

where H = H(S,K,a') = inf (g_(z,Ç) | z € K,Ç 6 a'}, g_(z,Ç) is

S s

Green's function for the region (t ^ S$ k(c (C \ [a,b] means that K is a compact subset of (t ^ [a,b].

Lopez, in the papers [14], [15] from 1978, interpolates to functions of

oo

the type f(z) = ^J . He shows a generalization of Carleman's 0

condition and he also proves that a sufficient condition for convergence

2n 1

of the [n-l/n ] rational interpolants is that lira E ~TZm__ ^{= 00}» n->°° k=l /1 a J

2n 9

here {a n, K. K.— JL - denotes the interpolation points, which are real and belong to [-°°,a], a < 0 and fix.

2. Definition and properties of the classes M(a,b) and M*(a,b)

Definition 1: If a,b are extended real numbers, and if

b do(t)

f ( z ) * S z - t a

b

for some bounded, non-decreasing function a(t), |a(t)| = Jda(t)>^

we say that a

f(z) € M(a,b).

Definition 2: If a,b are extended real numbers, and if g(z) - ;

&v J a 1 + zt

b

for some bounded, non-decreasing function ß(t), |ß(t)|= J dß(t)>0, we say that a

g(z) € M*(a,b).

Definition 3: We say that x G I(a,b) when x € |R ^ [a,b] and we say that X 6 I*(a,b) when - ^ 6 |R ^ [a,b].

Property l.If f(z) € M(a,b) then - — f(- —) € M*(a,b). > z z

Remark 1. European authors often use the class M*(a,b) of series of Stieltjes while russian authors use the class M(a,b), the functions of which they call Markov-functions.

Property 2:If

f (z) £ M(a,b) , f (z) t 0, then

f(x) ^>0, x _> b; f (x) <^0, x _< a.

This implies that

f(xj0*f(x2) _> Q * where ^x^>^x2 belong to the same side outside the interval [a,b].

Property 3;If

f(z) 6 M*(a,b), 0 < a < by

then

f(x) ^0, X > - f(x) _< 0, x ~ •

Property 4:If

f(z) € M(a,b), then

f(z+x^) G M(a-x^,b-x^ when x^ 6 |R.

Proof: f(z+x ) - J ÜÜi . J <tiÜ> , bjXl

I al J Z+X,-t a z^{- i t}t -x,) . 1 X Î a-x, ^J 1 z-u * u=t-x^

V «"i> •

V ¹ ^

1 Property 5:If

f

mco

i

R

f(z+x1) e M*(0, x ) when xx £ (-R ,°° ) .

Proof: This proof is taken from Baker [2, p. 819].

Let _{_} 1/R r da(t) d (t)

f(z) - / ITIF" •

By the change of variable of integration v = t

1 + tXi t

we have

1/R+x, v j ,-r---— )

f(2t ) , J 1 (l"VX1)dq(l~VX1

0 1 + zv

1/R+x^

f (z+x^) = J d ß(y) n 1 + zv

Remark 2: Property 4 is important for the calculations in section 6.

The fact w real axis.

The fact we need is that x-^ can be chosen from the entire

Property 6:If f(z) € M(a,b), a,b extended real numbers, then b

0 < lim iy f(iy) = / da(t) < 00 .

y-X» a

Proof: Since f(z) € M(a,b), we know that

b b b

cf \ • f da(t) r iy-t j /.v . r tda(t) iy f(iy) = iy* J . \ = J . _ da(t) + J -—'

7 J a iy-t a iy-t J a iy-t.

Hence

lim iy f(iy)

y

Since

b y-*» / a

/ da (t) + lim / tda(t) . iy-t

i. r tda(t)

lim J =

iy- t y-*» a

we see that tda(t)

Um j • iim ; <*2<ïi, R £ix>

y~ |t|<R ^ y~. 11] >R ly_t

b Iim

/

y

Now, and

lim

y-X» It

iy - t _< lim J

y-*» t<R

y-x» 1 1 1 < R

11 da(t)

xy-t -»• 0 , R fix,

t[da(t)

iy-t < / da(t).

1 1 1 > R

tda(t) iy-t hence from choosing R large enough we see that lim

y-Xx>

To characterize a function f(z) € M(a,b), we use

Lemma 2.1 f(z) € M(a,b) <=> f(z) satisfies (i) - (iii).

(i) f(z) is holomorphic in T+ = {z | Im z > 0} and maps T+ into

T = {w|lmw < 0}.

=0.

(ii) f(z) satisfies the inequality sup |yf(iy)| < C < 00 (z=x+iy).

y>l (iii) f(z) is continuous on IR ^ [a,b].

Proof of ^ : From Achiezer [1, p. 93] we see that (i) and (ii) imply

00

f(z) = / for some bounded, non-decreasing real function a(t)

Z L

—oo

We now need

Stieltjes-Perron^s inversion formula (Stieltjes [23, p. 72-75], Perron [18 , p. 188-90]).

Let

dg(t)

g ( z

) = J £1

z > o),

where 3(t) is a real function of bounded variation in every finite interval, such that

r ^dß(t> < co

J —00 ï^ltT I I ' Then

ß(d+0) + ß(d-O) ß(c+0) + 8(c-0) _ 1 f ^T

2 2 - " Hfl? J Im g(5+irV>d£- C

From (iii) we see that f(z) is continuous on |R \ [a,b], and Stieltjes-Perron"s inversion formula now tells us that a(d+0) + a(d-O) = a(c+0) + a(c-O)

for any interval (c,d) c |R \ [a,b]. This means that a(t) is constant on (-°°,a) and on (b,°°).

Proof of => : The calculation in Achiezer [l, p. 93] shows (ii). (i) and (iii) follow easily from the definition of f(z).

To characterize a function f(z) £ M*(0,~) , we use R

Lemma 2.2 f(z) £ M*(0, <=> f(z) satisfies (i) - (iii) (i) f(z) is holomorphic in (C \ (-°°,-R].

(ii) There exists a constant M < < » such that 0 < f (x) < M, x > 0.

(iii) Im f(z) < 0 when Im z > 0.

This lemma is proved by von Sydow [26] for R = 0, and from this we see that (i) - (iii) gives

00 r da(t)

f(z) - J I+ztT '

Now oo

1 r/ lx ⁼ r da(t) z z J z - t '

and since f(z) is continuous on we see that — f(- —) is

^ z z

continuous on (-°°,0) and on . K •

Lemma 2.1 now says that 1/R

/ 0

hence

;£(-:)• /—r. da(t) . z - t

pk(z)

Definition 4: When we write h(z) = ^q » we mean that ^Pfc(z), are polynomials of degree < k, < n respectively.

Definition 5: A rational function R (z) of type (n,m) has for n,m numerator a polynomial of degree ^ n and for denominator a polynomial of degree _< m.

A rational function of exact type (n,m) has for numerator a polynomial of degree n and for denominator a polynomial of degree m.

The following lemma is used frequently in the sequel:

pn_i(^z)

Lemma 2.3 Let R - (z) = ——r- n-l,n qn(z) be a rational function of exact type (n-l,n), where the leading coefficient of p^ ^(z) is positive as is that of q (z). '

Let the zeros of R , (z) interlace the poles, all of which n-l,n ——2 belong to (a,b).

Then

R n-1 y n (z) £ M(a,b).

Proof:We can write

Pn-l(z) n Xk R , (z) = ⁿ,\ E

n_1'^{n q}n^(z) k-1 ^Z ~ °k ' where a, < ou < ...< a € (a,b), and X, G |R. I l il k because R _ (z) must be real on the real axis.

n-l,n

We know that R n (x) is positive when x because

n-l,n ^r

the leading coefficient of *"s Posit:^ve anc* so ^-s t^at

of qn(z).

If we start from x = + 00 and approach the greatest zero of Qn(z) >

R - (z) is positive and at the zero a we have

n-l,n n

X

R (a +) ~ — > 0f ( a + = a + 0).

n-l,n n a + - a ^rvn n ^J

9 n n

From this we conclude that A > 0. n

When approaching the next zero of q^(z), we must first pass a zero of Pⁿ«j(^z) > ^anc* since ^Rn_^ ⁿ(^z) changes sign at zeros and poles, we again must have

X

R , (a .+) ~ —-2-^ > 0.

n-l,n n-1 a+ - - a n-1 n-1

Hence

A - > 0.

n-1

By continuing in this way we see that X^ > 0, k = l,2,...,n.

By inspection we now see that R .. (z) satisfies the conditions

J r n-l,n

(i) - (iii) of Lemma 2.1.

Definition 6: We write H(Œ) to denote thè. set of functions holomorphic in the region fì.

3. The method of Baker.

Using determinant theory Baker [2, p. 815] has shown that if f(z) € M*(0,1/R), then g(z) € M*(0,1/R), where g(z) is defined by the relation

f < 2> * T ^ r r ^ r <³-^l>

He then uses Property 5 of section 2 to show that if f(z) € M*(0,1/R), then g(z) € M*(0,1/R), where g(z) is defined by the relation

f(2) ' 1 • S-! )|W xo€(-R,.). (3.2)

O

He iterates this procedure, getting

f(z) = ^£<xo> (3.3)

1 + (z-X^Q)g¹(x¹)

1 + (z-x¹)g²(x²)

g8(

V

1 * '^z-^xs^)gs+l^(z)'

where gi.(^z) £ M*(0,1/R), k = 1,2,...,s and x. € (-R,⁰⁰) , i = 1,2,...,s.

Since g^tz)6M*(0,l/R), Property 3 of section 2 shows that g^(x^)>0, k=l,2,. .,s, Remark 1: When x^ = 0 for all i the expansion (3.3) gives a continued

fraction of the type (lb) (see section 1).

Baker reexpresses (3.3) as

A (z) + (z-x )g (z)-A (z)-C

f(z) = B (z) + (z-x )g (z).B j^.C ' (3,4)

S s s+1 s-1 s

where A (z), B (z), A - (z), B -(z) are polynomials in z, and C is S S S J. S JL S a constant.

Letting ês+i(z) run through M*(0,1/R), Baker is investigating the inclusion region for f(z), z fix. He says, that in any compact domain interior to the cut the sequence A (z)/B (z) converges uniformly s s towards an analytic function. This function is a series of Stieltjes.

Barnsley, [4, p. 301], uses (3.2) in the following way:

First we terminate (3.3) by putting 8^S+^ - Then we start from the bottom of the chain and define

(3.5)

gs-l(xs-l)

y«>-1 • ,)•«,(,.)

s-1 ^{s s}

We can assume that -R<x-<x1 — 2 — ⁰<...<x. — s We know from (3.2) that

V l^M • U) ^e M*(0,1/R). (3.6)

si. s

Since, g (z) € M*(0,1/R), g!(x) < 0 when x € (-R,°°).

s s

Hence

1 + (x-xs_i)gs(xs) > 1 + (x"xs-l)gs^{(x) > °'} X > XS*

From Property 3 of section 2, we know that g -(x -) > 0, hence a

S~"JL S-1

comparison between (3.5) and (3.6) gives

0 < h^(x) < gg_1(x), ^{X > Xg(} _> xg_1). (3.7)

We now define

v • i

:

(^

jH U) • <

-

>

s-z I We know that

V2^(z) • ^£ M«(0,1/R). (3.9)

Since 0 < h (x) < g - (x) when x > x , we see that 0 < 1 + (x-x _9)h..(x)<

J. S~1 S S""Z 1

< 1 + (x"xs-2)gs-l(x)' x > xs( - ^XS-1 - XS-2)* Remembering that êg_2^xs-2^ > we 8et from (3.9)

0 < g^s_²(x) < h²(x), X > x^s(> x^s_¹ > x^s_²).

If we define

WVk'

\^(z) " 1 - (z-x^h^U) ' ^k " ^{2,3 g}o^{<2) £(Z)}' and use the fact that g , (z) £ M*(0,1/R), we get

S~"K.

0 < h2k

-l

l

'

V

0 < gs-2k^ < h2k^' x > xs' k =

s+1 2

[ f ]

(3.10)

(3.11)

If s = 2n, n > 1, we see that ^n^ interpolates to f(z) at the 2n+l points xq,x^,.. ., x0 and

2n def p_(z)

ho„U)

n

where pn(z), <In(z) a^e polynomials of degree n.

Furthermore we get from (3.11)

pn^<x)

0 * ^£<x) ' Tiri ' ^x > ^x2n±^x2n-l - - V

nn

If s = 2n+l, n _> 1, we see that def P (z) (z)

2n+1 <Wz)

interpolates to f(z) at xq,x^, ... »x2n+^; *n^Z^'^n+l^ are P°-'-ynomia''-s

of degree n and n+1 respectively.

Furthermore we get from (3.10)

Pn(x)

° " Q ,(*) ^{< f(x)}' ^x =• ^x2n⁺l-^x2ni ••• ^^xo'

*n+l

We thus see that f(x) is bounded from above by ^n+l^ an(* ^rom below by h0 (x) when x > x0 - > x0 > ... > x .

2n 2n+l — 2n — — o

To get bounding properties for f(z) on the intervals x^ <_ x <_

i = 1,2,...,2n-l, we can repeat the procedure above.

Barnsley uses (3.4) when derivating f(z) with respect to gg+i^2)»

thus obtaining the best bounds that can be imposed on f(x) for

X € (-R,00). He says that A (z)/B (z) can be regarded as an extremal

S s

case of a series of Stieltjes and hence is one.

We are now going to prove:

Proposition 1: If f(z) € M*(0,1/R), then

i) g(z) € M*(0,1/R), ijf g(z) is defined as in (3.2).

ii) The rational approximant A /B in (3.4) belongs to M*(0,1/R). ____________________ g g

The proof of (i) does not involve any determinantal conditions as does the proof of Baker [2, p. 815].

Proof of i): For the proof we need the following lemma due to von Sydow [26 , p. 8].

if and only if g satisfies (a) - (c),

(a) g(z) is holomorphic in the complex plane cut along the negative real axis.

(b) There exists a constant K < «> such that 0 £ g(x) M, x > 0.

(c) Im g(z) jc 0 when Im z > 0.

From (3.2) we have

00

Lemma 3.1: g(z) = / f°^r some bounded, non-decreasing function a(t),

fCx^ - f(z)

g(z) = (z-Xl)f(z) ' x^e (-R,°°)

i • . . . r

(t)

and since f(z) = I

n 1+zt 0

we get

tda(t)

(1+x,t)(1+zt)

(3.12) / da(t)

0 1 + zt

g(z) = 1/R

i

t(1+zt)da(t) (1+x^) 1+zt

|£U>|

1/R i

(l+zt)da(t) 1+zt

t(l+zt)da(t) o^jR (1+zt)(l+x^t)dot(t) (1+x^t)I1+zt Ó ⁽¹+x^{^t) 1}+ztl

|f (z)

1/R tda(t) l/R(l+x1t)da(t)

£ (1+x^) |l+z 11² j^Q (1+x^) |l+zt

|f (z)

1/R. 2 l/R(l+x,t)da(t)

f t da(t) g.f 1 o z t> (1+x^t) |l+zt) ^ (l+x¹t)| l+zt|

|f (z)

,¹ f^Rtda(t) 1 /fRt(l+Xlt)da(t)

5 Q |l+zt|^(l+x^t) (1+x^t) |l+zt f(z)

1/R j_2 1/Rt(1+x,t)da(t)

9 r t dot (t) 0 r i o

|z I (1+Xj^t) Jl+zt I 5 ( 1+x^t) jl+ztj

|f (z)

TM o(-7Ì = " ^{Im 2} • f ¹f^R 2 ¹f^R

I f (z) 2 1 L o (1+xl^ |l+zt I J (l+xjt) |l+zt ^{2 -}

/ 1/R

- ( i

•)1

Thus g(z) satisfies condition (c) of Lemma (3.1).

From (3.12) we see that g(z) satisfies (a), and also that g(x) 0 when X > - R.

Now

da(t) .^fR t^da(t) ~ +*fR tda(t) ^ (Rtdot(t) ^ jj (1+xt) jjj (1+x^) (1+xt) ^ (1+x^) (1+xt) J0 (1+xt)

g'(X) = 2

( ^ da(t) \

\ J0 (1+xt);

1/R (1+x.. t) r 1 (1+xt) o . ft da(t) .2 . . 0 _rt(lW1. +xt)dar-\ (t) 0 J0 (1+^Xlt)((l+Xt> . J (1+^Xlt) (1+xt) ^J (l+^X]Lt)(l+xt)^Z

f2(x)

t(l+xn t)da(t)

- f i 2

* (1+x^t)(1+xt) f²(x)

2 _ \H^R l'da(t) ⁰ . ^r t²da(t) „ (^l r^R tda(t) X\

[ ^ (1+x^t)(1+xt) (1+x^t)(1+xt) \ ^ (l+x¹t)(l+xt)7j

f2(x) _< (Holders inequality) <_ 0.

g(x) <_ g(0) = / da(t) < » , X _> 0. 1/R 0

Lemma 3.1. shows that g(z) = / ^«(t) for a bounded, non-decreasing

0 on

d (t) function a(t). After the transformation — g(- —) = / —^

z ⁶ z ^Jq z-t we can, as in section 2, use Stieltjes-Perron inversion formula to oonclude that

1/R

1+zt *

»<*> - T ^ 0

A ( 2 \

Proof of ii): When we prove that — oG M*(0,1/R), we use (3.3)

• "«vs 2/

and start from the bottom, putting g^s+]/^z) = 0.

We define

gs-l(xs-l)

" °^{U )} - ^{( 3}'^{1 3 )}

We know that

h M ^{e M}*⁽⁰-^l/R)- ^<3-¹⁴⁾

Si. s

and also that g (z) £ M*(0,1/R).

S

This means that g .(x), g (x) > 0 when x > -R.

s-i s —

Since g , °s-l s-1 (x ,) > 0, we see that

0 < 1 + (x-x ,)g (x) < 1 + (x-x ,)g (x ), -R < x < x .

S J- S S J- S S s

A comparison between (3.13) and (3.14) now reveals that

0 < h (x) < g - (x) o °s-l j - R < x < x , s (3.15) and that

h (x) >0, o — ' x > x . s (3.16)

From (3.13), (3.15) and (3.16) we see that 0 < h (x) < K < oo x > ^{0 ,}

—- o —

for some constant K. We also notice that h (x) is continuous on (""R*00) • o

Now,

. Im h (z) = „ ,, 8s-i(Vi'-<"Im z)W = .

|1 + (z"xs_1)gs(xs)|

Since g _1 (x ) and g (x ) > 0, we see that

S JL S"" J. S S

Im h (z) < 0 when Im z > 0. o —

Furthermore, hQ(z) is holomorphic in $ ^ (-°°,-R' 1.

Lemma 3.1 now allows us to write

oo

hQ(z) = J 1+zt ^°r some b°unded, non-decreasing function ß(t) As before, using Stieltjes-Perron^s inversion formula, we see that hQ(z) € M*(0,1/R).

We now have need for the following lemma, the proof of which we postpone until the proof of ii) is completed.

Lemma 3.2. Let

8i(z) . ,

1 + (z_Xi+l3+1 2 )g2(z) x^j+le(-R,°°), h2(z) = —

1 + (z"xj+i)hl(z)

where g^(z), g^(z)⁹ h^(z) € M*(0,1/R) and h^(z) interpolâtes to — ^xl' ^xj ^ (""K»⁰⁰) • Furthermore, we let

0 < h-(x) < g⁰(x) when -R < x < min x.

i-l,2,ì..,j

Then t^Cz) € M*(0,1/R), and h^(z) interpolates to g^(z) aX x2,...,xj+1 e ^(-R,oo)

Furthermore, 0 < h⁰(x) < g-, (x), -R < x < min x.

i-l,2,*..,j+l

We use

. . _ gs-l-k(xs-l-k)

gS-l"k ^ » 8n(z) = f(z)»

1 ⁺ (z-x^s_i_^k)g^s_^k(z) o to define

x gs-l-k^^Xs-l-k^

h, (z) = , k = l,2,...,s-l.

1 + <z

"Vi-k

k-i

Starting from hQ(z) using Lemma 3.2 we produce a new approximant h^(z), satisfying the conditions of the lemma. Going upwards in the

expansion formula (3.3), we see that h^s-1(z) € M*(0,1/R).

Since A /B in (3.4) is the same function as h - (z), the proof s s s 1 of ii) is completed. p|

Proof of Lemma 3.2. We first calculate

Im Si(xi+l)(1+ (z-x4+i)-h (z)

Im h (z) = =* =*—5—^L-

|1 + (z-x^^h^z)!

Since h^(z) €

ÏÏTÛT - ^L;^{R (1}*"^:)d^^t) ,

1 0 [l+zt|²

and consequently

_ l/R Cz+|z|2t - X. - X . -Zt)

(z-X. )h (z) = J —— ± ^{1 — d$(t)}

J 1 - Q |l + zt|' ^{I 1} J. ^{„4- 1}

Now

l/R 1+x. t

Im h (z) = - Im z • f ^—*— d$(t)- g, (x. ,

2 0 11+ztI 1 J+1

and since

g^,(Xj+^) >0, 1 + ^ 0 when xj+^ ^ ("R>°°)> we have

Im 1*2(z) £ 0 when Im z > 0.

We have assumed that

h^(z) € M*(0,1/R); 0 < h^(x) < g2(x), - R < x < min x^, i and hence

1 + (x-xj+^)g2(x) < 1 + (x-xj+^)h^(x), - R < x < min x^. (3.17)

8i(xi+i)

Since g^Cx) = - > 0, x € (-R,⁰⁰) and 1 + (x-xj⁺¹)g²(x)

gl

h_(x) = - , we get from (3.17) 0 < h„(x) < g (x), 1 + (x-xj+1)h1(x)

- R < x < min x. i . Ì = 1 9 2 9 • • 9 J ⁺1

Since h^(x) is non-increa s ing on 1 + (x-x^+^)h^ (x) has its smallest value on -R < x < min x^ , hance t^x) ,> 0,

X > min x.. ¹

— . i i

From this we conclude that 0 £ l^Cx) < M when x > 0.

From Lemma 3.1 we know that h^(z) € M*(0, » ).

Since t^Cx) is continuous om we can use Stieltjes-Perron"s inversion formula to conclude that

h2(z) € M*(0,1/R).

The proof of lemma 3.2 is now terminated.

In this section we have noticed the importance of a relation of the following type;

f (x, )

f(z) ì .

1 + (z-x1)g(z)

We have proved that if f(z) € M*(0,1/R) and x^6 (-R,⁰⁰) then g(z) € M*(0,1/R)»

It was this property that made possible the use of Lemma 3.2.

If we want to extend the results to functions belonging to M*(a,b) where a#b < 0, we need a similar relation.

We use the function f(z) = f t-~t—^ to show that the relation above -1 1 + zt

does not produce a function belonging to the same class.

f (*-. ) Ex. 1 f(z) =

1 + (z-x^gCz) f(x..) - f(z) g(z) =

(z-x¹)f(z)

Now, ^

£W • .{ IÏÏT • hence j t dt

(1+Xj^t) (1+zt) g(z) = ï

J dt 1+zt

Choosing x^ = 0, z = 0, we see that g(0) = 0.

Hence g(z) £ M*(-l,l).

From this we see that the [0,1] Padé approximant is equal to the constant function f(0).

In [3, p. 246-48] Baker compares the relation

CM „ t v » a .. H(o)

w , . 1 + z*a^ - z g(z) 2 , • 1 £(0) •

with the associated (Perron [18]) or Jacoby-type (Wall [27] )

continued fraction. ₀₀

From this comparison he concludes that if f(z) = J ,

oo — oo

then g(z) = f - _{JL • Zu} > where a(t), 3(t) are bounded, non-

—oo

decreasing functions.

f(xi>

Trying to use the relation f(z) =

1 + (z-x1)a1 - (z-x1)z-x2)g(z) where f(z) € M*(a,b), 0 < a < b, ^x^»^x2 ^ I(^a>b),

f^) - f(x2)

a, =

-,

g(z) = ¹

(z-x0)f(z) Z - X l

which can be written

f(z) - f (x-j )

+ a¹f(z)\ ,

g(z) = 1 T j t2 da(t) j 1 da(t)

f(z)f(x²) a (1+Xj^t) t)(l+x(l+x²²t) t)(1+zt) a (1+x^t)(l+x(1+zt) a (1+x^ ²t)(1+zt) 9

r n n i r i \ i

(3.18) _ H tda(t) \1 <

a (1+x^t) (l+x2t) (1+zt)

From Holder's inequality we see that the parenthesis on the right hand side of (3.18) is non-negative when z £ E ^ a b .

However, if we choose x„ < - — we know that f(x„) < 0 and

Z â z

since f(x) 0, x > - -g, we conclude from (3.18) that g(x) £ 0,

* > - £ • "

From this and Property 3 of section 2 we realise that g(z) t M*(a,b).

If we use the relation f(xj f(x„)

f(z) = , X^X- e I(a,b), (3.19)

f(x2) + (z-x1)a1 - (z-Xj^) (z-x2)g(z) f(x1) - f(x2)

where a^T = ,

1 x2 ~ X1

it is possible to prove that g(z) € M*(a,b) if f(z) 6 M*(a,b).

We omit the proof, because we are not going to use relation (3,19) for functions belonging to M*(a,b). This is mainly for two reasons:

i) When interpolating to f(z) at complex conjugated points we need Property 4 of section 2. Property 5 is too weak to be usable in the calculations in section 6.

ii) From ex. 1 where f(z) = J » we saw th^at the constant function f(0) was the [0,1] Padé approximant to f(z). Thus

the "Pade table of f(z) need not be normal. This property causes difficulties when defining rational functions ^(z) a manner similar to that earlier in this section.

In section 4 we are using relation (3.19) to show:

If f(z) € M(a,b), then g(z) 6 M(a,b).

4. The Rational Interpolant Belongs to M(a,b).

The aim of this section is to show

Theorem 1: Let f(z)€ M(a,b), a,b extended real numbers, and let P , —— (z), Qn(z) be polynomials of degree n-1, n respectively, which

satisfy the relations

(1) f(z)Qn(z) - pn_]_(z) * A(z) • (z-x.^) ... (z-Xj^ ) (z-z^) (z-z^) ... (z-zk ) (z~zk

(2) f(z)Q (z) - P 1 (z) = B(z)»zn k3 1

n n-l

s

where A(z) , B(z) E H(t ^s [a,b]), B(z) is bounded at infinity, x^,x²,...,x^k

£ lR [a,b], z^,...,zk e (C ^ |R and k^+k^+2k^ = 2n. We only use (2) * when a and b are finite.^

Then

p _1 (z)

<rSr

<

>

>-

xn

Remark 1: If a = - 00 , b = 00 we do not interpolate at z = » or at any real point. Thus k^ = k^ = 0 and now the condition for pn_i(z)>

Qn(z) to satisfy is

f(z ) Qn(z) " pn_i(z) = A(z)•(z-z1)(z-z1)...(z-zn)(z-zn), where A(z) € H((fc ^ R) and € t ^ |R, i = 1,... ,n.

Remark 2: If a and b both are finite numbers we can interpolate at z = 00 , and if we put k^ = k2 ~ 0 the relations (1) and (2) transform into the relation

f(z)Qⁿ(z) - ^pn_j_(z) = B(z) *z ^ⁿ⁺¹\

where B(z) € H(<C "*•* [a,b]) and is bounded at infinity.

Pn-l(z)

Remark 3: ^ , ^N— is unique.

V

To see this we assume that P^_^(z), Qⁿ*(z) also satisfy (1) and (2).

Hence

fCzjQ^z) - P^JJ^Cz) = A1(z) (z^) (z-x2) ... (z-Xj^ ) (z-z1) (z-z1) .. . (z-zfc ) (z-zfc

f(z)Qⁿ*(z) - P^Cz) = A²(Z) (Z-X^ (Z-X2) ... (z^{-XJ^} ) (z-z^ (z-i^) ... (z-z^fc ) (z-z^fc

(ii)"

n-k -1 f(z)Q (z) - Pn (z) = B (z).z

ii n-1 1

n-k -1 f(z)Q *(z) - P * (z) = B_(z)*z

^ n n—i z

where A^(z), A^{z) , B^(z), B²(z) € H(<E ^ [a,b]) and B^z)^^) are bounded at infinity.

From this we get the relations

(iii)P* (z)Q (z) - P i (z)Q*(z) = A„(z) n-1 n n-l n J • (z-x ) l (z-xz ⁹) (z-x ) (z-zⁿ i ) (z-z" ).. i . (z-z )(z-z^k ),

2 2

2n-k -1 (iiii)P*_¹(z)Qⁿ(z) - Pⁿ_i(z)Q*(z) = B3(z)-z ,

where A^(z) , Bß(^z) € H((C ^ [a,b]) and Bß(^z) is bounded at infinity.

From (iiii) we see that (z)Qn(z) ~ Pn_^(z)Q*(z)) is a polynomial of degree _< 2n-l-k^ and from (iii) we see that it has k^+2k^ finite zeros.

Remembering that k^+2k2+k^ = 2n,we have a polynomial of degree

_< 2n-k^-l which has 2n-k^ zeros. Hence it ifc the zero polynomial, P , (z) P* (z)

Ä Ä n—1 n—1

• Qn(z) Q*(z) '

Using the following example we will show that the interpolant need not belong to M(a,b) when interpolating at arbitrary complex points.

Ex. 1. Let

«« - Jw • ⁱⁿ Å

and let Pg(z) = aQ> = ^0+Z P°iynomiais which satisfy the relations

f(i)q1(i) - PQ(i) = 0

f(l+i)q1(l+i) - pQ(l+i) = 0

Then q^(z) = z - ^ ^ *(ln/2 - i( - In 2)) and from this we see that

po^(z)

^ ( «0,1).

Remark 4: In this section we are constructing a rational function

where A(z) , B(z) € H(<t ^ [a,b]), B(z) bounded at infinity, x¹,x²,... ,x^fc € IR \ [a,b], ^zl>^z2> • •^# >^zk £ t ^ ft, ^ki^+2k2^+k3 ^{= 2n#}

To prove Theorem 1 we will divide section 4 into five parts:

4.1 Interpolation at complex conjugated points.

4.2 Interpolation at finite real points belonging to [ a,b L 4.3 Interpolation at infinity.

4.4 Interpolation at infinity and one real point.

4.5 Interpolation at arbitrary points.

Pn-1

which actually interpolates to f(z) at all interpolating points, i.e. satisfies the relations

Pⁿ ,(Z)

,f(z) - Qn(Z) = B(z) * z

The reason for this division is to make the reading easier and to make visible the method of proof.

4.1 Interpolation at Complex Conjugated Points.

When we want to interpolate to f(z) € M(a,b) at the complex conjugated points z^,z^, z2» z^,...,z^, z^, we start by defining functions

hj (z), j = l,2,...,n+l, by the relations (compare (3.3))

hⁿ(z) = f(z)

12 h^(z) = I W L '

hl(zi) + (z-z )bi - (z-z )(z-. jh^z)

|H2(Z2)|2

h²(z) —

h²(z²) + (z-z²)b² - (z-z²)(z-z²)h³(z) '

l^hn(zn⁾I

hn^{(z) =} ~ = '

hn⁽zⁿ) + (z-zⁿ)bⁿ - (z-zⁿ)(z-zⁿ)hⁿ⁺¹(z) h. (z.) - h.(z.)

where b. ⁼ ^ -—-—, j = 1,2,...,n. (4.1.1)

^ Z. - Z.

J J

Now we use the following lemma, which will be proved in section 6.

Lemma 4.1.1: If h^(z) € M(a,b), where a,b are extended real numbers, and if h²(z) is defined by the relation

I M O I2

h¹(z)=— ± ,

HL^ZL^ + ^Z-ZI^BI " (^z~^zi^(Z-Z¹)H2(Z)

h. (zJ-h. (z )

where b^ = —— , z^ £ t ^ |R,

Z1 - Z1 then h²(z) € M(a,b).

From this lemma and the definitions (4.1.1) we see that hj(z) € M(a,b), j = 1,2,...,n+l.

Next we use

IW I2

h (Z) _ (4.1.2)

h(n n ^z) + (z-z)b - (z-z )(z-z)h (z) n n n n n+1

to define 0

|h (z ) I

R0 ,(Z) 1 • (4.1.3)

V«n>

We know that

h ( z ) - h ( z ) b d o , ( t ) b d a ( t ) b = = / T? > 0, because h (z) = / ——- ,

n — z - z n n a z i ^ . | 2 ¹ n - t ¹ n a z - t

for some bounded, non-decreasing function ^an(^t)•

Hence

_ b (z-t)da (t)

h (z ) n n + (z-z ) n n b = f a i z - t ^ i £. 1 , n

and is real on |R ^ [a,b].

Since hⁿ(b) >0, if b is finite, we see from (4.1.2) that

0 < h (z ) + (b-z )b - |b-z |²h (b).

n n n n ¹ n¹ n+1

b dan+l^

Now h ,,(z) = f n+1 ^J a z-t , because h - (z) 6 M(a,b), and hence n+1

hn+^(b) > 0, which implies that

0 < h (z ) n n + (b-z)b - |b-z |n n 1 n1 ²h (b) < h(z) n+1 n n + (b-z )b . n n

A comparison between (4.1.2) and (4.1.3) gives

0cR⁰ -j^(b) < hⁿ(b), if b is finite. (4.1.4)

A similar estimation at z = a reveals that

0 > hn(a) > Rq ^(a), if a is finite. (4.1.5)

Since hⁿ(z), hⁿ⁺^(z)€ M(a,b), Property 6 of section 2 gives

0 < lim iyh (iy) < 00 ; 0 < lim iyh^+^(iy) < °° . (4.1.6

y-K"o y->oo

From (4.1.2) we get

l^HN<^Zj|2

0 < lim iyh (iy) = . (4.1.7)

y-*» bn - lim iyhn+1(iy)

y-X»

Remembering (4.1.6) we now know that

0 < b - lim iyh -(iy) < b . n _y-X» ^J n+1 ^J n (4.1.8)

Since

Ihn(zn}12

lim iyRⁿ -(iy) =

0,1 b

y-H» n

we see from (4.1.7) and (4.1.8), that

0 < lim iyR^Q ^(iy) < lim iyh^(iy) < ⁰⁰ (4.1.9) Y-**} * Y-XX)

From (4.1. 4) and (4.1.5), we see that the pole of RU, 1 n - (z) belongs to (a,b) This and (4.1. 9) are enough to make Rq ^(z) £ M(a,b).

To be able to use R^ ^(z) we will use

Lemma 4.1.2; Let

,2 HX(Z) = I <zx) I

HL^ZL^ + (Z-ZI^BI " (Z-Z^ (Z-Z1)H2(Z) l¹»¹(z¹) I²

k+1 — —

' ^hl^^zl^ ⁺ ~ (z-z¹) (z-z¹)R^k_^{1 k}(z) H1(Z1)-H1(Z1)

where h^z), h²(z), R^fc_^{1 k}(z) € M(a,b) , b-j^ = ⁼ , € <C ^ |R.

zrzi

Furthermore we let j^2) be a rational function of exact type (k-l,k) which satisfies the inequalities

O < lim k(z) £ lim iyh2(iy) , y-yoo ' y-x»

0 < RK_-J. K^) < H2^B^ ' — B *S FINITE ,

0 > t^Ca) > ÜL ^a is finite .

Then ^+1^ ^ M(a,b) and k+l^ is a rational function of exact type (k,k+l) satisfying

0 < lim iyR^ k+1(iy) lim iy^Ciy),

y-x» 9 y-K»

0 < (l>) < h^(b), jLf b is finite,

0 > h^(a) > k+i(a)> ìÉ. a is finite.

This lemma is proved in section 6.

Since R_ .. (z) satisfies the conditions of the lemma, we realise that U y 1 iterated use of thè lemma gives

I h _ • (z .)|²

R (z) ^{= n J n}"J ⁼ 6 M(a,b) ,

h n

satisfying

0 < lim iyR. ..-.(iy) i lim iyh .(iy), j = 0,1 n-1, y-^o J > J+ I y^«o ^N J

0 < R. . , (b) < h .(b), if b is finite, j = 0,1,...,n-1, j ,j-f-l n-j

0 > h .(a) > R. . , (a), if a is finite, j = 0,1 n-1.

n-j j ,j +1

Thus we see that Rⁿ_-^ ⁿ(^z) £ M(a,b) and interpolates to f(z) at the points ,Z^ ,,z^,...,z^,z^.

R n (z) also satisfies the relations above.

n-l,n

Realising that Rn_^ n(z) satisfies the relations (1) and (2) of Theorem 1, V ^k3 " '

complete.

k^= = 0, we see from the unicity of the interpolant that the proof is

4.2 Interpolation at Finite Real Points Outside [a, b ].

2n

We are interested in real, finite interpolation points such that x-<x„<... <x. <a<b<x.,-<... <x_.

1 - 2 - — J J + l - - 2 n Now we distinguish between two cases.

Case 1. j = 2k, k € {0,1,...,n}.

We begin by defining functions g^(z), g²(z),...,gⁿ⁺^(z) by the relations

g^z) = f (z)

• ^8l(x )g (X )

gx(z)

gl(x2) + (z-x^aj^ - (z-xi)(z-x2)g2(z) g2(x3)g2(xi>

g²(z)

g2(x4) + (z-x3)a2 - (z-x3)(z-x4)g3(z)

gk(x2k-l)gk(x2k)

%(2)

8k(x2k> + <2"x2k-l)ak" (z-x2k-l)(z-x2k)8k+l<z)

, , ^gk+l^(X2k+l^igk^(X2k+2⁾

8k+l

8k+l^x2k+2^ + 'Z_x2k-H>ak+1 ~ 'z_x2k-n"z *2k+2^8k+2f'Z)

8n^(x2n-l⁾⁸n^(x2n>

g„(z) = "11

8n^(x2n> ^{+ (z}"^x2n-l^)an " ^(z"^x2n^>8n.H^<2)

8i^(x2i-H⁾ " ⁸i^(x2i^>

where a. = -J—=J_J: 2—±J- , j = 1,2 n. (A.2.1)

1 x2j - ^x2j-l

The usefulness of the definitions (4.2.1) is seen from

Lemma 4.2.1: If f(z) € M(a,b), where a,b are extended real numbers, and if g(z) is defined by the relation

f(x )f(xj f(^z) =

f(x9) + (z-x-)a.. - (z-x..) (z-x0)g(z)