Multipoint Padé approximants used for piecewise rational interpolation and for interpolation to functions of Stieltjes' type

3 A

Multipoint Padé approximants used

for piecewise rational interpolation and for interpolation to functions of

Stieltjes^ type.

by

Jan Gelfgren

No 8 1978

UNIVERSITY OF UMEÅ

DEPARTMENT OF MATHEMATICS

S-90187 UMEÅ

SWEDEN

r

Multipoint Padé approximants used

for piecevri.se rational interpolation and for interpolation to functions of

Stieltjes' type.

by

Jan Gelfgren

No 8 1978

i— j

Akademisk avhandling soni med tillstånd av Rektorsämbetet vid Umeå universitet för avläggande av filosofie doktorsexamen

framläggs till offentlig granskning fredagen den 19 januari 1979 kl 10.15 i Hörsal C , Samhällsvetarhuset.

Multipoint Padê approximants used for piecewise rational interpolation and for

interpolation to functions of Stieltjes^y type.

By Jan Gelfgren

Department of Mathematics,University of Umeå,Sweden.

Abstract.

A multipoint Padë approximant,R,to a function of Stieltjes¹ type is determined.The function R has numerator of degree n-l and deno­

minator of degree n.

The 2n interpolation points must belong to the region where f is analytic,and if one non-real point is amongst the interpolation points its complex-conjugated point must too.

The problem is to characterize R and to find some convergence results as n tends to infinity.

A certain kind of continued fraction expansion of f is used.From a characterization theorem it is shown that in each step of that expansion a new function, g, is produced; a"function b£ thé: same 's> • ^ jtype £s. .f.The function g is then used,in the second step of the expansion,to show that yet a new function of the same type as f is produced.

After a finite number of steps the expansion is truncated,and the last created function is replaced by the zero function.

It is then shown,that in each step upwards in the expansion a rational function is created; a function of the same type as f.

From this it is clear that the multipoint Padê approximant R is of the same type as f.From this it is obvious that the zeros of R inter­

lace the poles,which belong to the region where f is not analytical.

Both the zeros and the poles are simple.

Since \ both f and R are functions of Stieltjes ' type the theory of Hardy spaces can be applied (p less than one )to show some error formulas.

When all the interpolation points coincide ( ordinary Padé approximation) the expected error formula is attained.

From the error formula above it is easy to show uniform convergence in compact sets of the region where f is analytical,at least wien the interpolation points belong to a compact set of that region.

Convergence is also shown for the case where the interpolation points approach the interval where f is not analytical,as long as the speed qî approach is not too great.

Key words:Continued fractions,Hardy space,Padê approximant, series of Stieltjes.

Papers summarized in this dissertation:

[A] Gelfgren, J.: Piecewise rational interpolation, Research Report, University of Umeå, Dept. of Mathematics, no 9, 1975.

[B] Gelfgren, J.: Rational interpolation to functions of Stieltjes"

type, Research Report, University of Umeå, Dept. of Mathematics, no 6, 1978.

Acknowledgements• I want to express my gratitude to Professor Hans

Wallin for his guidance and encouragement during my work with this thesis. I also want to thank Mrs Margareta Brinkstam and Mrs Anita Lidén for typing the manuscript.

0. Introduction.

If f(z) is defined at the points z^ £ <fc, i = 1 ,2,...,n+m+l, we can make the following definition (Karlsson 17]» is calling it "best interpolating rational function of type [n,m] to f at {z^}?^j¹⁺''" ").

Pn(«> n+m+1

Definition. The rational function . Q ( z ) ^{/ x} is called the {z.}. .. 1 1

multipoint Padë approximant to the function f(z) if Pⁿ and are polynomials of degree at most n and m, respectively, $ 0, and Pⁿ» Q^JJJ satisfy the relation

(fC^ - Pⁿ)(^z£) =0» i = ¹>² n+m+1. (1)

(If some of the points z^ are equal, we demand that the corresponding derivatives of (1) are equal to zero at that point.)

Remark 1. When z^ = 0, i = 1,2,...,n+m+1, we get the usual [n/m] Padë approximant.

Remark 2. When f(z) is analytical in some region containing the points , (1) can be written

fVz) ~ pn(z) = A(z) (z-zi(z_z2^ • • • (z-zn+m+l^ ' ^

where A(z) is analytical in the same region.

Readers interested in Padë approximants may read the papers [3] by Chui and [4] by Gilewicz^ach containing a long reference list

1.

1. Pieeewise rational interpolation.

When f is a formal power series at z =0, the [n/m] Padé approximant can be defined to be the unique rational function which has contact of highest order at the origin; P^ and are polynomials of degree n amd m respective!}' (see Baker [l, p* 4-8]).

If f is analytic in a small region containing the origin, we can use Hermite's interpolation formula there to get a n estimate of the error e(z) = I (f - Pn/Qjjj) (z) I •

If^fhowever, the function f is to be approximated on a real interval the picture changes. If we use just one Padë approximant, we cannot in general use Hermite^s interpolation formula to get a good error estimate in the whole region.

In the paper [A] we partition the interval and in one typical subinterval [x^, ^we determine the symmetric [n/m] Padë approximant P^ £ to f jat x^ and from the relation

f^^m i^(z) ~ ^Pn i^{(2) =} (^z_xi^^P ^^z~^zi+i^^P' (n+m+l=2p), where A(z) is bounded at x. and x.-.

i l+l

Furthermore, P ./Q . is unique (see [A, p. 2]).

n,i in, i

Since the poles of ^ can be situated inside [x^,x^⁺^] we see that the error e^(z) = |f(z) - P^{n can} l^ar8^{e at} some points of the interval [x^,x^+^].

In order to be able to formulate a theorem we need the following definitions [A, p. 3-4].

Let I = [a,b] be a real interval and let r be a real number which is greater than zero. Let Ü = {z | d(z,I) <_ r} , where d i s the usual distance function. Let I have the partition T: a = Xq < X^ < ...<X^ = b, where every subinterval, A^ = [x^,Xj⁺^], has equal length, A = |Aj| =

= ^xj⁺^ ~ ^xj f°^r J ⁼ 0,1,...,N-l and A < r.

2.

Let P ./Q . be the symmetric [n/m] Padé approximant to f at th e n,j m,j

endpoints x j , ^xj⁺^ ^^xj^,xj+i^*

In this way we get a piecewi.se rational function ^m(i)(z).

We let G(A) = {z = x + iy | x £ I, |y| _< (3/4)A} and formulate

Theorem A.l ([A, p.5]): ^Let f be analytic in Ü , and let I have the partition T. Let N > 0 be arbitrarily chosen. Then it is possible to find a real

number k such that, when n >_ k*m and z € G(A), we have /Ts"¹ A ⁿ""km+l I (f — S (T)) (Z) I < K(A ) max | f | (/yr r )>

1 n,m 1 —

except on a set F of capacity < r\.

If f is real valued on [a,b] we get the following stronger result for piece""

wise polynomial approximation (^p-l^^ ^a PÎ^ecewi^se polynomial function).

Theorem A.2 ([A, p.5-6]): Let f be analytic in Œ and real valued on [a,b]. Let [a,b] have the partition t. Then, for x € [a,b], we have

mF l^fl n P ^ - 4 ^ P - j

K '^{0 ,}- s & ( o « | < A²-

K ^r f J (2p -j) ^{F J}

where j = 0,1,2,...,p-l and (2p)^ = (2p-j)î '

We see that now there is no exceptional set. From this we can conclude that in general it is not better to use piece wise rational functions than to use piece wise polynomial functions.

When we used piece wise rational approximation to get error estimation in the complex plane, we could not tie together the différant rational functions that arose, and the approximation was not very good.

3.

2. Rational interpolation to functions of Stieltjes" type.

In [B] we use rational functions to interpolate to functions of the b

type f (z) = / z-t where a,b are extended teal numbers and a(t) a

is a bounded, non-decreasing function.

In the beginning of [B] there is a short historical review of the problem.

The main theorem of [B] is

Theorem B.l ([B, p.26]): Let f(z) = f , where a,b are

• ——— Z L. """""**""

a

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

Let P ,(z), Qⁿ(z) be polynomials of degree < n-1, <_ n, respectively, satisfying the relations (k^ + 2k² + = 2n)

(1) f(z)Qⁿ(z) - ?ⁿ_¹(z) = A(z) (z-Xj^) (z-x²) ... (z-Xj^ ) (z-z^ (z-z.^ .. . (z-z^fc ) ) n-k -1

(2) f(z)Qⁿ(z) - P^U) = B(z)z

V

where A(z), B(z) are analytic in B(z) bounded at infinity, x^,x2,...,x^ € |R X [a,b], z^.z^,...,z^ £ Ç x |R. fwe only use (2) when a and b^are finite/) ^

Then

Pn-l(z) br dß(t) . ^ ,

q /^zy— = J — for some bounded, non-decreasing

n 'a

function ß(t).

Remark 1: We see that this theorem contains three types of interpolation points

i) Real finite points outside [a,b]

ii) Complex conjugated points iii) Infinity.

P^R_ ^ ^ ^ d ß ( t )

Remark 2: Since ^<._^N = / __ i , Q_(z) has its poles inside

—— U ( Z J Z "" L XI

Hn a

[a,b], and from (1) and (2) we see that we have ordinary Hermite interpolation. When interpolating to arbitrary functions in a manner like (1) and (2), this is not always the case, which can be seen from

Example 1, Let f(z) = z 2 and let

f (Z)Q²(Z) - P^z) = A z(z-l) (z + ¹²'¹—• ) (z + ¹⁺²¹'^ )

where A is a constant.

Then P^z) = z, Q²(z) = z 2 and

V ^z) i

^ , v = — has no zero at z = 0.

Q 2 ( z ) z

By using Theorem B.l and some H^-theory (the idea of which I had from G. Lopez) we can show one convergence theorem for each of the following cases:

i) a and b are real and finite

ii) a = - ⁰⁰ and b is real and finite iii) a = - °°, b = 00

i) a and b .are real and finite

Theorem B.2 ([B, p. 51]). Let f(z) = J > ^a> k finite, real Z L

a

numbers, and let P^_^(z), Qn(z) be polynomials of degree < n-1, _< n, respectively, satisfying the relations

* — —

f(z)Qⁿ(z) - = A(z) (z-Xj^) ... (z-x^k ) (z-z^ (z-z^ ... (z-z^fc ) (z-z^fc ) n-k -1

f(z)Qⁿ(z) - Pⁿ_¹(z) = B(z) z ,

where x^,x^2>.. .-.x^ € |R ^ [a,b], z^z^..-^^ ,z^fc € <fc ^ |R, k^+2 k²+k^=2n B(z) bounded at i nfinity and A(z) bounded at x¹,x^2>...,x^ ,z^,z^,...,z^

5.

Then, for z €.C ^ [a,b], we have

|f (z) ^Pn-l^(z) < K

max (Iz-aI,Iz-bI) 1 - kv(z) I

1 ip (z) - tp (x.)

V V 1

n

i—1 1 - ip (x.)4i (z) V I V

2 ip (z) - \p (z.)

v v j

j-i

n

\(z) - ^(Zj) 1 - ip (z.)ip (z)

V J V

^^v(z) - ^^v(°°) 1 - I|I^V(«)I|I^V(Z)

(2.1)

where \p (z) =

—— v

k _/zzk z-a v-a

1EI +/r±

z-a v-a

is a conformai mapping of (t ^ ta,b]

onto the interior of the unit circle U such that one of the interpolation points, v, is mapped onto zero, is a constant depending on the

point v and f but not on n.

Remark 3: If we just interpolate at infinity we get

P -i (z)

I £(Z)

I ^ K -

inax( I z-a |, | z-b | ) l-|^(z)|

k(z) I 2n

• £± - i where K is a constant and iKz) = z-a

z-a » i

From this remark we get, putting z , GÜ

P

_ ì f(_ Ì ) _ n-i "

CO 03 -

-„>0 (- i ) n a)

< K 0)

lumax ( 11+wa |, | l+wb | )

2n

Putting a ?= 0, b = 1, we get

1

P£ i <^w>

£*⁽»> - h

< K

max(l, |l+ü)|) 1~Icp(tü) I

2n

|<p(u) I j

where <p(w) =

/- . 1 j f » P* i (to) vl+o) - 1 m, \ r da(t) n-1 -j=—r • ^{£ M} • I se 'WT

v l+ii) +1 0 ^xn

dß(t) l+ut '

This can be compared to the error formula by von Sydow [9], P* (u)

n-1

- WÙ < K

/l+z j 1 -|(p(ai) k(«o)|; 2n

and we see that Theorem B.2 gives the correct error formula for Padé approximation.

ii) a = - » and b is real and finite

Theorem B.3. Let f(z) = / ^ancj i^et p . (z), Q (z) be

J z - t n-1 TI —

— 00

polynomials of degree _< n-1, _<• n, respectively, satisfying the relation

f(z)Qⁿ(z) - = A(z) (z-x^)... (z-x^ ) (z-z¹) (z-zj^)... (z-z^k ) (z~z^k ),

where x^, x^ 6 (b,°°), € <t ^— |R, k^+2k2=2n, and A(z) £ H(t ^ (-<*>,b]).

Then ^ffor z € t ^ (-°°,b] j we have

f (z) - 1

V ^z)

^v(z) - ^v(xi>

n

max (1,| z-b | ) 1 - 1(z) | i=l 1 - I/J (x.)ip (z) V 1 V

• n

4» (z) - \p (z.)

v v j

1 - ^^v(zj)^^v(z)

il> (z) - ip (z.)

v v j

1 - ip^v(zj)ip^v(z)

(2.2)

, , / \ /z-b - /v-b where ip (z) = ——

/z-b + /v-b

v is an interpolation point, and K

is a constant depending on this point and on f, but not on n.

Remark : If a is finite and b = + ⁰⁰ , we only have to change the constant K , replace ttie factor ^ with 1

max(l,jz-bj) max(l,|z—a|) and change the conformai mapping ^ (z) accordingly.

7.

iii) a = -oo and b = °°

OO

Theorem B.4 Let f(z) = I and let P (z), Q_(z) be

z - t n-1 n

—OO

polynomials of degree n-1, _< n, respectively, satisfying the equation

f(z)Qⁿ(z) - Pⁿ_]_(z) = A(z) (z-zj^) (z-zj^) ... (z-zⁿ) (z-zⁿ> ,

where z^jz^,..., € {z | Im z > 0} and A(z) € H(<t ^ R).

Then, for z £ {Im z >0}, we have K i (z)

\cf \ ^n-l

•^£(z) " Qⁿ(z)

n

< K

— v

1 - U (z) , . . . |i=l ^ll - i|> ( z . ) i p (z) V ^{1 T}V 1 V

(2.3)

Z —V

where Im z > 0, \p = , v 6 {z-,z⁰,... ,z } and K is a constant

Tv — 12 n v

z-v

depending on this point and on f but not on n.

Remark. Since f(z) = f(z) and

P -i (z) P -, (z)

n-1 n-1

Qn^(z) Qⁿ(z)

the same convergence formula holds in the lower half plane, {z I Im z < 0}, if we change z^ to z^, i = l,2,...,n and V £ {z-,z ,...,z }•

1 I n

8.

3. The idea behind the proof of Theorem B.l ([B, p. 26]).

We start by writing l f < z ) I²

f(z) = % (3.1)

fÖ^) + (z-z1)b1 - (z-z1) (z-z1)g1(z) f(z¹)-f(z¹) _

where b^ = —— and ^zi>^z\ ^ IR.

z r ^z i

b J r b dß. (t)

From [B, p. 29] we know that if f(z) = / —— , then g1(z) = / —_

Z ~~ L 1 Z "" L

a a

(a(t), ß^(t) are bounded, non-decreasing functions).

Since g^(z) is of the same type as f(z) we write

I^sl<^z2⁾ |² gj/z) -

gl<z²) + (z-z²)b² - (z-z²(z-^z2)g²(z)

gl^(z2⁾"^gl^(z2⁾

where b² = —— — and ^z2^,z2 ^ ^ ^X

Z2 - Z2

Since g2(z) is of the same type as g^(z), it is of the same type as f(z) and we see that we can expand f(z) in the following way

i^f<^zi> r ,2

fCzp + (z-z¹)b¹ - (Z-Zj^) (Z-Zj^) |g^z²)

f(z) = — 2 ⁽³*²⁾

— — 2

gl(z2) + (z-z2)b2 - (z-z2)(z-z2)|g2(z3)I

'⁸n-i⁽ⁱtì I² Vi ^(zn^)t(z-^zn> V⁽²"^zn> ^(z_zn> ^sn^fe)

hr ^dßk^(t)

where g^Q(z) = f(z), g^fc(z) = f ^z - ^{t and} \ ⁼ Z ) z. - z.

k k k = 0,1,...,n.

9.

The expansion (3.2) can be divided into n relations

l

(

i)|

f(z) = — ⁼ , (3.3)

f(z¹)+(z-z¹)b¹-(z-z¹)(z-z¹)g¹(z)

lg1(z2) I2

g (z) = L_J , (3.4)

g1(z2)+(z-z2)b2~(z-z2)(z-z2)g2(z)

|g2<^] 2

g2(z) = ,

g²(^z3)⁺(^z~^z3)^b3"(^z~^z3)(^z2^_z3^^g3^^z^

'

s ⁿ _ ² <*> -

!=

^L

g„ ,(z .)+(z-z i)b _i"(z-z )(z-z ,)g ,(z) n-z n-1 n-1 n-1 n-1 n-1 n-1

Vl(«) l8n-l(zn>l 2

(3.n+2) Vl ^<2n' * ^(z"²n⁾ V ^(z-zn' ⁸n^(z)

fU^-fdj) 8^k-l<^Ik>-«^k-l⁶'k^>

where b- = , b. =

1 — k —

zi - ^zi ^zk - ^zk

From [B, p.29] we know that gjc(z) is °f the same type as f(z), i.e.

b dek(t)

g^(z) = J _ • —, for some bounded, non-decreasing function 3j^c(t), a

k = l,2,...,n. We now use (3.3) - (3.n+2) in the following way:

i) We put 8ⁿ(^z) = 0 in (3.n+2), thereby replacing 8ⁿ_^(^z) by a rational function Rq ^(^z) which interpolates to gⁿ_^(^z) at zⁿ and "z^, and we prove that Rq ^(z) is of the same type

as gn-l(z)*

ii) We substitute Rq ^(z) for gn_^(z) i-n ( 3.n+l) thereby replacing gR_2(z) with a rational function R^ 2^ which interpolates to g „(z) at z ,,z .,z ,z and we prove

en-2^v n-1 n-1 n n ^r that R^ 2(z) is of the same type as §n_2^z^*

iii) Continuing in this way we finally get a rational function

Rn-1 n^ ^whi^ch interpolates to f(z) at z^z^z^ä^ zⁿ>^zn» and R . (z) is of the same type as f(z). n-1,n jr

When interpolating at infinity we use an expansion formula different from (3.2) and yet another when we interpolate at finite real points.

We see that the proof contains two iterative procedures, one to show that in each step downwards in (3.3) - (3.n+2) we produce a function of the same type as f, and one to show that in each step upwards in

(3.n+2) - (3.3) we produce a rational function of the same type as f.

11.

4. A comparison with some papers by Goncar and Lopez.

In the papers [5] by Gon?ar, [6] by Gondar and Lopez, [3] by Lopez, it i s shown that the multipoint Padé approximant to

f(^z)

= J

E

A, • >

J z - t n-l,n ' k

a z - a,

k and G (a,b). This is equivalent to Theorem 1 in [B].

To show this they use the theory of orthogonal polynomials and the Gauss-Jacobi quadrature procedure, which is far quicker than the method used in [B].

In [6] Goncar and Lopez also use the Gauss-Jacobi procedure to estimate k - (z)I and to make this estimation they cannot let the inter-

rn-l,n ^{1 J}

polation points be arbitrary close to [a,b]. When this condition is satisfied they get uniform convergence for the sequence ^to f in compact subsets of

(C

00

Lopez, in [8], interpolates to the function f(z) = f • - [_ at rea l 0 z - t

points belonging to the negative real axis. Since he too is using the Gauss-Jacobi procedure when estimating |Rn_^ 19 ^is interpolation points must belong to [-«>, aj where a < 0 is fixed.

Carleman E2] has proved that the condition

00 - l/2n

E ( — ) = < » (y is the n:th moment of f (z))> is enough to make 1 ^Pn

the [11-l/n] Padë approximants converge to f(z) when n ⁰⁰.

In [8] Lopez shows a generalization of that condition. As a corollary Lopez shows: (^a^> k = l,2,...,2n are the interpolations points)

If lim

E

then the sequence converges uniformly to f in compact sets of <C v [0,»).

12.

Since a - z 1 - az

i - a-H

>(H°l

>,

i l - a z I

we see that a - z 2 1 - az

< e

I"" _l² —1—9 (1- a ) /T I l ²\ ll-äzl²

- (^2)(X-|.|2)

< e (4-1)

Now we define 2n a - z

b (z)

= n

e u.

n

k 1

k z

From (4.1) we easily get:

2n

If lim

Z

then {Bⁿ> converges uniformly to zero in compact sets of U.

From (2.1) we then have 2n

Lemma 1 : Let lim E 1 - U (a.) = where ^ (z) is the

1=1 ¹ v i ¹ v

n-H» ¹¹

conformal mapping in (2.1). Then converges uniformly to f in compact sets of (C ^ [a,b].

Obviously, a similar lemma is true when we have the conformai mapping used in (2.2) (or (2.3)).

JÏ ^—

When a = - ⁰⁰ and b = 0 we have i|> (z) = and thus

v /z + /v

(tö:- /v \ 1

1

- i*v

i>"

7-T-TT—T7T7=^\

l«.| + Ivi • <

-

>

2 2 Re(

I °t£ I + I v I +2 Re^v'ôtTvÇ^ ^

'"i

13.

When ou > a > O, för alla i, (4.2) and Lemma 1 give:

2n 1

If lim X = 00 , then converges uniformly

n-x» 1 1 + /o7 '

i

to f in compact sets of (C ^ (-°°,b]. This is essentially the corollary of Lopez.

From (4.2) and Lemma 1 we see that the interpolation points can approach the interval [a,b] if the speed is not too fast.

If Goncar and Lopez had used another way of estimating |ⁿ(z

)J

Since Gonïar and Lopez have shown the important fact that the interpolant R - (z) is of the same type as f(z) the new results, in [B] are those

n-l,n ^Jtr

of the following natureÎ

Let f(z) = / rïa(t) , where a, b are extended real numbers^-and a(t) is Z L

a

a bounded, non-decreasing function.

Let g(z) be defined by the relation f(x¹)f(x²) f(z) =

f(x²) + (z-x²)a¹ - (z-x^ (z-x²)g(z)

f^) - f(x²) where x, ,x„ € tR [a,b] and a, = — Ì .

12 1 x² - x^

h in / . \

Then g(z) = J — f o r a b o u n d e d , n o n - d e c r e a s i n g f u n c t i o n ß ( t ) . Z "" L

a

14.

REFERENCES.

Baker, G. A. Jr., "Essentials of Padé approximants", Academic Press, New York 1975.

Carleman, T., "Les fonctions Quasi-analytiques", Gauthier-Villars, Paris 1926.

Chui, Ch. K., Recent results on Padë approximants and related problems,

"Approximation Theory II" (Lorentz, Chui and Schumacher ed.), Academic Press, New York 1976, pp. 79 - 115.

Gilewicz, J., "Approximants de Padé", Lecture notes in mathematics 667, Springer-Verlag, Berlin 1978.

Goncar, A. A., On the degree of convergence for rational approximation v to certain analytical functions (Russian), Mat. Sb., 105 (147) (1978), 147 - 163.

Goncar, A. A., Lopez, G., On Markov's theorem for multipoint Padé

approximation (Russian), Mat. Sb., 105 (147), (1978), 512 - 524

Karlsson, J., Rational interpolation with free poles in the complex plane Preprint University of Umeå, No 6, 1972.

Lopez, G., Conditions for convergence of multipoint Padé approximants to functions of Stieltjes" type (Russian), Mat. Sb., 107 (149) (1978), 69 - 83.

von Sydow, B., Error estimates for Gaussian quadrature formulae, Numer. Math., 29 (1977/78), 59 - 64.