Best approximation by rational functions and by meromorphic functions with some free poles

BEST APPROXIMATION BY RATIONAL FUNCTIONS

AND BY MEROMORPHIC FUNCTIONS WITH SOME FREE POLES*

By J. L. WALSH

in College Park, Maryland, U.S.A.

I have recently indicated [1, 2, 3, 4] some cases of best approximation of

a meromorphic function f(z) by rational functions R.~(z) of a given type

(n, v) having some free poles (that is, poles not prescribed in position), where

it is proved that the free poles approach necessarily (n ~ ~) the poles of

f (z) . The object of the present paper is to indicate (w that the methods

already introduced for the case that the prescribed poles of the R.v(z) lie at

infinity admit extensions that apply to the more general case that the pre-

scribed poles of the R.v(z) do not lie at infinity nor in fact in a finite number

of points. We study also (w the problem of approximation by meromorphic

functions, bounded with the exception of v free poles.

w R a t i o n a l F u n c t i o n s .

A rational function Rnv(Z ) is said to be of type (n,v) if we have

R.~(z) a~ + ' " + a " ~]bk] # 0 . =-bozv+blz v-l +. . .+b~ '

If the function f(z) is continuous on the point set E which has no isolated

points, there exists for each type (n,v) a function R.~(z), of type (n,v) such

that the uniform (Tchebycheff) norm

lira(z)- R.X:)II = [maxl f (z ) - Rn (z)l, z on E]

* Sponsored (in part) by U.S. A i r Force Office of Scientific Research,

359

360 J.L. WALSH

is minimum; this function R.~(z) is not necessarily unique, but any particular

determination suffices for our purposes. Some poles of R.~(z) may be prescribed

in position.

We shall establish the following theorem:

Theorem 1. Let E be a closed bounded set whose boundary B consists

of mutually disjoint Jordan curves B1,B2,. . . ,B ~. Let C be the union of

mutually disjoint Jordan curves Co, C1, "", C u disjoint from E, of which Co

contains in its interior E + C - Co. For each curve Cj (j > O) let there exist

a Jordan arc which joins Cj to E without cutting the other curves Ck.

Let the function U(z) be equal to zero on B, equal to unity on C, and

otherwise harmonic and continuous in the extended plane. Let D be the

union of the regions where U(z) is not constant, D, the union of the regions

where O < U(z) < a( < l), F~ the locus U(z) = tT, O < a < l .

Let the given points ~.~,~.2,'",~.. lie on the set where U(z) = 1, the given

points fl.l,fl.2,'",fl,,,.+ l lie on the set where U(z)=-O, such that we have

(A > O)

(1) lim (z--fl"!~)'"(z----fl"'"+O I '/" . ~ o o ( z - ~ . ~ ) . . - ( z - ~ . . ) = e x p I - A ( 1 - u ( z ) ) ]

on D, uniformly on every compact subset of D.

Let finally f ( z ) be a function analytic on E, meromorphic with precisely

v poles in D o (p < 1) and let the sequence of functions R.,.+~(z) be of type

(n,n + v) with v poles free and the n others restricted to lie in the points ~.k.

Then i f we have

(2) limsup [max If(z) - R.,.+~(z)[, z on E] 1/" < e -Ap , n - + o o

(a condition which is satisfied by the functions R.,.+~(z) of type (n,n + v)

with v poles free and the n others in the ~.k, of best Tchebycheff approxima-

tion to f ( z ) on E) the v free poles of the R...+~(z) are finite (for n sufficiently

large) and approach respectively the v poles o f f ( z ) in Dp. The sequence

R.,.+~(z) converges to f ( z ) throughout Op except in the poles o f f ( z ) , and

BEST APPROXIMATION BY RATIONAL FUNCTIONS 361

on each closed set S in the closure of D~(O < ~ < p) and containing no pole

o f f ( z ) we have

(3) limsup [maxlf(z ) - -R. , . .~(z)[ , z on S ] ' / ' < e -a(p-~). ?l--~ O0

If the Bj and Cj are given, and satisfy the geometric conditions enumerated,

then the fi.k and a.k can be determined to satisfy (1); it is sufficient [-5, w

if these points are distributed on B and C respectively, uniformly with respect

to the harmonic conjugate of U(z). If the curves B i and Cj are given, they

can be transformed by a suitable conformal map [6 3 so that the fl.k can be

defined as uniformly distributed on B, and the a., distributed among # + 1

suitably chosen fixed points with multiplicities designated in advance.

We shall point out some of the properties of the expansion of an arbitrary

analytic function obtained with the aid of (1), for the inequalities obtained

are to be employed in the proof of Theorem 1; further properties are to be

derived at the beginning of w

If equation (1) is satisfied, and if the function q~(z) is analytic in Dp + E,

then the rational functions r..(z) of type (n, n), determined by the fact that

the poles lie in the points ~.k and that r..(fl.k) = q~(fl.k), satisfy [-5, w167 8.7]

(4) limsup [max [ q~(z) - r..(z) 1, z on E] '/" < e -ap. n ~ o o

This inequality follows also by (19) below.

'Under the conditions of the theorem, let g(z) =- z*+ ... be a polynomial

whose zeros are the poles o f f ( z ) in Dp counted according to their multipli-

cities. The function f ( z ) . g ( z ) is analytic in Dp + E, and by (4) with

~(z) -- f ( z )g ( z ) we deduce

(5) limsup [maxlf(z) - r,,.(z)/g(z)[, z on E] 1/" < Cap. t l ~ o O

The function r,,.(z)/g(z) is of type (n,n + v), of which n poles are formally

in the points ~.k. Then by comparison with (5), it follows that the functions

R...+~(z) of best (Tchebycheff) approximation to f ( z ) on E satisfy (2).

362 s.L. WALSH

We continue the proof of Theorem 1 in several steps. By starting with (1)

or even in deriving (1), we may write [5, w167 8.7, 9.11]

(6) lim [(z -- ft.,) .-.(z -- fl.,.+x)[ 1/" = ~2(z), exterior to B, t l --* OO

(7) lim ](z - e . 0 . . . ( z - e..)l 1/" = ~ l ( z ) , interior to C, B---~ o0

uniformly on each compact set; to simplify the exposition we have chosen

here E as the interior of a Jordan curve B, and the region D bounded by B

and a second Jordan curve C O containing B in its interior. Comparison of

(1) with (6) and (7) yields

(8) r =_ Oz(z)/Cbl(z ) = exp [ - A ( 1 - C(z)] , A > O,

and A is equal to (1/27:) times the total variation along a locus F~

monic function V(z) conjugate to U(z).

We can now write

o f t h e har-

(9) S.(z) - R., .+v(z) - r. , .+v(z),

where the two functions of the second member are respectively the functions

of Theorem 1 and the functions r. . (z) /g(z) of (5); the latter function has v

poles fixed in the v poles o f f ( z ) in Dp, for n sufficiently large. The two functions

in the second member of (9) are each of type (n, n + v) with n poles formally

in the points e.k, SO S.(z) is of type (n + v, n + 2v). Inequalities (2) and (5)

give

(10) limsup Emaxls.(z) 1, z on E]'/" =< e -a~ n - ~ o o

As a matter of convenience, we first discuss the case v = O, and shall prove

L e m m a 1. I f v = O , we have

(11) limsup [maxlS. (z) l , z on D ~ + E ] I / ' < e -A(p-~), 0 < z < l .


To study the functions Sn(Z) exterior to B, we cannot consider merely the

quotient S,,(z)/m,,(z), where e).(z)is the rational function whose absolute

value occurs in (1), for this quotient may well have singularities on B. We

set then

T.(z) - S . ( z ) ( z - ~ . 1 ) " " ( ~ - ~ . . ) [~2(z)]"

a function whose absolute value is that of a function analytic (not necessarily

uniform) with no singularity in D nor exterior to B even at infinity. The poles

of S,(z) lie on or exterior to Co, hence exterior to BR if R (> 1) is suitably

chosen; we use the notation B R to indicate the image of ]w] = R when

w = 4J(z) maps conformally the exterior of B onto I w] > 1 with ~ = ~ ( ~ ) .

It follows [-5, w Lemma 1] that if 1 < Z < R and if F~(a > 0) lies interior

to Bz we have by (10), with 0 < a < z < 1,

(12) limsup [max lS,(z)] , z on F~]I/" < e-AP[(RZ- 1 ) / ( R - Z)] , n - - * oO

limsup [max[S.(z) l, z on F J '/" n---~ oo

< limsup [max]r . (z) l , z on r j 1/". e - a (1 -~ n - * o o

< limsup [maxlT.(z) l, z on F~]x/".e - a (1 -~ n - - * oo

< limsup E m a x l S . ( z ) l, z on F . ] t / " ' e -a( t -O" e a(t-~ n- -+ 0 0

Here we replace the first factor in the last member by the second member

of (12), and allow a to approach zero and Z to approach unity, which yields

(11).

It may be noticed that this proof of (11) does not require the analyticity

of the Jordan curves B and Co.

In Lemma 1 we have chosen for simplicity the boundary of D to consist

of merely two Jordan curves, but clearly the boundary of D may consist of

any finite number of such curves, with only minor modifications in the proof.

364 J.L. WALSH

Indeed, several components of B and C may be Jordan arcs instead of Jordan

curves.

We remark that Lemma 1 does not require that the functions R,,,+~(z)

be defined for every n; it is sufficient if these functions form an infinite se-

quence. A similar remark applies to Theorem 1. Thus Lemma 2 (below)

applies to any subsequence of the original sequence involved.

Suppose now v > 0. One can assume that each free pole either lies at in-

finity or is bounded; such a pole cannot be finite yet become infinite with n

[-compare 3, 4, 9]. A finite number of bounded free poles of S,,(z) or T,,(z)

cannot affect a limit or superior limit such as we have used in the proof of

Lemma 1, except in the vicinity of the limit points of such poles. Thus we

have [compare 3, 4]

L e m m a 2. I f v > O, the sequence S,,(z) defined by (9) converges to zero

throughout Dp except in the limit points of the free poles. Let S be a closed

set in Dp + E which contains no limit point of the free poles; we have

(13) limsup [max lS.(z)[, z on $31'"< e x p ( - A ] - p - m a x U ( z ) ] ) , n --~ oo

where max U(z) is maximum on S.

The proof of Theorem 1 now follows the proof [3, 4] for the case that

all the prescribed poles of R.,.+v(z) lie at infinity. Each subsequence of

R...+.(z) admits a new subsequence whose poles approach those o f f ( z ) in

Dp. Each pole o f f ( z ) in Dp is the limit of free poles of the R.,.+.(z), of the

same total order. The free poles have no limit finite or infinite other than

the poles o f f ( z ) in Dp.

Inequality (4) implies 115, w

limsup ['max] ~b(z) - r..(z)], z on D.] x/" <_ e -a(p-a), 0 < tr < p. n- -* o o

Then for the set S of (3) we have

limsup [maxl f (z ) - r..(z)/g(z)[, z on S] 1/" <_ e -a(p-~) , n- -~ oo

and (13) with (9) yields (3). Theorem 1 is established.


We add the remark, whose proof is essentially contained in that of [4,

Theorem 7], that if ~ is a pole of f (z ) of order k in Dp, then for each free pole

~, of the k poles of R,,,,,+~(z) which approach ~ we have

limsup 1cr - ~.11/. __< e-a(p-,O/k, ~1--~ O0

where ~ lies on F, .

Theorem 1 has been based on (1) rather than on (6) and (7), although we

might have chosen the opposite. Equation (1) may hold even if the ~,k and

fl,,k are not uniformly distributed on C and B respectively; for instance, if

the curves Cj and Bk are analytic, it is sufficient to distribute the C(.k uniformly

on a suitably chosen locus U(z) = const > 1, and the fl,,k on a suitably chosen

locus U(z) = const < 0.

We continue with some complements to Theorem 1. Although Theorem 1

does not require the R,,,,,+~(z) to be defined for every value of n, Theorem 2

makes that hypothesis.

T he o rem 2. With the hypothesis of Theorem 1, let the points

~.1,o~.2,...,ct.. be independent of n, so that all the prescribed poles of

R..n+v(z) are also prescribed poles of R.+1,.+~+l(z); this is possible [5,

w167 8.7, 8.8]. Let p(0 < p < 1) be the largest number such that f (z) is

meromorphic with precisely v poles in Dp, and let the R.,.+~(z) be defined

for every n. Then the equality sign holds in (2) and (3).

Suppose the first member of (2) less than or equal to e -Apl, 1 > p~ > p;

the locus U(z)= Pl lies in D and we have

(14) limsup [maxlRn+l,.+v+l(z)- R...+v(z)], = on E]I/"-__ e -ap'. 1 1 - ' 0 0

The function S~ whose absolute value appears in (14) is of type

( n + v + l , n + 2 v + l ) , with n + l poles in the points ~nk either on C or

separated from E by C, and with 2v poles which approach in pairs the v poles

of f (z ) . Lemma 2 deals with just such a function except for notation and

(together with Lemma 1) shows that (10) implies (13). Precisely that same

method of proof commencing with (14) shows that the present sequence S~

366 J . L . WALSI-I

satisfies the analogue of (13) with p replaced by Pl , on an arbitrary set S

in Dp, + E containing no pole of f(z) in Dp. It follows that the sequence

Rn,,+v converges uniformly, necessarily to the analytic function f(z) as limit,

in an annular region or finite number of annular regions containing Fp in

their interior, which contradicts the definition of p.

If the strong inequality holds in (3), we again reach a contradiction by the

method of proof of Theorem 5 below, and this completes the proof of

Theorem 2.

The condition of Theorem 2 that the ~nk be independent of n is indispensable

for the validity of this proof. Without this condition (or a similar one) the

function S~ is no longer of type (n + v + 1, n + 2v + 1), but rather of

type (2n + v § 1, 2n + 2v + 1), and the proof does not hold. But it is not

essential for the proof that the fl.k be independent of n.

Further, more refined, properties of the functions R.,.+~(z) in (2) can be

established by methods already developed [2, 3, 4], but we shall not consider

them here.

w Meromorphic Functions with Free Poles .

We turn now to the application of the preceding methods and similar ones

to the study of approximation by more or less arbitrary functions mero-

morphic in D + E whose significant poles are v (or fewer) in number and

unprescribed in position. The auxiliary functions used in the proofs of

,Theorems 1 and 2 remain auxiliary functions that disappear in the final re-

sults. In our methods we may change D and E into sets that are more con-

venient. Here two possibilities present themselves: (i) to use D and E in their

present form, which has the disadvantage that the functions r.,(z) of Theorem

1 are not bounded in D, even when v = 0, for the majority of their poles lie

on C; or (ii) choose a new function U(z) appropriate to the region D, then

choose the Jordan curves Cj bounding D as analytic (which can be accom-

plished with help of a suitable conformal map), extend harmonically the

function U(z) from D across those curves so as to be harmonic in a new

region Do, and then make use of the methods used to prove Theorem 1. We

shall continue with the method (i), but it is to be noticed that an arbitrary


region D + E of Theorem 1 can be mapped conformally onto a region

A = D, + E(0 < z < 1) also satisfying the conditions of Theorem 1.

We choose the geometric configuration of Theorem 1, then, except that

we may assume D to be connected. We wish to approximate on E to a function

q~(z) analytic on E, meromorphic with precisely v poles in Dp, by functions

tk,(z) each meromorphic with a number not greater than v of poles in

A = D , + E , 0 < p < z < l , where z is henceforth fixed. We denote by

o~,(z) the rational function of z whose modulus appears in (1). The auxiliary

rational functions ~b,(z) that make the approximation will have their poles

in the points ~,k uniformly distributed on C (thus exterior to A), and (if v = 0)

are equal to ~b(z) in the points fl,,k uniformly distributed on B. In this case

if v = 0, as in (4), the r are defined by the two equivalent equations

(15) 1 fco.(~)r r - r - ~ r - z ) ' z in D, + E .

F ~

1 1 z # ~ k . (16) r - 2rci w.(t)

F

In (15) we choose e, 0 < e < tr < p < z; by (1) and (15) there follows

(17) limsup [max ]r - r l, z on F~] 1/" < e x p [ - A ( t r - e ) ] . n---~ O0

Inequality (17) holds when z lies on E; if we let e approach zero we have

(18) limsup [max I r r z on /Z] ~/" ___ e x p ( - A a ) , n.--~ oo

and if we further let tr approach p ,

(19) limsup [maxlr r z on E] x/" =< exp(--Ap). n--~ oO

I f we allow tr to approach p in (17) we have

(20) lim sup [max I r - r z on F~] ~/" < e x p [ - A ( p - e)]. /i--# O0

368 J.L. WALSH

In (16) we take z on F,; when tr approaches p there results

(21) l imsup [max] q~.(z), z on F.] i / . < e x p [ A ( z - p)]. n- -+ o o

The functions ~b,(z) are analytic in A; if we suppose them to be defined for

every n, then (19) and (21) yield for tr = 0 and tr = z

(22) lim sup [max] ~b. + l(z) - qS.(z) 1, z on F.] 1/. < exp [A(a - p) ] . n - - + ~

The first member of (22) is a convex function of tr for 0 < tr < z, which has

a value not greater than e x p ( - A p ) when tr = 0, a value not greater than

A(z - p) when tr = z, and which cannot be negative when a = p, if we choose

p the largest number such that f ( z ) is meromorphic with precisely v poles

in Fp. Thus the equality

(23) limsup ['max[4~.+x(z) - ~b.(z) I z on F.] x/" = exp[A(cr - p)] B--+ OO

holds for all values of o-, 0 < ~r < z.

The sequence q~.(z) that occurs in (15)-(23) is analytic in A, therefore not

immediately suitable for approximation to meromorphic functions. To

introduce approximation by meromorphic functions we say that the sequence

of functions f . (z) meromorphic in a region A o is of class (M,v) in A o if each

function is of the form f . l(z)/ f .2(z), where f . l (z ) is analytic in A o with

(24) limsup [supl f , x(z)[ , z in Ao]l/" < M, n - - ~ OO

and where the denominator is a polynomial zU+ ... of degree p ( < v). Con-

sequently, if two sequences are of class (M, v) in Ao, the new sequence formed

as their term-by-term sum is of class (M, 2v) in Ao.

T h e o r e m 3. Let the function f ( z ) be analytic on E (notation of Theorem 1),

meromorphic with precisely v poles in Dp, 0 < p < z ; then there exists

a sequence of funct ionsf . (z) of class (e a( '-") ,v) in A ( = D r + E) such that

we have


(25) limsup [ m a x l f ( z ) - f . ( z ) [ , z on E] ~/"< e -a", tl.-~ O0

(26) limsup [maxlf.l(z) ], z in A ] l / " < e x p [ A ( z - p ) ] , 11"--*oo

(27) f . ( z ) - f . l ( z ) / f . ~ ( z ) , L 2 ( z ) - z " + . . . , /~ <= v.

Let g ( z ) - z~+ ... be the polynomial whose zeros are the poles o f f ( z ) i n

Dp. We have (25) by settingf.(z) -- r..(z)/g(z) in (5), and (26)is a consequence

of (21), with r - r..(z). A sequence of functions f .(z) satisfying (25) and (26) having extremal

properties for each n is not difficult to provide. We write e. = [max If(z) -f .(z)] ,

z on El , where f . (z) - r..(z)/g(z) in (5). Consider now the set of all admis-

sible functions F . ( z ) - F.l(z)/F.z(Z), where F.l(z) is analytic in A with

[F.l(Z)[ _-< [-max [r..(z)[, z in A] for all z in A and Fnz(Z ) =-- z p -I-'. '. Choose

a minimizing sequence of admissible functions FC.1)(z),F(.2)(z), ... , namely such

that lim [-max[f(z) - F~.k)(z)[, z on E] = inf[maxlf(z) - r . (z ) [ , z on E l , n - ~ o o

where the infimum is taken over all admissible Fn(z ). The functions ( k ) __ (k ) (k) F. ( z )= F.1 (z)/F~z(z) form a normal family in A, because the F(.~)(z) are

analytic and uniformly bounded in A, and no zero of the F~.k2)(z) can be-

come infinite unless the sequence F~.k)(z) approaches zero (k ~ ~ ) uniformly

on E and hence (Vitali) also in A; in the latter case we may set F~.k)(z)- 0

in A for every k. Consequently the sequence F~.k)(z) admits a subsequence

which converges uniformly in A, necessarily to an admissible function F~.~

This "last function is extremal in the sense that for an arbitrary admissible

F.(z) we have

[max lf(z ) -F.(~ z o n E] __< [max l f (z ) - F.(z) l, z o n E].

In particular the first member is not greater than e.. The sequence of functions

F~.~ n = 1,2, . . . , satisfies (25) and (26). Incidentally, our condition on

[F.l(z) l in A can be broadly generalized in the light of (26).

We develop now some results that lead toward the converse of Theorem 3;

the hypothesis consists of (25) and (26); we seek the properties off . (z) .

370 J.L. WALSH

L e m m a 3. Let the functions ~.(z) be analytic on E, of

(e a~'-p'),v) in A = D, + E, 0 < Pl < z < 1. Suppose also

(28) limsup [maxl~.(z) l, z on E ] ~/" < e-aP' ; tl---~ O0

class

thus (~n(Z) has no pole on E when n is sufficiently large. Suppose the finite

poles of the ~.(z) uniformly bounded, and S a closed set in the closure of

D. + E, 0 < a < Pl, which contains no limit point of the poles of the r

Then the sequence ~.(z) converges to zero uniformly on S, with

(29) limsup [maxl~.(z) 1, z on S]1/"< e A ( a - p D .

W-.~ QO

We set r -~ . l ( z ) / r r - z " + "".

We have l~.2(z)] > m , ( > O) on S, and when n is sufficiently large

(30) [r I < Ir l/ml z on S.

With z on E, we have ] O.2(z) l < M1, for the zeros of ~b.2(z ) are uniformly

bounded; thus

(31) l~pn,(Z)l/M1 <= I~.(z)l , z on E.

The hypothesis of Lemma 3 implies

(32) limsup [maxl~ . l (z ) ] , z in A] l f"< e A('-p'), n- -~ o o

and (28) with (31) gives

(33) limsup [maxlO.l(z) I , z on E] I/" < e -a~ ?l--+ O0

By the convexity of the first member of

(34) limsup [max IO.11, z on F~] ~/" < e A('-p~) , n- -~ OO

0_<a<_~,

with respect to a, (34) is a consequence of (32) and (33).


We can now write successively, by (30), by S as a subset of D, + E, and

by (34)

limsup [maxlr z on S] 1/" ?l.--i, r

< limsup[maxl~.l(z), z on S] 1/" n---~ O0

=< l imsup[max[r z on F~] x/" =< e ~(~-~ ", n " * O0

thus (29) follows, and Lemma 3.

Lemma 3, like Theorem 4 below, does not require that the functions r

or f . (z) be defined for every value of n; an arbitrary infinite sequence suffices.

Theorem 4. Let D, E, D, be as in Theorem l, let z be fixed, 0 < z < l ,

and let A be the region (or regions) D r + E. Let the function f ( z ) be analytic

on E, meromorphic with precisely v poles in Dp, 0 < p < z < 1. Let the

sequence of functions fn(Z) be of class (eaC'-P),v) in A, and such that

(35) limsup [max] f ( z ) - f . ( z ) ] , z on E] ~/" < e -ap n --i, O0

(a condition satisfied by certain functions f .(z) of best approximation

to f ( z ) on E); thus f . (z) has no pole on E, for n sufficiently large. Then

for n sufficiently large the v free poles of f . (z) are all finite; they approach

(n ~ oo) respectively the v poles o f f ( z ) in the region Dp. The functions f . (z) approach f ( z ) throughout the region A ' = Dp + E minus the v poles o f f ( z ) in Dp, and on each closed set S in A' and in the closure of D~(tr < p) we have

(36) limsup [maxlf (z) - f~(z) [ , z on SIal"< e a(`-~ n - - ~ O0

The function f ( z ) is meromorphic in Dp, and its poles in Dp are the zeros

of a polynomial g ( z ) - z ~ + .... Then the function q~(z)=_f(z).g(z) is

analytic in Dp, and inequality (5) is satisfied, where the function r,,(z)/g(z)

is rational. These functions r , , ( z ) - ~,(z) are identical with the functions

q~.(z) of (15) and (16), with q~(z) as already indicated. Our equivalent to (21)

is unchanged (p < ~)

372 J . L . WALSH

(37) limsup [-max I q~.(z)l, z on F~]l/" < e a('-p). n " * O0

By (20) it follows that c~.(z)/g(z) approaches f ( z ) uniformly on any closed

set in Dp containing no pole o f f ( z ) .

We have now two sequences of functions, the dp.(z)/g(z) of (15) and (16)

and the f .(z) of Theorem 4, each sequence of class (e a('-p), v) in A, by

(37) and by the hypothesis of Theorem 4. Then the difference (I)n(Z) -- Cb.(z)/g(z) --f .(z) is of class (e ~ '-p) , 2v) in A, when the v poles of thef.(z) are bounded

as n becomes infinite. By (25) for the functionsf.(z) - r..(z)/g(z) of Theorem 3,

and (35) for the functionsf.(z) of Theorem 4, we have

(38) limsup [maxlr z on El ' I"< e -ap . n. -~ O0

Lemma 3 applies to the sequence ~.(z).

The proof of Theorem 4 henceforth follows the proof of I-4, Theorem 3].

Suppose the v poles of the f . (z) bounded as n becomes infinite, and let z =

be an arbitrary pole o f f ( z ) in Dp of multiplicity 2, hence a pole of qb.(z)/g(z) when n is sufficiently large. There exists an annulus in Dp whose center is

which contains in its lacuna no other pole o f f ( z ) , which contains in its lacuna

at least 2 poles of each term of a suitably chosen new subsequence of any

subsequence of the f . (z) of Theorem 4. It follows that these functions f .(z) possess respectively v poles in Do, when n is sufficiently large, which approach

respectively the v poles o f f ( z ) in Dp. The case that the poles of the f .(z) are

not necessarily bounded as n becomes infinite is easily handled, and it remains

merely to prove (36).

Lemma 3 with (38) gives (29) in the form

(39) limsup [max ldpn(Z)/g(z) - f . ( z ) [ , z on S] 1/. < ea(.-p), n---~ oO

and (20) gives

(40) limsup [max ldP(z)/g(z)- dp,,(z)/g(z)], z o n S]l/n~ e acr n " ~ O0

with (o(z)/g(z)=f(z), which yields (36).


As a complement to Theorem 4 we have

T he o rem 5. In Theorem 4, let the f,(z) be defined for every n, and let

p(< z) be the greatest number such that f (z) is meromorphic with precisely

v poles in Ep. Then the equality sign holds in (35) and if S is a continuum (not a single point), holds also in (36).

It is clear that the conclusion of Theorem 5 can fail if one considers all

possible norms and if the f,(z) are not defined for every n. In fact, even with

the Taylor development of a function (which can be determined by best poly-

nomial approximation in the sense of least squares over a circumfernce with

center the origin) the superior limit that occurs in (35) is not necessarily a

limit, and cannot be a limit if the series has sufficiently large gaps.

We consider the sequence f . + l ( Z ) - - f . ( z ) , of class (ea(~-P),2v) in A, of

which the v pairs of poles approach the v poles o f f ( z ) in D o respectively. If

the strong inequality sign holds in (35), we may write (p < Pl < z)

(41) limsup [ m a x [ f ( z ) - f . ( z ) [ , z on Ell~"< e-aP~< e -ap, 1 1 - ' 0 0

(42) limsup [max[ f .+l (z ) - f . ( z ) ] , z on E]I/"< e -Apl. . - - ~ oo

It follows then from Lemma 3 that the sequence f . + i ( z ) - f . ( z ) converges

uniformly in an annular region (or in several such regions) which contain

Fp in their interiors, therefore annular regions in which f (z) is analytic, contrary to the definition of p.

It remains to consider (36). The sequence f .+ l ( z ) - f . ( z ) is analytic and

possesses the function A [ U ( z ) - p] as harmonic majorant [8] in A except

in the neighborhoods of the poles o f f ( z ) ; for the f.(z) are of class (e a(~-p), v)

in A by hypothesis, and on E we have (42) with pl replaced by p. The v poles

o f f .+ l(z) and the v poles off . (z) have no effect on this harmonic majorant,

except in the union Vo of the neighborhoods of the v poles o f f (z ) in D o. For

at each point of D~ except in Vo we have uniformly

(43) limsup If.+l(z) - f . (z) [1/ . < eatV(z)-d; . " ~ OO

this inequality holds uniformly on E and (in terms of boundary values) on

374 J.L. WALSH

F,, and also (by (36)) on the boundary of Vo. In fact, one can define a func-

tion Ul(Z) harmonic in D, - Vo, equal to U(z) on the boundary of D~, equal

to �9 on the boundary of V0; this function Ul(z) approaches U(z) when the

diameters of the components of Vo approach zero, uniformly in D~ minus

a fixed Vo. Hence (43) holds with U(z) replaced by Ul(Z), and even without

this replacement, uniformly in D , - Vo.

If the first member of (36) is less than e a~`-v), we apply [8, Corollary to

Theorem 1]: I f V(z) is a harmonic majorant of the sequence [F,(z)] 1/" in a

region R, and i f for a continuum Qo in R consisting of more than one point

we have

limsup [maxlF.(z) l 1/", z on Qo]a/" < [maxe v~z), z on Qo], f l - - ~ 0 0

then this strong inequality holds for every continuum Qo in R. It follows

then from (43) that its first member is less than unity throughout a complete

neighborhood of Fp, contrary to the definition of p.

We add a remark relative to the principal part O(z) of a pole ~ of f (z) in

Do, and to the sum O,(z) of the principal parts of the poles of f ,(z) which

approach ~. If y is a circumference with center ~ whose closed interior lies

in Dp but contains no other pole off(z) than ~, we may write

(44) O(z)-O.(z)=-2~ t f [f(t)---f~-(t)]dt, zexterior to y.

?

If U(~) = tr, 0 < tr < p, and if the radius of y is allowed to approach zero,

we deduce by (44) and (36)

(45) limsup [max I O(z) - O . ( z ) ] , z on S , ] x/. < ea~,-p), n---r o o

where S t is any closed set exterior to y.

Theorem 6. Under the conditions of Theorem 4, let $1 be a closed

set not containing the pole z = ~ off(z) in Dp, and suppose U(~) = t7. Then

(45) is valid.


I f we consider the re la t ion (45) for all the poles o f f ( z ) in Dp, and i f each

o f these poles lies on or in ter ior to F , , 0 < a < p , then for any closed set

$1 conta in ing no pole o f f ( z ) in Dp we have

(46) l i m s u p [-max] ]~ [ - 0 ( z ) - 0 , ( z ) ] ] , z on $111/"<= e A~'-p). n--.r oo

Theorems 3-6 clear ly have proper t ies o f invar iance under one- to-one con-

fo rmal t r ans fo rmat ion o f the configurat ions involved. There exist extensions

o f these resul ts : (i) to an a rb i t r a ry ffh power n o r m over E(0 < p < ~ ) ; (ii)

where the n o r m is a rb i t r a ry and hypotheses and conclusions refer to more

refined proper t ies than (35) and (36). These extensions are reserved for ano ther

occasion.

REFERENCES

1. J. L. Walsh, Note on the convergence of approximating rational functions of prescribed type. Proc. Nat. Acad. Sci. (U.S.A.) vol. 50 (1963), 791-794.

2. - - - , The convergence of sequences of rational functions of best approximation. 'Math. Annalen. vol. 155 (1964), 252-264.

3. - - - , The convergence of sequences of rational functions of best approximation II. Trans. Amer. Math. Soc. vol. 116 (1965), 227-237.

4. - - , The convergence of sequences of rational functions of best approximation with some free poles. Proceedings of Symposium on "Approximation of Functions" held by General Motors Corp. in Detroit, Sept. 1964. Ed. by H. L. Garabedian, pub. by Elsevier (Amsterdam); pp. 1-16.

5. - - . , Interpolation and Approximation. Amer. Math. Soc. Coll. Pubs. No. 20,1935. 6. ~ , A sequence of rational functions with application to approximation by

bounded analytic functions. Duke Math. Jour. 30 (1963), 177-190. 7. - - , Sur l'approximation par fonctions rationelles et par fonctions holomorphes

born6es. Annali di Mat. (4) vol. 39 (1955) 267-277. 8. ~ , Overconvergence, degree of convergence, and zeros of sequences of

analytic functions. Duke Math. Jour. vol. 13 (1946), 195-234. 9. - - , An extension of the generalized Bernstein lemma. Colloquium Mathe-

maticum (Wroclaw), Leja Jubilee Volume. To appear.

UNIVERSITY OF MARYLAND,

COLLEGE PARK, MARYLAND, U.S.A.

(Received March 4, 1966)

Documents

Best approximation by rational functions and by meromorphic functions with some free poles