Note on Degree of Convergence of Sequences of Rational Functions of Prescribed Type

Note on Degree of Convergence of Sequences of Rational Functions of Prescribed TypeAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 67, No. 3 (Nov. 15, 1970), pp. 1188-1191Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/60433 .

Accessed: 05/05/2014 21:45

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 194.29.185.253 on Mon, 5 May 2014 21:45:26 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nas

http://www.jstor.org/stable/60433?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Proceedings of the National Academy of Sciences Vol. 67, No. 3, pp. 1188-1191, November 1970

Note on Degree of Convergence of Sequences of Rational Functions of Prescribed Type

J. L. Walsh*

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARYLAND, COLLEGE PARK, MD. 20740

Communicated August 5, 1970

Abstract. This note studies the degree of convergence of various sequences of rational functions of given types which are of best approximation to a given function analytic on a given point set and meromorphic on a larger set. The sequences are conveniently represented by the analog of the Pade table.

A rational function rmn(z) of form

( aozm + azm

- + - ... + am

Ebil | 0 bozn b0+ blzn-1 + +

bn

is said to be of type (m,n). There exists' for a more or less arbitrary closed point set E of the z-plane and a function f(z) continuous on E a rational function Rmn(Z) which minimizes the Tchebycheff (uniform) norm

[max| f(z) - rmn (z)f, z on E] (1)

among all rational functions of type (m,n). These minimizing functions can be arranged in a table analogous to that of Pade:

Roo, Rio, R20, . .

Roi, Rii, R21, ... (2)

R02, R12, R22, .

The first row consists merely of polynomials. It is appropriate to study the convergence of various sequences formed from this table. Convergence of diagonal2, of rows3, and of columns4 has been variously studied, and the present note has the purpose of considering more general sequences, where the function

f(z) is meromorphic (for instance) at every finite point of the plane and E is an arbitrary closed bounded point set whose complement K is connected, and regular in the sense that Green's function g(z) for K with pole at infinity exists. We denote by E<, generically the locus g(z) = log -(>0), and by D, the interior of E,.

It is to be noted that the norms corresponding to the function f(z) have the monotonic property

lf(z) - Rmn(Z)l > 1f(z) - Rjk(z)ll (3)

1188



VOL. 67, 1970 SEQUENCES OF RATIONAL FUNCTIONS 1189

whenever m < j, n < k. We proceed to prove

THEOREM 1. Let E be a closed bounded point set whose complement K is connected and regular. Let the function f(z) be analytic on E, meromorphic on Dp, 1 < p < , with poles al, a2, .. ., a,, in DP and enumerated according to their mul- tiplicities. Then there exist rational functions rmn(z) of respective types (m,n) with poles in the ai, such that for the norm on E

lim sup {If(z) - rmnn(z)I l/m < l/p. (4)

Consequently for the extremal functions of table (2) we have also

lim sup I1f(z) - Rnn(z)) II /m _ 1/p. (5)

Inequality (5) follows from (4) and the mninimizing property of the Rmn(z). Inequality (4) follows at once from the known5 degree of approximation on

E by polynomials qm(Z) of respective degrees m to the function

F(z) = (z - Ol)(Z - ac2).. .(z - an)f(z), which is analytic on E and throughout Dp:

lim sup ]lF(z) - qm(z)lI/m < I/p.

Under the conditions of Theorem 1, suppose the poles of f(z) lie on the respective level loci Ep,, Ep2,. .., where pi < p2 -' p3 .... Then Theorem I does not necessarily apply successively to each line of (2), for we may have for instance P1 < p2 = p3, in which case E2 = EP3 passes through two poles of f(z), and n = 2 is not possible. However, we clearly have, by (3)

IIf(z) - Rm2(Z)\ < lf(z) - R51(z)\\,

lim sup flf(z) - Rm2 () / < lim sup flf(z) - Rmr (Z) l/m. m-- co m-+ o

More generally, we obviously have by (3)

lim sup lf(z) - Rm,n+l (z) 1/m < lim sup I f(z) - Rmn (Z) lI/rm

m-->? co m-- co

< lim sup lf (z) - Rm,n-1 (z)l 1/mn (6)

even if Dpn contains more than n zeros of f(z), that is to say, even if Theorem 1 does not apply directly to p == pn.

THEOREM 2. Suppose f(z) in Theorem 1 is modified so as to be meromorphic with an infinite number of poles in Dp. Then we have

lim sup l1f(z) - Rnm(rZ)\l i/m _ 1/p. (7)

There exists a sequence of numbers pi < P2 < . . ., n- p such that Eop passes through no pole of f(z) in Ep. Denote by Pn the number of poles of f(z) in Dpn.

Then we have by (3) for m sufficiently large

11f(z) -

Rmm (Z) 11 - !{f(z) Rmrn,p ,(Z)



1190 MATHEMATICS: J. L. WALSH PROC. N. A. S.

whence by (5)

lim sup 1(z) -Rmm(Z)l1 lim sup IIf(z) - Rmrn.P(z) l1/M < l/pp., m---* o M--4 oo

which yields (7) as pp -- p.

COROLLARY 1. If p = oo in Theorem 2, we conclude

lim l|f(z) - Rmm (Z)11/- O. (8) m-->00

Inequality (8) is of particular importance, for (ref. 5, ?9.3) it implies con-

vergence of Rmmr(Z) to f(z) (or to its analytic extension) in any region containing no limit point of poles of the Rmm(Z) and which can be joined to E by a new

region containing no such limit points. We turn now to sequences in (2) other than diagonal and rows. If we set

N = max (m,n), then rmn(Z) is said to be of degree N. We prove for an arbitrary

sequence (m,n) COROLLARY 2. Under the conditions of Theorem 2, (i) with n > m, N = n,

1 > m/n > T > 0, we have

lim sup fIf(z) - Rmn(z) \11N < l/p, (9)

and (ii) with n < m, N = m, n/m > r > 0, we have

lim sup 1f(z) - Rmn(z)) 11N < I/pT. (10) mo

With (i) we have

|lf(z) - Rmn(Z)ll < I|f(z) - Rmm(Z)11,

lim sup l1f(z) -- IR (Z) \11N < lim sup (lif(z) - Rr(z)lIl\m)m/ n /< 1/PT.

r---? co Mr--- oo

With (ii) we have

lif(z) - Rn(z)11 < 11lf(z) - Rnn(Z)l, lim sup Ilf(z) - R z) lim sup (1z n) - R z)nn Ii/nIn/ < /

Of course, in Corollaries I and 2 the sequence Rmn(z) need not be defined for every m or for every n, so a given sequence (m,n) can be separated into two subsequences which satisfy (i) and (ii) respectively. Sequences parallel to the diagonal are included in both (i) and (ii), and below.

If we have in Corollary 2 the important case p = co, then the second members of (9) and (10) become zero.

In case (i) of Corollary 2, the condition m/n > r > 0 can be replaced by

lim inf (m/n) r > 0,

and in case (ii) the condition n/m > r > 0 can be replaced by

lim inf (n/m) > r > 0. n--1 oo



VOL. 67, 1970 SEQUENCES OF RATIONAL FUNCTIONS 1191

There are two other comments relative to sequences found from table (2) with the hypothesis of Theorem 2. First, we consider a sequence Rmnn(z) where m -- oo but n is bounded; we suppose lim infm,-0o n = no, lim supm-,o, n = ni, then for m sufficiently large we have no

<

n < ni. By (3) we conclude for n sufficiently large

1f(z) - Rmni(Z)\ II lf(z) - Rmn(Z)I1 I 1f(Z) - Rmnol\,

l/pnl = lim sup lIf(z) -- Rmn(z)ll/m < l/pno. (11) m??--- -oo

Second, we consider a sequence Rmn(z) where m is bonded but n --a oo. For n sufficiently large we have mo < m <_ mi, where lim inf,no_ m = m0, lim supnco m = mi. Here the analog of (11) depends heavily on the zeros rather than the poles of f(z). We suppose that F(z) - ./f(z) is analytic and different from zero on E, meromorphic with precisely k poles in Ex,, X1 < X2 < .. - ~ X < co. Then4 we have for the rational functions rnm(z) of best approximation to F(z) on E with m= k

lim sup FlF(z) - rnrm(z)1/n <- 1/X. (12)

The functions l/rnm(z) can4 be used as comparison functions for the degree of convergence of the functions Rmn(Z) of best approximation to f(z) on E. There results as in (11)

I/Xr, < lim sup If(z) - Rm,(z)l/n _ 1/Xmo, n---- co

where the mi are the respective numbers of zeros of f(z) in the Em,.

Key Phrases: approximation, rational functions, degree of convergence, analog of Pade table.

* This research was sponsored (in part) by the U.S. Air Force Office of Scientific Research, grant no. 69-1690.

1 Walsh, J. L., Math. Zeit., 38, 163 (1934). Walsh, J. L., Acta Math., 57, 411 (1931).

3 Walsh, J. L., Math. Ann., 155, 252 (1964). 4 Walsh, J. L., Proc. Nat. Acad. Sci. USA, 50, 791 (1963). 5 Walsh, J. L., Interpolation and Approximation (Amer. Math. Soc. Colloquium Pub., 1935),

vol. 20.



Documents

Note on Degree of Convergence of Sequences of Rational Functions of Prescribed Type