Note on the Convergence of Approximating Rational Functions of Prescribed TypeAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 50, No. 5 (Nov. 15, 1963), pp. 791-794Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/71921 .

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Proceedings of the

NATIONAL ACADEMY OF SCIENCES Volume 50 ? Number 5 ? November 15, 1963

NOTE ON THE CONVERGENCE OF APPROXIMATING RATIONAL FUNCTIONS OF PRESCRIBED TYPE

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated September 3, 1963

By a rational function of type (j, k) we understand a function of the complex variable

aozj + alz-1 +z + a. j i r(Z) bozk + lZk -1+ . + b

If the function f(z) is continuous on a closed bounded set E, there exists under suitable conditions a rational function of type (j, k) of best approximation to f(z) on E, namely, minimizing the Tchebycheff (uniform) norm

[max I f(z) - ri(z) , z on E] (1)

among all functions of type (j, k). These minimizing functions can be arranged in a table' which is analogous to that of Pade:

r00, r10, r20 . .

r01, r11, r21, . .

r02, r12, . . .

The first row consists merely of polynomials. It is appropriate to study the con- vergence of various sequences formed from this table. I have recently studied2 3 the convergence of certain rows of the table and now wish to indicate some re- sults concerning the convergence of columns.

To be more specific, we have3

THEOREM 1. Let E be a closed bounded set whose complement is connected, and regular in the sense that it possesses a Green's sfunction G(z) with pole at infinity. Let

Ev denote generically the locus G(z) = log o-(> 0). Let the function f(z) be analytic on E, meromorphic with precisely v poles interior to Ep, 1 < p < o*. Let the rational functions rn,(z) of respective types (n, v) satisfy

lim sup f(z) - rn^(z) l1/ l/p, (2) n--- coo

where the norm is as in (1). Then for n sufficiently large the function r,,(z) has pre- cisely v finite poles, which approach respectively the v poles of f(z) interior to Ep. If

791

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792 1MATHEMATICS: J. L. WALSH PROC. N. A. S.

D denotes the interior of Ep with the v poles of f(z) deleted, the sequence rn,(z) converges to f(z) throughout D, and for any closed bounded set S in D and in the closed interior

of E,, I1 < < p, we have

lim sup [max I f(z) - rn,(Z) |, z on S]1/ aC/p. (3) n--.-- co

If the rn,(z) are the rational functions of type (n,v) of best approximation to f(z) on E in the sense of Tchebycheff, then (2) is satisfied.

Thus far, the rn(,z) need not be defined for every n, but below they shall be so defined. Whether the rn,(z) are extremal or not, let p be the largest number such that f(z) is

meromorphic with precisely v poles interior to Ep. If (2) holds, then (2) holds with the

equality sign, as does (3) with S = Er, provided 1 < ao < p, and provided no pole of f(z) lies on E,.

As a complemnent to Theorem 1 we shall proceed to prove THEOREM 2. Let E and Er be as in Theorem 1. Let the function F(z) be analytic

and different from zero on E, meromorphic with precisely v zeros interior to Ep, 1 < p _ oo. Let the rational functions Rvn(z) of respective types (v,n) satisfy for the

Tchebycheff norm on E

lim sup II F(z) - R,n(z) l/n Ilp. (4) n---> oo

Then the poles of the R,n(z) interior to E, approach either Ep or the respective poles of F(z) interior to Ep, multiplicities included. If D1 denotes the interior of Ep with the

poles of F(z) deleted, the functions R,,(z) converge to F(z) throughout D1, and for any closed bounded set S in Di and in the closed interior of E, we have

lim sup [max IF(z) - R,,(z) I. z on S]l/ < a-/p. (5) n-->- co

If the R,n(z) are the rational functions of type (v,n) of best approximation to F(z) on E in the sense of Tchebycheff, then (4) is satisfied.

Thus far, the Ryn(z) need not be defined for every n, but below they shall be so defined. Whether the Ryn(z) are extremal or not, let p be the largest number such that F(z) is

meromorphic with precisely v zeros interior to Ep, I < p ? oo. If (4) holds, then (4) holds with the equality sign, as does (5) with S = E,, provided 1 < Oa < p, and provided no pole of F(z) lies on E,.

Theorems 1 and 2 are essentially equivalent to each other with interchange of the roles of zeros and poles.

We proceed to prove Theorem 2. We write on E

1 1 _ R,n(z) - F(z)

F(z) Rvn(z) F (z)Rn ) (6)

For n sufficiently large, the denominator F(z)R,n(z) is bounded from zero, by (4)' Then with the notation f(z) = 1lF(z), rn^(z) = 1/R,^(z) inequality (2) follows from (4) and (6), so Theorem 1 applies. If a is an arbitrary zero of F(z) interior to Ep, let C be a circle in D1 whose center is a, which contains on or within it no

pole of F(z) and no zero of F(z) other than a. Then a is a pole of f(z) interior to

Ep, but f(z) is analytic and different from zero on C, as is rn,(z) for n sufficiently large; rn,(z) converges to f(z) uniformly on C. The functions F(z) and R,n(z) are for n sufficiently large analytic on and within C; hence, by the equation

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VOL. 50, 1963 MATHEMATICS: J. L. WALSH 793

1 1 r (z) - f(z) (7)

f(z) rv (z) f(Z)rnV(Z)

R,n(z) converges uniformly on and within C to F(z). For n sufficiently large the functions Rn(z) have precisely v finite zeros, which approach, respectively, the v zeros of F(z) interior to Ep. The functions R^, (z) converge to F(z) throughout Di.

By Hurwitz's theorem, each zero of f(z) interior to Ep is approached by an equal multiplicity of zeros of rn,,(), so each pole of F(z) interior to Ep is approached by an equal multiplicity of poles of R,,n(z). No point (not even a pole of f(z)) interior to Ep not a zero of f(z) is approached by zeros of the respective rn,(Z) (compare Lemma 1 of ref. 2), so no point interior to Ep not a pole of F(z) is approached by poles of the respective R,n (z).

For such a point set as the circumference C just considered, inequality (3) ap- plies, where S is C. By equation (7), inequality (5) likewise applies, where S is the closed interior of C. In order to prove (5) in Theorem 2, we choose r, a < r < p, so that no zeros or poles of F(z) lie on E. 'With center each zero of F(z) on S, let a circle C (as before) be constructed so that the union S' of S and the closed interiors K of those circles lies interior to Er. Then by (3) for z on S-K and by (7), and by (5) for z on K with a replaced by r, we have

lim sup [max IF(z) - R,(z) r, z on S']l/n 7r/p, n-,- o

an inequality valid also for z on S. If we :replace S' by S and allow r to approach o, inequality (5) follows.

Again, by (7) and (2) we have for the Tchebycheff norm

lim sup IF(z) - 11n < /p, (8) n-- oo rnr(z)

where the rn,(Z) are now functions of types (n, v) of best approximation to f(z) on E. Since (8) holds for certain functions l/r,,(z) of types (v, n), it holds also for the functions Rn,(z) of best approximation to F(z) on E; that is to say, (4) hclds for the latter functions.

Under the conditions of the last paragraph of Theorem 2, suppose (4) holds with the inequality sign. Then by (6) it follows that (2) holds with the inequality sign, which contradicts Theorem 1. Under the same hypothesis, it follows by the method just used that (5) holds with the equality sign and with S = E,, provided no pole or zero of F(z) lies on E,; the corresponding values of a are everywhere dense in the in- terval 1 < a < p. If for a particular E,,, interior to Ep and passing through a zero but no pole of F(z), the first member of (5) with S = E,,, is ao/p, less than arl/p, we choose a neighboring r, 1 < r < ao, so that no pole of F(z) lies between E,, and ET, and that no zero or pole lies on ET. Then for all E,, r < a- < o-, the first member of (5) with S = E, is not greater than [max (ao, r) i/p, -which is less than il/p; this contradicts (5) with S = E, and with [max(ao, r)] < o- < o-1 as just proved when no pole or zero of F(z) lies on E,. The proof of Theorem 2 is complete.

It is indicated3 that with the hypothesis of the last paragraph of Theorem 1, and if f(z) = 0 on no component of E, each point zo of Ep is a limit point of zeros of the rnP(z); consequently with the hypothesis of the last paragraph of Theorem 2, each

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794: MATHEMATICS: F. E. BROWDER PRoc. N. A. S.

point z, of Ep is a limit point of poles of the R,,(z). This conclusion is true evenl if

p = oo, for then the only finite limit points of the poles of the Rn,,(z) are the poles of F(z), where -the multiplicity of the pole of F(z) equals the multiplicity of the

poles of the Rn,,(z) approaching it. As has been indicated by the statements of Theorems 1 and 2, in the former

there are few poles of the r,,(z) and their limit points are the poles of f(z) interior to Ep; in the latter there are few zeros of the R,^(z) and their limit points are the zeros of F(z) interior to Ep. If the poles aj of f(z) can be so ordered that G(al) <

G(C2) <. . . < G(ac), and if f(z) has these but no other singularities interior to ET,, then for each v not greater than u, the limit points of the poles of the r,,(Z) are the first v poles a,, a, . . . , a,. If the zeros aj of F(z) interior to .E can be similarly ordered, and if F(z) is meromorphic interior to E,, then for each v not greater than g the limit points of the zeros of the Rn,,(z) are the first v zeros a., a2, . . . , a.

The present note deals primarily with the Tchebycheff norm, but under suitable circumstances the methods of reference 2 apply to a pth power norm and yield an analogue of Theorem 2.

This research was sponsored (in part) by the Air Force Office of Scientific Research.

1 Walsh, J. L., Math. Zeit., 38, 163-176 (1934). 2 Walsh, J. L., "The Convergence of Sequences of Rational Functions of Best Approximation,"

in preparation. 3 Walsh, J. L., "The Convergence of Sequences of Rational Functions of Best Approximation,

11," in preparation.

VARIATIONAL BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR ELLIPTIC EQUATIONS, II*

BY FELIX E. BROWDER

DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY

Communicated by A. Zygmund, July 29, 1963

In two preceding notes under the same title' (which we shall refer to below as (I) and (II), respectively), we have proved existence and uniqueness theorems for solutions of variational boundary value problems for quasi-linear elliptic differential

equations of order 2m (m > 1) of the form

Au =E lal< m DaA(x, u, ..., Dmu) =f. (1)

In the present note we shall give a method of establishing the existence of solutions of such problems under drastically weaker hypotheses than those of (I) and (II) but with the loss of a proof of uniqueness. The method which we shall apply is a variant of our preceding method, which was an orthogonal projection argument involving nonlinear operators in Hilbert space satisfying very weak continuity conditions. This variant applies in addition a form of the Leray-Schauder theory of the degree for completely continuous displacements in a Banach space.2

Section 1.-We apply the notation of our preceding notes.' On the open set Q of Rn, we consider the quasi-linear differential operator A defined by (1) above,

794: MATHEMATICS: F. E. BROWDER PRoc. N. A. S.

point z, of Ep is a limit point of poles of the R,,(z). This conclusion is true evenl if

p = oo, for then the only finite limit points of the poles of the Rn,,(z) are the poles of F(z), where -the multiplicity of the pole of F(z) equals the multiplicity of the

poles of the Rn,,(z) approaching it. As has been indicated by the statements of Theorems 1 and 2, in the former

there are few poles of the r,,(z) and their limit points are the poles of f(z) interior to Ep; in the latter there are few zeros of the R,^(z) and their limit points are the zeros of F(z) interior to Ep. If the poles aj of f(z) can be so ordered that G(al) <

G(C2) <. . . < G(ac), and if f(z) has these but no other singularities interior to ET,, then for each v not greater than u, the limit points of the poles of the r,,(Z) are the first v poles a,, a, . . . , a,. If the zeros aj of F(z) interior to .E can be similarly ordered, and if F(z) is meromorphic interior to E,, then for each v not greater than g the limit points of the zeros of the Rn,,(z) are the first v zeros a., a2, . . . , a.

The present note deals primarily with the Tchebycheff norm, but under suitable circumstances the methods of reference 2 apply to a pth power norm and yield an analogue of Theorem 2.

This research was sponsored (in part) by the Air Force Office of Scientific Research.

1 Walsh, J. L., Math. Zeit., 38, 163-176 (1934). 2 Walsh, J. L., "The Convergence of Sequences of Rational Functions of Best Approximation,"

in preparation. 3 Walsh, J. L., "The Convergence of Sequences of Rational Functions of Best Approximation,

11," in preparation.

VARIATIONAL BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR ELLIPTIC EQUATIONS, II*

BY FELIX E. BROWDER

DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY

Communicated by A. Zygmund, July 29, 1963

In two preceding notes under the same title' (which we shall refer to below as (I) and (II), respectively), we have proved existence and uniqueness theorems for solutions of variational boundary value problems for quasi-linear elliptic differential

equations of order 2m (m > 1) of the form

Au =E lal< m DaA(x, u, ..., Dmu) =f. (1)

In the present note we shall give a method of establishing the existence of solutions of such problems under drastically weaker hypotheses than those of (I) and (II) but with the loss of a proof of uniqueness. The method which we shall apply is a variant of our preceding method, which was an orthogonal projection argument involving nonlinear operators in Hilbert space satisfying very weak continuity conditions. This variant applies in addition a form of the Leray-Schauder theory of the degree for completely continuous displacements in a Banach space.2

Section 1.-We apply the notation of our preceding notes.' On the open set Q of Rn, we consider the quasi-linear differential operator A defined by (1) above,

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