Note on Approximation by Bounded Analytic Functions

Note on Approximation by Bounded Analytic FunctionsAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 37, No. 12 (Dec. 15, 1951), pp. 821-826Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/88523 .

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VOL. 37, 1951 MAITHEMATICS: J. L. WALSH 821

NOTE ON APPROXIMATION BY BOUNDED ANALYTIC FUNCTIONS

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated October 10, 1951

The primary object of this note is to establish a result on the topic indicated in the title:

THEOREM 1. Let C be an analytic Jordan curve, which together with its

interior lies in a region D. Let the sequence of functions fn(z) be analytic in

D, satisfy the inequality (R > 1)

Ifn(z) I < A1R", z in D, (1)

and likewise for some function f(z) satisfy the inequality

If(z) - fn(z) A2/np +

a, z on C, (2)

where p is a non-negative integer and we have 0 < a < 1; here and below, all

numbers Aj represent positive constants, independent of n and z. Then the

pth derivative f() (z) satisfies on C a Lipschitz condition of order a if we have

O < a < 1, and is of class A* with respect to arc length on C if we have a = 1.

We assume that D is a Jordan region bounded by a curve Cp, namely the image of the circle y,: Iwl = p(>l) when the exterior of C is mapped conformally by z = 4l(w) onto the exterior of -: w I = 1, so that the points at infinity correspond to each other; this assumption obviously does not diminish the generality of the proof. Let Po(z), Pi(z), P2(z), . . ., be the Faber polynomials for C; any function F(z) analytic and bounded interior to Cp can be expanded interior to Cp:

co Iv- Z'a F[41(w)] dw

F(z) = akPk(z), ak = - (3) k = o0 2ri J W w

where in the integral the Fatou boundary values are used on ,p. The

polynomials Pk(z) are uniformly bounded on C. Thus if we write

fn (z) = anoPo(z) + anlPl(z) + an,2P2(Z) + .., (4)

we have by (1) and (3)

ank A1Rn/pk,

and if Sn, N(z) denotes the sum of the first N + 1 terms of the second member of (4), we have by the boundedness of the Pn(z) on C,

co

Ifn(z) -

Sn, N() < E A,Rn/pk = A4Rn/pN, z on C. (5) k=N+l



822 MA THEMA TICS: J. L. WALSH PROC. N. A. S.

We now choose the positive integer X so that px > R, whence

Ifn(z) - Sn, xn(z) I A4(R/pX)', z on C, (6)

with R/ip < 1. From (2) and (6) we may write

f(z) - Sn, x,(z) I < A,/nP + a, z on C. (7)

The polynomials S,. x, of respective degrees Xn are not defined for every degree, but we may set

P (z) = Sn, n(Z), Xn < k < X(n + 1).

Then the polynomials Pk(z) are defined for all positive integral values of k, we have from (7)

I f(z) - k(z) I < A6/kP + a, z on C, (8)

and the conclusion followsl, 2 from (8). In determining the polynomials Sn, xn(z) of degrees Xn in (6) we have

used for convenience the Faber polynomial expansion of the given functions

fn(z). Various other polynomial expansions may also be used, not neces-

sarily of the form ZckPk(z) in terms of given polynomials Pk(z). It is

sufficient if there can be determined polynomials Sn, xn(z) of respective degrees Xn approximating on C to an arbitrary function f,,(z) analytic and in modulus not greater than Mn interior to Cp, with measure of approximation indicated by the inequality

Ifn(Z) - Sn, xn(z) I - A7,Mn/p,, z on C.

Numerous kinds of polynomials, defined for instance by interpolation, satisfy this condition;3 it is essential that C possess suitable continuity properties, but C need not be analytic; Theorem 1 admits a corresponding extension.

To Theorem 1 we add an easily established converse: THEOREM 2. Let C be an analytic Jordan curve, which together with its

interior lies in a bounded region D. Let f(z) be analytic interior to C, con-

tinuous in the corresponding closed region, and let f(P)(z) satisfy on C a Lip- schitz condition of order a(0 < a < 1) or be of class A* with respect to arc-

length on C (a = 1). Then there exist polynomials f,(z) of respective degrees n so that (1) and (2) are satisfied.

The existence of polynomials fn(z) satisfying (2) is known.l1 2 For these polynomials we have by (2) and by the boundedness of f(z) on C

Ifn(z) I < A8, z on C. (9)

We choose p(>l) so large that, in the notation already introduced, D lies interior to C,. Then (9) implies by the generalized Bernstein lemma4



VOL. 37, 1951 MATHEMATICS: J. L. WALSH 823

Ifn(z) I < Aspn, z interior to Cp,

so (1) follows with R = p. The discussion just given also includes the

COROLLARY. If the polynomials fn(z) of respective degrees n are given so

that (2) is valid, and if D is bounded, then (1) is also valid.

If C, D, f(z) satisfy the conditions of Theorem 1, and if a positive number

M is given, it follows from Montel's theory of normal families that among the functions analytic and in modulus not greater than M in D, there exists

at least one function FM(z) of best approximation to f(z) on C, namely, such

that

AM = max. [If(z) - FM(z) I, z on C] (10)

is least. Theorems 1 and 2 are of significance in the study of such

functions:

THEOREM 3. Let D be a bounded region containing in its interior the

closed interior of the analytic Jordan curve C, and let the function f(z) be

analytic interior to C, continuous in the corresponding closed region. For

each positive M, let FM(z) be the (or a) function analytic and of modulus not

greater than M in D, of best approximation to f(z) on C in the sense that

/M given by (10) is least. Then a necessary and suficient condition that

f(P)(z) satisfy a Lipschitz condition of order a(O < a < 1) on C or be of class

A* (a = 1) for arc-length on C is that

log M -M1/(p a) (11)

be bounded as M --O o.

If (11) is bounded, we choose the sequence M = et, whence for the

functions FM(z) and a suitably chosen A9

' /(P + a) < A /n, M < A 9 + a/np

+ a,

and the conclusion follows from Theorem 1.

Conversely, let f(z) be given with the requisite properties on C. We

compare the measure of approximation AM with the corresponding measure

of approximation for the polynomial f,(z) of Theorem 2, for specific values

of M not less than the modulus of f,(z) in D. We obviously have /M <

A2/np + ", provided n is defined as a function of Mby the inequalityAl Rn<

M < A Rn + 1, whence

M/(p + a) < A21/(P +

t)/n, log M < log A, + (n + 1) log R,

from which it follows that (11) is bounded. Theorem 3 is established. An alternate form of the condition on (11) is to require that

log ILM + (p + ca) log log M

be bounded as M becomes infinite. If (for 0 < a < 1) the function f(p)(z)



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

satisfies a Lipschitz condition of order a but of no greater order, or (for a = 1) if f(p)(z) is of class A* but f(P +

1)(z) does not satisfy a Lipschitz condition of any order, then p + a is the greatest number y for which

log ty + Y log log M (12)

is bounded as M - X oo, where PM is the measure of approximation for the function FM(z) of best approximation. Indeed, p + a is the greatest number y for which (12) is bounded as M becomes infinite, not passing through all values but passing through an arbitrary sequence of values of the form M = AoRin, n = 1, 2,3, ....

Alternative (and simpler) proofs of Theorem 1, and hence of Theorem 3 can be given by mapping the interior of C onto the interior of 7: w = 1, and making use of the special properties of the Taylor development in the closed interior of y. Of course for the unit circle the Faber development is identical with the Taylor development, so this alternate proof of Theorem 1 is not greatly different in form from the proof already given. This alternate proof has no immediate analog, however, if we replace the

Jordan curve C by a Jordan arc. Theorems 1, 2 and 3 extend to the case where C is an analytic Jordan arc. THEOREM 4. Let C be an analytic Jordan arc, let D be a region containing

C, and let the functions f,(z) be analytic in D and satisfy both (1) and (2). Then f(P)(z) satisfies a Lipschitz condition of order a if 0 < a < 1 and is of class A* for arc-length on C if a = 1, on any closed subarc of C containing no

endpoint of C.

M ap C conformally by z = a(w) onto the line segment S: -1 ? w < 1; conditions (1) and (2) in suitably modified form persist. We approximate in the w-plane on S to the functions fn['l(w)] by polynomials in w which

interpolate to fn[p(w)] in the zeros of the Tchebycheff polynomials for S.

The second (outlined) proof of (6) making use of polynomials defined by

interpolation is then valid.3 Introduce now the classical transformation w = cos 0 which maps S onto the axis of reals - oo < 0 < oo and transforms

a polynomial in w of degree n into a trigonometric polynomial in 0 of order

n. From a theorem due to S. Bernstein and de la Vallee Poussin in the

case 0 < a < 1 and from a theorem due to Zygmund in the case a = 1, it now follows that f['(cos 0)] has the requisite properties as a function of

6. The class A* is invariant under suitable transformation,2 so Theorem 4

follows. If C is of length 2c, and arc-length s is suitably measured on C alge-

braically from the mid-point, it readily follows by a slight extension of the

reasoning just given, that the function f(z) = F(s/c) F(cos o) has on

C a pth derivative with respect to w which satisfies a Lipschitz condition of order a(0O < a < 1) or is of class A*(a = 1) with respect to w. The

reciprocal of this reasoning is also valid, including the proof of (1) and (2).



VOL. 37, 1951 MA THEMA TICS: J. L. WALSH 825

THEOREM 5. Let C be an analytic Jordan arc which lies in a bounded

region D, and let the function f(z) be given on C. For each positive M, let

FM(z) be the (or a) function analytic and of modulus not greater than M in

D, whose measure of approximation to f(z) on C as given by (10) is least.

Let C be of length 2c and let arc-length s on C be measured algebraically from the mid-point; we set f(z) = F(s/c) = F(cos w) on C. Then a necessary and sufficient condition that dPF(cos c)/dco satisfy a Lipschitz condition of order a(O < a < 1) or be of class A* (a = 1) with respect to o is that (11) be

bounded as M becomes infinite. A further result is similar to Theorems 4 and 5: THEOREM 6. Let f(x) be a real or complex-valued function of the real

variable x, periodic with period 2ir. Let d(>0) be arbitrary but fixed. For

each M (>0), among the functions analytic and in modulus not greater than

M in the region D: I y < d, periodic with period 2rr, let FM(z) denote the

(or a) function of best approximation to f(x) on C: - oo < x < oo in the

sense that

P.a = lub[[f(x) - FM(x) I, - < x < oo ]

is least. Then a necessary and suficient condition that f(P)(x) satisfy a

Lipschitz condition of order a (O < a < 1) or be of class A* (a = 1) on C

is that (11) be bounded as M - eo.

The sufficiency of this condition follows from Theorem 4 or Theorem 5. To prove the necessity of the condition, we employ as comparison sequence as in the proof of Theorem 3, the known sequence of trigonometric polynomials.f((z) of respective orders n which satisfy (2). For this sequence inequality (1) is readily established by setting w = eiz, which maps C onto the circumference I w = 1: a trigonometric polynomial in z of order n is a

polynomial in w and l/w of degree n, and it is convenient to apply a lemma5

concerning the latter type of polynomial. Theorem 6 remains valid if D is no longer the region I y < d but a subregion of the latter containing C, and also remains valid if the 7FM(z) are not required to be periodic.

If we modify the hypothesis of Theorem 1 by merely requiring that C be a continum (not a single point) interior to D, and by replacing the second member of (2) by A2/R2', R2 > 1, it follows that the sequence f,(z) converges uniformly and its limit f(z) is analytic in some subregion of D containing C [compare Walsh, Trans. Am. Math. Soc., 47, 293-304

(1940)]. A similar remark applies to Theorems 4 and 6. The results of the present note concern Problem a, namely the study

of the connection between measure of approximation to f(z) on C and con-

tinuity properties of f(z) on C. A further study involves Problem /, the connection between measure of approximation to f(z) on C and continuity properties of f(z) on a related point set such as C,; results on the latter

problem will be forthcoming.



826 MATHEMATICS: S. SHERMAN PROC. N. A. S.

Sewell, W. E., "Degree of Approximation by Polynomials in the Complex Domain," Ann. Math. Studies, 9 (Princeton, 1942).

2Walsh, J. L., and Elliott, H. M., Trans. Am. Math. Soc., 68, 183-203 (1950). The class A* was first introduced into the theory of approximation by Zygmund.

3Walsh, J. L., and Sewell, W. E., Duke Math. Jnl., 6, 658-705 (1940). See especially Theorems 3.2 and 4.7, and ?5.

4 Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex

Domain, New York, 1935, p. 77.

5 Page 259 of reference 4.

ON A THEOREM OF HARDY, LITTLEWOOD, POLYA, AND BLA CKWELL

BY S. SHERMAN

LOCKHEED AIRCRAFT CORPORATION, BURBANK, CALIFORNIA

Communicated by John von Neumann, October 24, 1951

Let U = {u} be a real vector space. Let a(b) be a measure on U with

a finite spectrum, Sa = {ui 1 < i < n} (S = {u 1 < j< nb}). We say that

a > b a>b 1

if and only if for each real convex sop U

a(uj) ,(ui) > E b(uj)q (uj).

We say that

a > b a>b 2

if and only if

b(u)uj = E pija(ui)u,

where

Pij > 0, pja(u) = b(uj),

and

E Pi,= 1. i

From the definition of convex function and the properties of pij it follows

immediately that

826 MATHEMATICS: S. SHERMAN PROC. N. A. S.

Sewell, W. E., "Degree of Approximation by Polynomials in the Complex Domain," Ann. Math. Studies, 9 (Princeton, 1942).

2Walsh, J. L., and Elliott, H. M., Trans. Am. Math. Soc., 68, 183-203 (1950). The class A* was first introduced into the theory of approximation by Zygmund.

3Walsh, J. L., and Sewell, W. E., Duke Math. Jnl., 6, 658-705 (1940). See especially Theorems 3.2 and 4.7, and ?5.

4 Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex

Domain, New York, 1935, p. 77.

5 Page 259 of reference 4.

ON A THEOREM OF HARDY, LITTLEWOOD, POLYA, AND BLA CKWELL

BY S. SHERMAN

LOCKHEED AIRCRAFT CORPORATION, BURBANK, CALIFORNIA

Communicated by John von Neumann, October 24, 1951

Let U = {u} be a real vector space. Let a(b) be a measure on U with

a finite spectrum, Sa = {ui 1 < i < n} (S = {u 1 < j< nb}). We say that

a > b a>b 1

if and only if for each real convex sop U

a(uj) ,(ui) > E b(uj)q (uj).

We say that

a > b a>b 2

if and only if

b(u)uj = E pija(ui)u,

where

Pij > 0, pja(u) = b(uj),

and

E Pi,= 1. i

From the definition of convex function and the properties of pij it follows

immediately that



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Note on Approximation by Bounded Analytic Functions