r„ i

NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 22 (1993), 1-20
B E R N S T E IN T H E O R E M S
T hom as B ag by
and
N orm an Levenberg
(Received September 1992)
Abstract. The classical Bernstein theorem states that a continuous function / on
[—1,1] extends to a holomorphic function on an open neighborhood of [—1,1] in C
if and only if lim s u p d j/” < 1, where
n —+oo
dn = in f{11/ - Pn || [—1 , 1 ] : Pn polynomial of degree < n }
and || • || denotes the supremum norm. We outline four proofs of this theorem, and
indicate how these m ethods can be used to obtain extensions of the theorem to
higher-dimensional problems involving holomorphic functions, harmonic functions,
and solutions of a general class of elliptic equations.
In tro d u c tio n
There is a close relation between the smoothness of a function / and the speed
at which / may be approximated by polynomials. To state results of this type we
introduce, for any continuous complex-valued function / on any compact set K in
the plane C, the approximation numbers
dn = dn{ f ,K ) = inf{||/ —pn \\K ■P n e Vn},
where Vn is the vector space of complex polynomials in z of degree at most n; in
this paper || •|| always denotes the supremum norm. The Weierstrass approximation
theorem states that lim dn = 0 for any continuous function / on [—1, 1], and it
n —>oc
is natural to ask for additional conditions on / which guarantee that dn converges
rapidly to zero. A beautiful result of this type is the classical theorem of Bernstein,
which states that / extends to a holomorphic function on an open neighborhood
of [—1,1] in C if and only if dn satisfies an exponential decay estimate
dn < Cpn
for some constants
C > 0 and
p G (0,1).
In fact, a sharp version of the Bernstein theorem relates the constant p to the
size of the open neighborhood of [—1,1] to which / can be extended. Walsh [W]
later gave an important extension of the Bernstein theorem in which the interval
[—1,1] is replaced by certain compact subsets of C. The theorems of Bernstein and
Walsh serve as a link between the classical ideas of approximation theory and some
higher-dimensional problems of current interest concerning holomorphic functions
of several complex variables and partial differential equations in R N.
The goal of this survey paper is to give four different proofs of the Bernstein
theorem, and in each case to explain how the technique extends to give informa
tion about higher-dimensional problems. In this way we hope to provide a gentle
1991 A M S M athem atics Subject Classification: 41A10, 41A17
2
THOMAS BAGBY and NORMAN LEVENBERG
introduction to some recent work on functions of several complex variables [Z2],
[S2j, [Bl], harmonic functions of several variables [A] [BL 1], [ZS], and solutions
of elliptic equations [BL 2]. The four proofs are given respectively in Sections 1, 2,
3, and 4.
There is still another proof of the Bernstein and Walsh theorems, due to Zaharjuta and his coworkers, using the technique of Hilbert scales [Zl], [Z2 ], [Z3], [ZS].
This technique makes use of a considerable amount of functional analysis, and also
leads to results concerning holomorphic functions of several complex variables and
partial differential equations. We will not discuss this technique here, but refer to
[Z3] for a recent report.
Finally, we recall the closely related theorem of Jackson [L] which states that for
any continuous function on [—1, 1] we have
dn < (1 + 7r2/2 )u ;/(l/n ),
where
a;/(/*)=
sup
\f(x + t) - f(x)\
x € [ - l ,l ] , |t|< h
is the modulus of continuity of / on [—1,1]. Extensions of the Jackson theorem
are known for holomorphic functions [W, Appendix A2] and harmonic functions
[R] [A], and are given for solutions of elliptic equations in a forthcoming paper by
L. Bos and the present authors.
1. T ru n catio n . B e rn ste in T h eo rem s in Several C om plex V ariables
An elementary approach to the theorems of Bernstein and Walsh is to regard
them as statements about the error in truncating geometrically convergent series
expansions. As the simplest example, let’s consider first the closed unit disk A =
{z : \z\ < 1} in C, and suppose that / is holomorphic on a neighborhood of A.
To be specific, we assume that / is holomorphic on the open disk {z : \z\ < R h
where R > 1, and we ask to what extent the size of the radius R determines the
rate of decay of the approximation numbers dn(f, A). To study this, we recall
that the Taylor expansion
akZk for / about the origin converges absolutely
and uniformly on compact subsets of {z : \z\ < R} to / . Applying the Cauchy
estimates to / on {z : \z\ < r}, where 1 < r < R, we obtain |an | < M /r"
with M = sup{|/(,z)| : \z\ < r}. Letting pn(z) = Y^k=oakzk be the n-th Taylor
polynomial for / , it follows that
< y / , s ) < n / - f t . i i A < ? i^ r i j .
This implies that lim supdn(/, A )1/ ” < 1/r, and we may now let r f R to conclude
n —>00
that
lim supdn (/, A )1//n < 1/R.
n —►oo
This proves the following equivalence in one direction.
3
BERNSTEIN THEOREMS
T h eo rem 0. Let f be continuous on A = {2 G C : \z\ < 1}, and R > 1. Then
lim supdn(/, A )1/” < l / R
(1)
n —►
00
if and only if f is the restriction to A of a function holomorphic in {z G C : \z\ <
R}.
We have already proved “if” . To prove “only if” we will use the fact that any
polynomial p(z) satisfies the Bernstein- Walsh inequality
IpMI < IIpIIa Pdeep.
1*1 < p;
(2 )
this estimate follows from applying Lemma 1 below, with g&(z) = log \z\, so for
the moment let’s assume (2) and complete the proof of Theorem 0. Let / be a
continuous function on A such that (1) holds; we will show that if pn is a polynomial
of degree < n satisfying dn = \\f —J9n 11A» then the series po + YlTiPn ~ P n- 1)
converges uniformly on compact subsets of {z : \z\ < R} to a holomorphic function
F which agrees with / on A. To do this, we choose R' with 1 < R' < R-, by
hypothesis the polynomials pn satisfy
M
II/-P » IIa < ;^ ,
,,
n = 0 , 1 2 ... ,
(3)
for some M > 0. We now let 1 < p < R', and apply (2) to the polynomialpn —pn- 1
to obtain
SUp \pn { z )
\A<P
-
P n-l(z)\
< pU\\pn ~ Pn—11|A < pn{\\Pn ~ / ||a + I I /- P » - i|I a )
-
H
R!
Since p and R' were arbitrary numbers satisfying 1 < p < R' < R, we conclude
that po + YlTiPn ~ P n- 1) is locally uniformly Cauchy on {z : \z\ < R}, and hence
converges locally uniformly on {z : \z\ < R} to a holomorphic function F; from (3)
we see that F = f on A, so Theorem 0 is proved.
L em m a 1. (Bernstein-Walsh property) Let K be a compact subset of C such
that C — K is connected. Suppose that C — K has a Green function g n ’, that is,
suppose there is a continuous function g x ■C —» [0, +oo) which is identically equal
to zero on K , harmonic on C —K , and has a logarithmic singularity at infinity in
the sense that gi<(z) —log \z\ is harmonic at infinity. Then
gK (z) = max jo,sup j ^ ^ l o g | p ( 2)| j j ,
(4)
where the supremum is taken over all non-constant polynomials p such that ||p|| k <
1. In particular, if R > 1 and
D r =={z : gK {z) < log R},
(5)
4
THOMAS BAGBY and NORMAN LEVENBERG
then
2 6 D R.
IpWI <
(6 )
It is easy to prove a weak form of (4). In fact, if p is any nonconstant polynomial
such that \\p \\k < 1> then the function V =
log \p\ —gx is subharmonic on
C —K , bounded at oo, and continuously assumes values which are at most zero on
dK . By the maximum principle we have V < 0 o n C U {o o } —K , which proves that
gK(z) is greater than or equal to the right side of (4). We will not give the proof
that gx(z) is actually equal to the right side of (4), but only mention that one can
construct a sequence of monic polynomials {pn} with degpn = n such that
locally uniformly on CU{oo} —K (cf., [W], Section 4.4); for example, the Chebyshev
polynomials for K will do. (We will soon discuss these polynomials in a special
case.)
Now let’s look at another example, I = [—1,1]. If / is holomorphic on an open
neighborhood of [—1, 1], it is natural to hope that some series expansion of / on a
neighborhood of [—1, 1] would yield truncations which are “asymptotically optimal”
approximations on [—1,1], and this idea leads to the Bernstein theorem. To state
the theorem we recall that the function 2 = $(iu) = |(u ; + ^ ) is a conformal map
from {w : |iu| > 1} onto C — /; and the circles S r = {w e C : |iy| = R > 1}
are sent to confocal ellipses E r with foci at —1 and 1. Note that C — I has the
Green function g i ( z ) = log |<$- 1(2)|. The ellipses E r are then the boundaries of
the regions D r defined by (5). The open ellipses D r will be used in stating the
following Theorem 1 of Bernstein, and in Lemma 2.
T h eo rem 1. (Bernstein) Let f be continuous on I, and R > 1. Then
lim supdn(/, Z)1/ ” < l / R
n —►
oo
(7)
if and only if f is the restriction to I of a function holomorphic on D r .
To prove “only if” we repeat the proof after the statement of Theorem 0, using
the Bernstein-Walsh inequality (6) in the case K = I. To prove “if” we need a
classical series expansion for holomorphic functions in the open ellipses D r , which
we give as Lemma 2. To state this expansion, we consider the conformal mapping
= <£-1 from the complement of I onto {w G C : |iu| > 1}. It turns out that the
function
is a monic polynomial of degree n, which has minimal sup-norm on I among all
such polynomials, called the n-th Chebyshev polynomial for I. It follows from (8)
that ||Tn ||/ =
for n > 1, and lim |T’n (z)|1/ n = p / 2 uniformly for z € E p,
n —►oo
p
> i.
BERNSTEIN THEOREMS
5
L em m a 2. If f is holomorphic on D r , then
OO
f ( z) = ^ 2 anTn{z),
Z € Dr ,
n —0
where
for 1 < p < R. The series is uniformly convergent on compact subsets of D r .
P ro o f. The fact that f is holomorphic on the difference D r —[—1,1] shows that
the composition / o <3? is a holomorphic function on the annulus {1 < |u>| < R},
with a Laurent series expansion X^^L-oo cn ^ n, where
for 1 < p < R. For each integer k > 0 the function z kf ( z ) is in fact holomorphic
on the entire ellipse D r, so for 1 < P < R we have f E z kf (z) dz = 0; applying to
this integral, with k = n — 1, the change of variables 2 = <&(w) = (w + l /w)/ 2, we
see that c„ = c_n for each integer n > 1, which proves Lemma 2.
From the formula for a n , we see that / holomorphic on D r implies that
limsup |an |1//n < 2/ R,
giving an analogue of the standard Cauchy-Hadamard theorem for the radius of
convergence of a power series ([D], Section 4.4; [W], Chapter III). Now to prove
“if” in Theorem 1, one repeats the proof in the first paragraph of this section, using
Lemma 2 in place of Taylor expansions.
As mentioned in the introduction, Walsh showed that Theorem 1 holds if [—1,1]
is replaced by certain compact sets K C C.
T h eo rem 2. (Walsh) Let K be a compact subset of the plane such that C \K is
connected and has a Green function gx- Let R > 1, and define D r by (5). Let
f be continuous on K . Then lim supdn( f , K ) 1^n < 1/ R if and only if f is the
n —►oo
restriction to K of a function holomorphic in D r .
To prove “only if” in Theorem 2 we repeat the proof after the statement of
Theorem 0, using the Bernstein-Walsh inequality (6). To prove “if” we need a
series expansion for holomorphic functions in the region D r , and we only outline
this construction. In this setting D r is a finite union of domains and can be
approximated by lemniscates; i.e., we can find a sequence of polynomials {qn} and
positive numbers {rn} such that the sets
6
THOMAS BAGBY and NORMAN LEVENBERG
are domains which increase up to D r . If we choose n sufficiently large so that
K c f i n and fin is “almost” all of D r , then we can expand / in a series of the
form
OO
/ ( 2) = ^ 2 uk(z ) M z )]k
k
where the Uk are polynomials of degree less than that of qn. Just as with power
series, the above series converges uniformly on each level set { z : \qn(z)\ < r},
for r < r n. By taking appropriate polynomials qn, truncations of these series again
provide good polynomial approximators for / . For details we refer to [W, Chapters
111,1V].
Theorem 2 has a natural extension to several complex variables. In this setting
many tools, such as conformal mapping and the Hermite remainder formula (dis
cussed in Section 4 below) are lacking; but the remarkable truth of the m atter is
that the exact same result holds. First, a few preliminary definitions and com
ments. Let K C CN be compact with connected complement. Let / be continuous
on K and define
dn( f , K ) = w l{ \\f - pn \\K :pn G
}
where V™ is the vector space of holomorphic polynomials of N variables of degree
at most n (a function g on an open set Vt C CN is holomorphic if it is holomorphic
in each variable separately). W ith the one-variable Lemma 1 as motivation, we
define
uK (z) = m a x |o ,s u p |^ ^ l o g |p ( ; z ) | j j ,
(9)
where the supremum is taken over all non-constant polynomials p such that \\p \\k <
1. Then uk is automatically lower semicontinuous, but need not be upper semicontinuous. We let
u*K (z) = limsupwK(C)
z
be the upper semicontinuous regularization of u k \ this is the smallest upper semi
continuous function greater than (or equal to) u k ■ Then u*K may be = -foo;
otherwise u*K is plurisubharmonic: the restriction of u*K to any complex line is
subharmonic or = —oo. We remark that if F is holomorphic, then u — log |F | is
plurisubharmonic. We say that K is L-regular if uk = u*K , that is, if uk is contin
uous. For example, if K is the closure of a domain in CN with C 1 boundary, then
K is L-regular; other geometric criterion for L-regularity can be found in [BL1],
[S2 ]. The reason for the “L” is that the class of plurisubharmonic functions u in
CN of logarithmic growth; i.e., such that u(z) — log\z\ < 0(1), \z\ —> oo, is called
the class L. The competitors
log |p(.z)| with \\p \\k < 1 in the supremum for
uk clearly belong to L; historically, for any Borel set E, the function
u e (z )
= sup{w(,z) : u 6 L, u < 0 on E}
was called the L- extremal function of E and it was proved later that for compact
sets, this upper envelope coincides with that in (9).
BERNSTEIN THEOREMS
7
If the compact set K C CN is L-regular, then for each R > 1 we define
DR = {z : uK (z) < logR};
(10)
then we clearly have the Bernstein-Walsh inequality
Ip(*)I < l b lk f ldeep,
*e dr
( ii)
for every polynomial p in CN (conversely, it is known that if K satisfies a BemsteinWalsh property, then K is L-regular [Z2], [S2], [K]).
A compact set K C CN is called polynomially convex if K coincides with its
polynomial hull
K = {z
e
CN : \p{z)\ <
\\p \\k
for all polynomials p}.
For example, every compact set K C R N = R N + iO C CN is polynomially convex.
If ./V = 1, it is easy to describe the class of all polynomially convex compact sets: a
compact set K C C is polynomially convex if and only if C —K is connected. With
the above definitions, Theorem 2 goes over exactly to several complex variables.
T h eo rem 3. Let K be an L-regular, polynomially convex compact set in CN . Let
R > 1, and let D r be defined by (10). Let f be continuous on K . Then
limsup dn( f , K ) 1/ n < 1/ R
71— KX)
if and only if f is the restriction to K of a function holomorphic in D r .
To prove “only if” we may repeat the proof after the statement of Theorem 0,
since K satisfies the Bernstein-Walsh inequality (11). The “if” proof, although
not hard, requires some deeper knowledge of several complex variables, and we
will only outline this proof. In analogy with the one-variable lemniscates used in
proving Theorem 2, we call a set of the form
Cl = {z = (z i , . . . , z N) : |*i| < 1, . . . ,|zjv| < l,|p i(z )| < 1, . . . APm(z)\ < 1},
where p i, . . . ,pm are holomorphic polynomials, a polynomial polyhedron. Since K is
polynomially convex and D r is a domain containing K , it is not hard to show that
one can construct polynomial polyhedra {fin} containing K and approximating
D r . But the deep theorem we need to replace the polynomial expansion from the
one-variable argument is the following.
Oka-W eil E x ten sio n T h eo rem . Let f be holomorphic in a polynomial polyhe
dron
ft = {z G CN : \zi\ < 1 ,... , \zN \ < 1, |pi(z)| < 1 ,... , \ P m ( z ) \ < 1}.
Then there exists F holomorphic in
AN+m
=
{(Z,W) = (Z i,...
,Z N , W i , . . . ,W m )
’ N , \Wj\ < 1}
such that
f ( z) = F( z , pi { z ) , . . . ,pm(z)),
2 e ft.
One can then use truncations of the Taylor series of F(z, w) =
ca(3Zaw13 re
stricted to w = (p\{z),... ,pm{z)) to get good approximating polynomials to / .
This is essentially the proof in Siciak [S2j; his original proof appeared in 1962
[SI]. Zaharjuta [Z2] gave another proof of Theorem 3, and Bloom [Bl] gave a neat
modification of the proof in [S2 ].
8
THOMAS BAGBY and NORMAN LEVENBERG
2. D uality. B e rn ste in th e o re m s for harm o n ic functions
To begin this section we prove “if” in Theorem 1 using duality. We suppose
/ is holomorphic on D r -, instead of actually constructing a good polynomial ap
proximator to / , we will try to rewrite the numbers dn = dn( f , I ) in such a way
that we can estimate them. To do this we let 1 < r < p < R. We need a global
C°° extension F of / that agrees with / on a neighborhood of / , so we let 0 be
a smooth cut-off function which is identically equal to 1 on D p and has compact
support in D r . We then set F = </>/ in D r and let F be identically 0 outside of
Dr.
If n is fixed, then by the Hahn-Banach theorem there exists a signed measure
fi = nn supported in / , with total variation \^\(I) = 1, such that // annhilates the
vector space Vn (that is, f j Pn dfj, = 0 for all pn G Vn) and
dn —
j
f d[i.
Since F = f on /, we can write
dn =
J
Fd[i = (n*F)(Q),
( 12)
where F(z) = F ( —z). Now
ri
f)
H * F = (n * F) * 6 = (p, * F) * — -E —
oz
oz
* (^ * -£')>
(13)
where 6 is the point mass at 0 and E ( z ) = ^ is the Cauchy kernel. We write
=
* £ )(* ) = ±
/
n Ji
C
the Cauchy transform of p. Note that Ji is holomorphic outside I. From (12) and
(13) we then obtain
where A is Lebesgue measure in C.
In order to utilize formula (14) for dn, we need estimates for Jl. We first note
that since |/i|(/) = 1, we have
\Jl(z)\ < A,
z e dDr,
(15)
for some constant A > 0 depending only on the distance from I to dDr. In addition
we have the growth estimate
l£(z)l = 0 (l/|z |" )
as |z| -* oo;
(16)
9
BERNSTEIN THEOREMS
— Ylk
uni
(k
= 0 for
this follows from noting that for 2 sufficiently large we have
formly for C £ I-, and then using the fact that p satisfies
0 < k < n. We now consider the function
u(z) = g i (z) + i log
>
where gi is the Green function for C — I. Using (15) and (16), we see that u(z)
is subharmonic in C — Dr, bounded above at 00, and continuously assumes values
which are at most logr on dD r. By the maximum principle we have u(z) < logr
on C — Dr] that is,
\Jl{z)\ < A[elogr~9l^ ] n,
z e C - Dr .
(17)
Prom (14) and (17) we conclude that lim supdn(/, I ) l^n <
n —»oo
We may now let
P
r I 1 and p f R to obtain (7), which completes the proof.
The preceding proof of “if” in Theorem 1 also proves “if” in Theorem 2, and in
fact this proof may be extended to yield a Bernstein theorem for harmonic functions
in R N, where N > 2. To state this we let
be the vector space of all harmonic,
real-valued polynomials of N variables of degree at most n. If / is a continuous
real-valued function on a compact set K C R N, we now define
dn( f , K ) = inf{||/ - hn \\K : hn €
T h eo rem 4. [A], [BL1], [ZS] Let K be a compact subset of R N such that R N—K
is connected.
(a) Let
be an open neighborhood of K . Then there is a constant p G (0,1),
depending only on K and Cl, with the following property: if f is harmonic on Cl,
then limsupn_+oc dn(f, K ) l/ n < p.
(b) Suppose that when we regard K C R N = R N + *0 C CN, the set K is Lregular. Let 0 < p < 1. Then there is an open neighborhood Q of K , depending only
on K and p, with the following property: if f is a real-valued continuous function on
K such that limsup n_ 00 dn(f, K ) l^n < p, then f extends to a harmonic function
on Cl.
A theorem of Plesniak [P], [BL1] shows that a compact set K C R N will satisfy
the L-regularity hypothesis of Theorem 4(b) if it is the closure of a domain in R N
with C 1 boundary. We remark that the boundary smoothness is essential; a simple
example of a compact set K c R2 C R2+ iR2 = C 2 which is the closure of a Jordan
domain in R2, but is not L —regular, can be found in [Sa]. To prove Theorem 4(b)
we may repeat the proof after the statement of Theorem 0, since each polynomial
pG
extends in an obvious way to a holomorphic polynomial on CN which then
satisfies the Bernstein-Walsh inequality (11) in CN.
To prove Theorem 4(a) we follow the outline of the duality proof above, replacing
the Cauchy kernel by the fundamental solution E (x — y) = c n \x — y|2~iV for the
Laplace operator A (here cn is a constant depending only on the dimension). The
10
THOMAS BAGBY and NORMAN LEVENBERG
details of this proof may be found in [BL1]; however, we will discuss here the main
new ingredient, which is the problem of estimating the Newtonian potential
( f i * E ) ( x ) = [ cN \x - y\2~N dniy)
Jk
on compact subsets of Vt—K, under the assumption that we have the decay estimate
|(/z * -E)(x)| = 0 (|x|- n )
as
|x| —►oo
(18)
analogous to (16). We may approach this problem with the help of the Kelvin
transform T ( x ) = x /|x |2, under which a harmonic function h(x) on a domain
G c R n - {0} is transformed into a harmonic function h(x) = \x\2 Nh(x/\x\2)
on T(G). Under this transformation the condition (18) of rapid decay at infinity
is transformed into a condition of flatness near the origin, and the problem of
estimating p * E under the hypothesis (18) is reduced to proving the following
Schwarz lemma for harmonic functions [BL1].
L em m a 3. Let Cl be a bounded domain in R N and let a € £1. I f K is a compact
subset of ft, then there exist constants C > 1 and p € (0,1), depending only on
K and ft, with the following property: if f is a harmonic function on ft satisfying
|/ | < 1 in ft, and if D af(a) = 0 whenever |a| < n, then ||/ ||k < Cpn .
We will now outline a proof of Lemma 3 based on techniques from the theory of
functions of several complex variables. We recall that in the preceding section we
introduced the function u k in CN as a substitute for the ordinary Green function
with pole at infinity in C1. For the purpose of proving Lemma 3 we wish to
introduce in CN a substitute for the ordinary Green function with a finite pole in
C1. Following Klimek [K], we define for each domain ft C CN and each point a 6 ft
the pluricomplex Green function
G~(z-,a) ee sup u(z),
U
where the supremum is taken over all nonpositive plurisubharmonic functions u on
ft such that u(z) — log\z — a| has an upper bound in some neighborhood of a. It
is known that (?-(•; a) is plurisubharmonic in ft (see [K], [BL 1]). This fact leads
to the following Schwarz lemma for holomorphic functions of several variables [Bi],
[BLlJ.
L em m a 4. Let ft be a bounded domain in CN and let a G ft. Let K be a compact
subset of Cl. If f is a holomorphic function on ft satisfying \f\ < 1 in ft, and if
Daf(a) = 0 whenever |a| < n, then ||/ ||k < pn, where
p = supexp[G~(-; a)] < 1.
K
11
The inequality p < 1 is clear from the fact that G~(-; a) is a nonpositive function
on ft which is subharmonic as a function of 2N real variables. The rest of the
BERNSTEIN THEOREMS
11
lemma follows from the fact that the function u (z) = ^ log \f{z)\ is one of the
competitors in the definition of G~(-;a).
Now how does this help us with the proof of Lemma 3? Well, a harmonic function
is real-analytic and thus, about each point in our domain Q C
we can get a
power series expansion of / which converges as a (holomorphic) function in CN
in a neighborhood of the point in CN. We now use the following well-known fact
[Ha], [ABG],
L em m a 5. For each R > 0, there exist positive numbers r < R and C with the
following property. If h is harmonic on a real ball B = {x e R N : \x—a| < R}, then
there is a holomorphic function H on the complex ball B = {z E CN : \z —a| < r}
which agrees with h on the real ball B ' = B fl B and satisfies ||# ||g
We may now prove Lemma 3 by covering the compact set K by finitely many
real balls B such that the concentric real balls B ' associated with the complex balls
B still cover K , and the union of the complex balls B gives a neighborhood Cl of
K in CN to which we can apply Lemma 4. This completes our outline of the proof
of Theorem 4.
Theorem 4 is of course less precise than Theorem 2; in Theorem 4 we do not know
a precise relationship between the rate of decay of dn(f, K) and the size of the region
to which / can be extended as a harmonic function. However, Nguyen Thanh Van
and R. Djebbar [ND] have used Theorem 2 to prove a precise result for harmonic
functions of two variables. A compact set K C R2 is said to satisfy a harmonic
Bernstein-Walsh property if for every t > 1 there exist a constant M — M(t ) > 0
and an open neighborhood U = U(t) of K such that ||p||t/ < M tdegp\\p\\K for every
harmonic polynomial p on R2.
T h eo rem 5. [ND] Let K be a compact set in the plane with connected com
plement, and assume that K satisfies a harmonic Bernstein-Walsh property. Let
R > 1, and define D r by (5). Let f be continuous on K . Then
limsup dn( f , K ) 1/ n < 1 / R
n —►
oo
if and only if f is the restriction to K of a function harmonic in D r .
We close this section with a proof that a precise form of Theorem 4 can also be
given when K is a closed ball in R N. Indeed, for each integer p > 1, we can prove
an analogous result for the iterated Laplacian Ap = A o ... o A. We recall that we
can take a fundamental solution for Ap of the form
f cN,p\x\2p~N
if 2p < N or N odd;
I civtP\x\2p~N log |x| if 2p > N and N even.
All distribution solutions of Apu = 0 are real-analytic, and the uniform limit of a
sequence of solutions must also be a solution. We let H %(p) be the vector space
of polynomials q of degree at most n in N variables which are solutions of the
equation Apq = 0; if p — 1, we use our old notation
• For each compact set
K C R n with connected complement, we now define
dn( f , K ) = inf{11y —P u \\k : P n € W^(p)}.
For each a € R N and r > 0 we let B(a, r) = {x € R N : |rr —a| < r}.
12
THOMAS BAGBY and NORMAN LEVENBERG
T h e o rem 6 . Let K = B( a , r ) C R N, and let R > 1. Let f be continuous on K .
Then
lim supdn( / , K ) lt n < 1/ R
(19)
n —►oo
if and only if f is the restriction to K of a function F satisfying APF = 0 in
B(a, Rr).
P roof. W ithout loss of generality, we may take a = 0, r = 1; that is, K = B ( 0,1).
We prove “if” in the case 2p < N; a slight modification is needed in the other
cases. We work with the fundamental solution E(x) = c n ,p \x \2p ~ n for Ap. If
y e R n - {0 } is fixed, then for all x near the origin we have the Taylor series
expansion
E{x - y) =
Q[y) ( x ) ,
(20)
1—0
where Qjy)(x) = ( - l ) z J2\a\=i D otE (y)xa/a \ € H^{p) (see [BL 2 ]).
We now make the following claim: if « > 1, then there exists a constant 7 = 7 (/c)
such that
(21)
for x € R n and y £ R N — {0}. To prove this claim, note that since each derivative
D aE is homogeneous of order 2p — N — |a|, it suffices to prove (21) for x, y in the
unit sphere S = {£ € R N : |£| = 1}; i.e., |a:| = |y| = 1. Fix such x,y; since then
—1 < (x, y) < 1, the quadratic polynomial z2 + 2(x, y)z + 1 has no root in the unit
disk A c C. Thus the holomorphic function
fx,y{z) = ( zxi + y \)2 + ... + (z x N + yN)2,
z € A,
never vanishes in A, and we can define a holomorphic function gx^y : A —►C of the
form
g = 9x,y(z) = cN,p[(zxi + y i )2 + ... + (z x N + yN)2](2p~N)/2
such that gx,y{z) = E( z x + y) for —l < z < l . The mapping (s , t , z ) —> \fs,t{z)\
is continuous on 5 x S x {z : \z\ < 1/k} and hence must attain a minimum value
m(/c) > 0; upon applying the Cauchy inequalities to g, we obtain
D “E(y)x°‘/c \ = \g{l)(0)/l\\ < Kl sup 1^)1 < 7 AC1,
M=l
\z \ < 1 /
k
where 7 = 7 (k) = cjv)Pm(«:)(2p N^ 2. This proves the claim (21).
Now that the claim is proved, we may prove “if” either by using the truncation
method of Section 1 or the duality method of Section 2; for simplicity we use the
former. We first note that the claim implies that (20) holds whenever |x| < M;
and (20) holds uniformly on compact subsets of {(x, y) € R N x R N : M < M}-
13
BERNSTEIN THEOREMS
Now let / satisfy Apf = 0 on i?(0, R ). Let 1 < k < p < R] and let F be a smooth
function which agrees with / on a neighborhood of B(0, p) and is supported in a
compact subset of B(0,R). Then
F (x)=
[
E {x-y)A *F {y)d\{y)t
J p<\y\<R
o<\ v\ <R
where A is Lebesgue measure in R N. From this we obtain, for |rc| < p,
Fix) = £
/
QiV\ x )ApF(y) dXly).
l= o j p < \ y \ < R
Since Q\y\ x ) € Hf* (p), we conclude that
d n ( f , K ) < sup F I * ) - Y , f
Q\V\ x ) A ”F ly )d y
|* |< i
i=o Jp<\y\<R
o°
.
= sup
/
QiV\ x ) A pF( y)dy
|* |< 1 i=n+1 j p<\v\<R
<
M
(
1 —K/p
*
r
\p J
sup
|APf(y)l)
p <\ y\ <R
where 7 = 7 («) is the constant of (21). From this estimate we conclude that
lim supdn(/, K ) l/n < —. We may now let k | 1 and p ] R to obtain (19).
n —*00
P
We now prove “only if” in the harmonic case (p = 1). We will prove the following
“strong” harmonic Bernstein-Walsh property for K = B (0,1). For any R > 1, there
exists c = c(N ) such that
HMb(o,.r) < cm 7v (n )iT n N - 1||/in ||K
(22)
for all hn e
where mN( n ) is the dimension of
■Once we have proved this,
one follows the proof of “only if” given after Theorem 0.
To prove (22), we use the spherical harmonic expansion of hn on S. Let {Y^}
be the classical orthonormal spherical harmonics in R N , i.e., Yk € ^d(k) (d(k) —
degree of Yk), Yk is homogeneous of degree d(k), f s Yj(x)Yk(x)da(x) — 6jk (<r —
surface area measure on 5) and we can write hn(x) =
akYk{x) where
ak = Js hn{x)Yk{x)d(j{x). It is known that
\\Yk\\B(o,R) = R d{k)\\Yk \\K < R d^ [ C Nd(k)N~1}1^2,
for some constant CV; and by the above relations we get \ak\ < ||^n||ft:[o'
which yields (22) and the theorem in the harmonic case (p = 1).
We now indicate very briefly the proof of “only if” for Ap when p > 1. In this
case one can prove a slight modification of (22).
14
THOMAS BAGBY and NORMAN LEVENBERG
For all e > 0, and all R > 1, there exist a constant Cn depending on n and N and
a constant Me depending on e and p , such that
lim cnrn = 0 ,
n —►oo
for all
|r| < 1
(23) (a)
and
(23) (b)
for all pn S
(p).
We outline the proof of (23). Fix e > 0 and R > 1, and let pn € H %(p). It is easy
to verify that the function
(24)
is a harmonic polynomial of degree n in R N+1 which satisfies
K n(x,0) = Pn(x),
x€Rn
(cf. [Av], Corollary 1.17). We can therefore apply (22) to K n and standard
estimates for harmonic functions (cf. [ABG, Lemma 1]) to obtain (23).
Once (23) is proved, one can modify the proof of “only if” following Theorem 0
to prove “only if” for Ap.
3. B alayage. B e rn ste in T h eo rem s for E lliptic E q u atio n s
Theorems 1 and 2 may be regarded as sharpened versions of the Runge approx
imation theorem: if a compact set K C C has connected complement, then every
holomorphic function on a neighborhood of K can be uniformly approximated on
K by polynomials in z. It is well known that the Runge approximation theorem can
be proved by the technique of “pushing poles”, or “balayage of poles” [W, Section
1.6]. It turns out that this “pole-pushing” technique may be refined to prove “if” of
Theorems 1 and 2 in a weak form; these techniques were developed by Andrievskii
(see [A] and the references given there). Since the details are rather technical, we
only outline this approach to Theorem 1, and then give further references.
We suppose / is holomorphic in an open neighborhood S7 of I, and we outline
the proof that there exists a number p < 1, depending only on fl, such that
limsup n_>00 dn(f, J )1//n < p. We may find a smooth function F, of compact support
on C, which agrees with / on an open neighborhood of I. Then
(25)
where A is Lebesgue measure on C. We may now break up the domain of integration
in (25) into the union of subdomains with small diameter; in this way we write F
as the sum of finitely many functions, each holomorphic on the complement of a
small disk. We let g be one of these functions, so that g is holomorphic off a small
BERNSTEIN THEOREMS
15
disk D, and we wish to show that g can be approximated on I by polynomials
pn(z) with errors which decay at an exponential rate depending only on D. To do
this we construct a sequence of points {Cj }^=o in C —/ , where Co is close to the disc
D, £j+i is close to Q, and (g is far from I. We next define a sequence of functions
{gj}sj=o which are holomorphic near 7, such that qo = g and ||<7j+i —Qj\\K is small.
These functions qj are defined inductively: we expand the function qj in a Laurent
series about Cj+i> and then truncate this series at a certain point to obtain qj+\.
The last function qa will be a rational function whose only pole is at
and we
take our polynomial approximator pn to be a partial sum in the Taylor expansion
of qs about the origin. The details of this proof may be obtained from [BL 2] in
the special case of the Cauchy-Riemann operator.
This pole-pushing technique was used in higher dimensions by Andrievskii [A]
to prove Theorem 4(a). The Laurent expansions in the preceding proof are then
replaced by “exterior” expansions in spherical harmonics; the Taylor expansion is
replaced by an “interior” expansion in spherical harmonics.
Both the pole-pushing technique and the duality method have recently been
used by the authors [BL2] to prove Bernstein theorems for solutions of elliptic
partial differential equations. To state this result we let p(x) = ]C|a|=m aa£a be
a non-constant homogeneous polynomial in R N, with complex coefficients, which
is never equal to zero on R N — {0}; here N > 2. Then the partial differential
operator p(D) = p( d / d x i , ... , d / d x n) is elliptic. We let Cn be the vector space
of polynomials q of degree at most n in N variables which are solutions of the
equation p(D)q = 0. If / is a continuous function on a compact set K C R N, we
now define
dn(f,K)
= in f {||/ - p n IIK - Pn e C n}.
T h eo rem 7. [BL2] Let K be a compact subset of R N such that R N — K is
connected.
(a) Let
be an open neighborhood of K . Then there is a constant p G (0,1),
depending only on p{D), K, and Q, with the following property: if f is a solution
of p{D) f = 0 on Cl, then lim sup^j^^ dn( f , K ) 1/n < p.
(b) Suppose that when we regard K C R N = R N + iO C CN , the set K is
L-regular. Let 0 < p < 1. Then there is an open neighborhood Q of K , depending
only on p(D ), K and p, with the following property: if f is a continuous function
on K such that limsupn_>00dn( f , K ) 1/ n < p, then f extends to a solution F of
p(D)F = 0 on Cl.
The proof of Theorem 7(b) is similar to the proof of Theorem 4(b). In the proof
of Theorem 7(a) by duality one can no longer utilize the Kelvin transform, but the
method outlined in Section 2 can be modified using another type of plurisubharmonic extremal function from several complex variables; full details may be found
in [BL2]. In the proof of Theorem 7(a) by pole-pushing techniques, one needs sub
stitutes for the Taylor and Laurent expansions used in the outline at the beginning
of this section. In place of the Taylor expansion one may use Lemma 5 above,
which is valid for solutions u of our homogeneous elliptic equation p(D)u = 0,
and ordinary power series expansions for functions of several complex variables. In
place of the Laurent expansions one uses the expansion for solutions of p(D)u = 0
16
THOMAS BAGBY and NORMAN LEVENBERG
near infinity given in Lemma 7 below. To motivate it we note that in a Laurent
expansion about the origin, each term which vanishes at infinity is actually some
constant times a derivative of the fundamental solution E(z) = ^ of the CauchyRiemann operator. A similar remark can be made about an “exterior” expansion
in spherical harmonics. In Lemma 7 we give a series expansion for solutions of
p(D)u = 0 near infinity in terms of derivatives of a fundamental solution for p( D).
There exists a fundamental solution E for p{D) which is a locally integrable
function on R N of the form E{x) = E\(x) + E 2{x) log |x|; here the restriction of
E\ to R n —{0} is real analytic and homogeneous of degree m — N, and E 2 is a
homogeneous polynomial of degree m —N if m > N and N is even (otherwise
E 2 = 0) (see [H, Chapter 7]). We let Q = {w e C°°{RN ) : p{D)w = 0 in R N },
and for each I we let Ji denote the set of all polynomials q on R N which are
homogeneous of degree I and satisfy p(D)q = 0 on R N.
L em m a 7. There exists a constant M > 0 with the following property: if r > 0
and f is a solution of p( D) f = 0 on \x\ > r, then there exists a unique sequence
w G Q, ho G Jo, h\ G 3 i, ... such that
OO
f ( x ) = w(x) + ^ 2 hk(D)E(x)
o
uniformly on |ar| > M r.
Lemma 7 is essentially due to Zemukov [Z]. The version stated here was obtained
later by one of the authors and applied by Dufresnoy, Gauthier and Ow to approx
imation problems for solutions of elliptic equations [DGO]. Related expansions
have been given by John, Balch, Harvey and Polking. The expansion in Lemma 7
is further developed by the authors in [BL2] to give the technical estimates needed
for the proof of Theorem 6 .
4. In te rp o la tio n
The proof of Theorem 1 in the present section is one of the simplest to give,
yet the most difficult to generalize; it uses polynomial interpolation to construct
good approximators. The key ingredient we need is the Hermite remainder for
mula for interpolation of a holomorphic function of one variable. Let z i , . . . , zn
be n distinct points in the plane and let / be a function which is defined at these
points. The polynomials lj(z) = Y[k^ { z - Zk)/ X[k^ j ( zj ~ zk), j = 1 ,... ,n, are
polynomials of degree n — 1 with lj(zk) = Sjk, called the fundamental Lagrange
interpolating polynomials, or FLIP’s, associated to z i , . . . , zn. Then the polyno
mial p{z) =
f ( zj ) h ( z ) is the unique polynomial of degree n — 1 satisfying
p(zj) = f(zj), j = 1 ,... ,n; we call it the Lagrange interpolating polynomial, or
LIP, associated to f , z i , . . . , zn. Suppose now that T is a rectifiable Jordan curve
such that the points z i , ... , zn are inside T, and / is holomorphic inside and on T.
We can estimate the error in our approximation of / by p at points inside T using
the following formula.
L em m a 8 . (Hermite Remainder Formula) For any z inside T,
...
. .
1 f u(z) f ( t ) ,
/ w - p ( 2) = 2- y r ^ ) ( F — } dt’
17
BERNSTEIN THEOREMS
where ui{z) = n ! c = i( z ~ z k)<jj(t)-u(z) f { t )
dt is clearly a polynomial of
rr
(t — z)
uj(t)
K ’ 2m Jr
degree < n — 1. Using the Cauchy integral formula for / , we see that
P roof. The function p(z) = ——: f
“ (*) f ( t )
dt
r uj(t) (t — z)
(26)
for 2 inside T. In particular, for each k we have f ( z k ) —p(zk) — 0, and hence p = p.
Now the lemma follows from (26).
We will now prove “if” in Theorem 1 using interpolation. We suppose / is
holomorphic on D r , and fix R! < R. We let n > 1, and recall that the zeros of
the Chebyshev polynomial Tn are simple and lie on /; we take p = pn to be the
LIP for / , x \ , ... , x n where Tn{x) = I I( x ~ xj)- We next note that by (8) we have
r„ i __ o l —n , and
1+
w 2n
for
2 e E r >,
where 2 = \{w + ^-). Now if \ f ( z ) \ < M on E r >, then for x £ I we have
m
2iriTn{x) L e r > Tn ( t) ( t
-dt
x)
2 \n
M
le n g th ^ )
< —21_n
2n
R ' J 1 - l / R ' 2n di st ( I , E r ,)
It follows that we have lim supdn(/, I ) l ^n < 1/R'. We may then let R' | R to
n —>oo
obtain (7), which completes the proof.
One may use a similar argument to prove “if” in Theorem 2: one can construct
good polynomial approximators by interpolation at appropriate points in K, and
the desired estimates come from a form of the Hermite remainder formula.
Although the method of interpolation is a natural tool for proving Bernstein
theorems for functions of one complex variable, it is less successful as a technique
for proving Bernstein theorems for problems in higher dimensions: multivariate
interpolation is more subtle than univariate interpolation, and there is no analogue
of the Hermite remainder formula. However, once we have proved Bernstein the
orems in higher dimensions by the methods discussed in earlier sections, we can
use interpolation as a practical method for constructing approximators which are
asymptotically optimal. To describe this in a typical situation, we let p(D) be an
elliptic operator of the type described before Theorem 7, with the corresponding
polynomial classes Cn defined there. We order a basis ei,C 2, ... for ( J £ n by in
creasing degree and use any ordering for those polynomials of the same degree.
Let m n — d im £ n, so that e i ,... ,e mn form a basis for Cn. Choose mn points
A n = {ani , ... ,a nmn} C K and form the generalized Vandermonde determinant
^n(-^n) —det[ei(anj)]i)j =i i... ,m n .
18
THOMAS BAGBY and NORMAN LEVENBERG
If Vn(An) / 0, we can form the FLIP’s
,
/
\
lnj{x) =
VnyClnli ' • • t *E»• • • 5®nmn)
.
---------- V j A ) ---------- ’
J =
In the one (complex) variable case, we get cancellation in this ratio so that the
formulas for the FLIP’s simplify as indicated above. In general, we still have
lnj{oini) = <$jj and lnj e Cn since lnj is a linear combination of e i , ... , emn. We
call
An = s u p 5 2 |^ n j(x)|
the n-th Lebesgue constant for K , A n. For / defined on K ,
{Lnf )(x) — ^ ^ / ianj)lnj{x )
3=1
is the Lagrange interpolating polynomial (LIP) for / at the points A n. We say that
K is determining for (J Cn if whenever h G (J Cn satisfies h = 0 on K , it follows
that h = 0. For these sets we can find points A n for each n with Vn(An) ^ 0. We
have the following elementary result [BL2].
T h eo rem 8 . Let K be determining for (J Cn and let A n C K satisfy Vn(An) ^ 0
for each n. Given f bounded on K , if lim sup A*'n = 1, then
n —►oo
limsup ||/ - L nf \\]^n = lim supdlJ n.
n —►oo
n —►oo
P roof. Fix £ ^ 0 and choose, for each rt, a polynomial Pn G jCji ^vith || f
d l/n + e. Since pn G Cn, we have L npn = pn and
11
^
11/ - Lnf\\K = 11/ - Pn + L npn - L nf \\K
< | | / - P n | k + A n | | / - p n ||K = ( l + A n) | | / - p n ||tf.
Using the hypothesis limsupAy™ = 1, we obtain the conclusion.
n —►
oo
Arrays of points {An}, n = 1, 2, . . . satisfying lim supA yn = 1 can be con71— ►OO
structed by taking A n to be a set of n-Fekete points for K : for each n, choose
An C K so that
m u -.\Vn(Xn)\ = |K,(A„)|.
Since |V^l(-X'ri)| is a continuous function on K mn, such points exist; by definition
they satisfy An < m n and, as can be readily shown, lim m lJ n = 1.
BERNSTEIN THEOREMS
19
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V. Andrievskii, Uniform approximation on compact sets i n R k, k > 3, preprint.
[ABG]
D.H. Armitage, T. Bagby, and P.M. Gauthier, Note on the decay of elliptic
equations, Bull. London Math. Soc. 17 (1985), 554-556.
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T. Bloom, On the convergence of multivariate Lagrange interpolants, Con
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[D]
P. Davis, Interpolation and Approximation, Dover, 1975.
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A. Dufresnoy, P.M. Gauthier and W.H. Ow, Uniform approximation on closed
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W.K. Hayman, Power series expansions for harmonic functions partial differ
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G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, 1966.
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W. Plesniak, On some polynomial conditions of the type of Leja in C71, in
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THOMAS BAGBY and NORMAN LEVENBERG
20
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Thom as Bagby
Rawles Hall
Indiana University
Bloom ington IN 47405
U.S.A.
Norman Levenberg
University of Auckland
Private Bag 92019
Auckland
NEW ZEALAND

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