bernstein theorems

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 22 (1993), 1-20

B E R N S T E IN TH EO R EM S

T hom as B a g b y a n d N o rm a n Le v e n b er g

(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 su p d j/” < 1, where

n —+oo

dn = inf{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 methods 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 nThere 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 —>ocis 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 information 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] [BL1], [ZS], and solutions of elliptic equations [BL2]. 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 Zahar- juta 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),

wherea;/(/*)= 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 runcation. B ernste in T heorem s in Several Com plex V ariablesAn 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 limsupdn(/, A)1/” < 1/r, and we may now let r f R to concluden —>00

thatlimsupdn(/, A )1//n < 1/R.

n—►oo

This proves the following equivalence in one direction.

BERNSTEIN THEOREMS 3

T heorem 0. Let f be continuous on A = {2 G C : \z\ < 1}, and R > 1. Then

limsupdn(/, 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 ~ Pn-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

MI I / - P » I I a < ; ^ , 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 ) - P n - l ( z ) \ < pU\\pn ~ Pn—11|A < pn{\\Pn ~ / | |a + I I / -P » - i |Ia )\A<P

- H R!

Since p and R' were arbitrary numbers satisfying 1 < p < R' < R, we conclude that po + YlTiPn ~ Pn-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.

Lem 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 gx ■ 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

Dr =={z : gK{z) < log R}, (5)


thenIpWI < 2 6 DR. (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 <0onC U {oo} — 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 heorem 1. (Bernstein) Let f be continuous on I, and R > 1. Then

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 Dr , 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—►oop > i .

limsupdn(/, Z)1/” < l /R (7)n—► oo


Lem m a 2. I f f is holomorphic on D r , then

OOf (z) = ^ 2 anTn{z), Z € Dr ,

n —0

where

for 1 < p < R. The series is uniformly convergent on compact subsets of D r .

Proof. 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

on the entire ellipse D r, so for 1 < P < R we have f E zkf (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 an , we see that / holomorphic on D r implies that

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.

Theorem 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

restriction to K of a function holomorphic in Dr .

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

for 1 < p < R. For each integer k > 0 the function zkf ( z ) is in fact holomorphic

limsup |an |1//n < 2/ R,

n—►oo


are domains which increase up to Dr . If we choose n sufficiently large so that K c f i n and fin is “almost” all of Dr , 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 (discussed in Section 4 below) are lacking; but the remarkable truth of the matter is that the exact same result holds. First, a few preliminary definitions and comments. 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). With the one-variable Lemma 1 as motivation, we define

uK(z) = m a x |o ,s u p |^ ^ lo 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 semicontinuous 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 continuous. 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

ue (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).


If the compact set K C CN is L-regular, then for each R > 1 we defineDR = {z : uK (z) < logR}; (10)

then we clearly have the Bernstein-Walsh inequality

Ip(*)I < lb lk f ldeep, * e d r ( i i)for every polynomial p in CN (conversely, it is known that if K satisfies a Bemstein- Walsh 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 heorem 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/R71— KX)

if and only if f is the restriction to K of a function holomorphic in Dr .

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 , . . . , zN) : |*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 Extension T heorem . Let f be holomorphic in a polynomial polyhedron

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 f t .

One can then use truncations of the Taylor series of F(z,w) = ca(3Zaw13 restricted 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].


2. Duality. B ernste in theorem s for harm onic functionsTo 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 approximator 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 Dp and has compact support in D r . We then set F = </>/ in D r and let F be identically 0 outside of D r .

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 jPn 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 — * (^ * -£')> (13)oz oz

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)


this follows from noting that for 2 sufficiently large we have — Ylk uniformly for C £ I-, and then using the fact that p satisfies ( k = 0 for0 < 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 dDr. 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 limsupdn(/, I ) l^n < We may now letn —»oo 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 RN, 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 RN, we now define

dn( f , K) = inf{ ||/ - hn\\K : hn €

Theorem 4. [A], [BL1], [ZS] Let K be a compact subset of RN such that RN—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 L- regular. 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 limsupn_ 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 RN will satisfy the L-regularity hypothesis of Theorem 4(b) if it is the closure of a domain in RN 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 p G 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) = cn \x — y|2~iV for the Laplace operator A (here cn is a constant depending only on the dimension). The


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)J k

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].

Lem 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], [BL1]). This fact leads to the following Schwarz lemma for holomorphic functions of several variables [Bi],[BLlJ.

Lem m a 4. Let ft be a bounded domain in CN and let a G ft. Let K be a compact subset of Cl. I f 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


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],

Lem m a 5. For each R > 0, there exist positive numbers r < R and C with the following property. I f h is harmonic on a real ball B = {x e RN : \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 heorem 5. [ND] Let K be a compact set in the plane with connected complement, 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 / Rn—► oo

if and only if f is the restriction to K of a function harmonic in Dr .

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 € RN and r > 0 we let B(a, r) = {x € R N : |rr — a| < r}.


T heorem 6 . Let K = B(a, r ) C R N, and let R > 1. Let f be continuous on K . Then

limsupdn( / , K ) ltn < 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).

Proof. Without 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 DotE (y)xa/a \ € H^{p) (see [BL2]).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 DaE 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) = (zx i + y\)2 + . . . + (zx 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 + ... + (zxN + yN)2](2p~N)/2

such that gx,y{z) = E(zx + 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 \M=l

= \g{l)(0)/l\\ < Kl sup 1^)1 < 7AC1,\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}-


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 o<\v\<Rp<\y\<R

where A is Lebesgue measure in R N. From this we obtain, for |rc| < p,

Fix) = £ / QiV\ x)ApF(y) dXly).l= o jp < \y \< R

Since Q\y\ x ) € Hf* (p), we conclude that

d n ( f , K ) < sup |* |< i

= sup|* |< 1

F I * ) - Y , f Q\V\ x ) A ”F ly)dyi=o Jp<\y\<R

o° .

/ QiV\ x ) A pF(y)dyi=n+1 j p<\v\<R

< M ( * r sup |APf(y)l)1 — K/p \ p J 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) < cm7v(n )iT nN -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) whereak = Js hn{x)Yk{x)d(j{x). It is known that

\\Yk\\B(o,R) = R d{k)\\Yk \\K < Rd^ [ 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).


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

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

(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. Balayage. B ernste in Theorem s for E lliptic EquationsTheorems 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 limsupn_>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

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

lim cnrn = 0, for all |r| < 1 (23) (a)n—►oo

and(23) (b)

(24)

is a harmonic polynomial of degree n in R N+1 which satisfies

K n(x,0) = Pn(x), x € R n

(25)


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 [BL2] 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 RN, 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/dx i , . . . , d / dx 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 RN, we now define

d n ( f ,K ) = in f { ||/ - p n IIK - Pn e Cn}.

T heorem 7. [BL2] Let K be a compact subset of R N such that RN — 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 limsup^j^^ dn( f , K ) 1/n < p.

(b) Suppose that when we regard K C RN = 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 substitutes 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


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 Cauchy- Riemann 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) + E2{x) log |x|; here the restriction of E\ to R n — {0} is real analytic and homogeneous of degree m — N, and E2 is a homogeneous polynomial of degree m — N if m > N and N is even (otherwise E2 = 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.

Lem 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 approximation 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 rpo la tionThe 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 formula 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 l j(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 polynomial 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.

Lem 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’


where ui{z) = n !c = i(z ~ z k)-

Proof. The function p(z) = ——: fK ’ 2m Jr

< j j ( t ) - u ( z ) f { t )dt is clearly a polynomial of

rr (t — z) uj(t) degree < n — 1. Using the Cauchy integral formula for / , we see that

“ (*) f ( t ) r uj(t) (t — z)

dt (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 Dr , 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 haver„ i __ o l —n , and

1 + w2 n for 2 e Er >,

where 2 = \{w + ^-). Now if \ f (z ) \ < M on E r >, then for x £ I we have

2iriTn{x)Lm -dt

< —21_n 2n

e r > Tn ( t) ( t x)

2 \ n M l e n g t h ^ ) R' J 1 - l / R ' 2n di st (I , E r ,)

It follows that we have limsupdn(/, I ) l^n < 1/R'. We may then let R' | R ton —>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 theorems 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,C2, . . . for ( J £ n by increasing degree and use any ordering for those polynomials of the same degree. Let m n — d im £n, so that e i , . . . ,emn 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 .


If Vn(An) / 0, we can form the FLIP’s

, / \ VnyClnli ' • • t *E» • • • 5®nmn) .

l n j { x ) = ---------- 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 = su 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 An. 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 An for each n with Vn(An) ^ 0. We have the following elementary result [BL2].

T heorem 8 . Let K be determining for (J Cn and let An C K satisfy Vn(An) ^ 0 for each n. Given f bounded on K , if lim sup A*'n = 1, then

n —►oo

limsup | | / - Lnf \\]^n = lim supdlJ n.n —►oo n —►oo

Proof. Fix £ ^ 0 and choose, for each rt, a polynomial Pn G jCji ^vith || f 11 ^ d l/n + e. Since pn G Cn, we have Lnpn = pn and

11/ - Lnf\\K = 11/ - Pn + Lnpn - Lnf \\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 con-71— ►OO

structed by taking An 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.


R eferences

[A] 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.[Av] V. Avanissian, Cellule d ’harmonicite et prolongement analytique complexe,

Hermann, 1985.[B] T. Bagby, Approximation in the mean by solutions of elliptic equations, Trans.

Amer. Math. Soc. 281 (1984), 761-784.[BLl] T. Bagby and N. Levenberg, Bernstein theorems for harmonic functions, to

appear, in Methods of Approximation Theory in Complex Analysis and Mathematical Physics.

[BL2] T. Bagby and N. Levenberg, Bernstein theorems for elliptic equations, to appear.

[Bi] E. Bishop, Holomorphic completions, analytic continuation, and the interpolation of semi-norms, Ann. Math. 78 (1963), 468-500.

[Bl] T. Bloom, On the convergence of multivariate Lagrange interpolants, Constructive Approximation 5 (1989), 415-435.

[D] P. Davis, Interpolation and Approximation, Dover, 1975.[DGO] A. Dufresnoy, P.M. Gauthier and W.H. Ow, Uniform approximation on closed

sets by solutions of elliptic partial differential equations, Complex Variables 6 (1986), 235-247.

[Ha] W.K. Hayman, Power series expansions for harmonic functions partial differential equations, Bull. London Math. Soc. 2 (1970), 152-158.

[H] L. Hormander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1983.

[K] M. Klimek, Pluripotential Theory, London Mathematical Society Monographs 6 , Clarendon Press, 1991.

[L] G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, 1966.[P] W. Plesniak, On some polynomial conditions of the type of Leja in C71, in

Analytic Funcitons Kozubnik 1979, Proceedings, Lecture Notes in Mathematics 798, Springer-Verlag, pp. 384-391.

[R] D.L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 41-53.

[Sa] A. Sadullaev, P-regularity of sets in CN, in Analytic Funcitons Kozubnik 1979, Proceedings, Lecture Notes in Mathematics 798, Springer-Verlag, pp. 402-408.

[Si] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.

[S2l J. Siciak, Extremal plurisubharmonic functions in CN, Ann. Polon. Math. 39 (1981), 175-211.


[W] J.L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Coll. Publ. Vol. 20, Third Edition, 1960.

[Zl] V.P. Zaharjuta, Isomorphism of spaces of harmonic functions, in Mathematical Analysis and its Applications, Volume III (Russian), Izdat. Rostov Univ., Rostov-on-Don, 1971, pp. 152-158.

[Z2] V.R Zaharjuta, Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein- Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. (Russian) 33 (1976), 137-148.

[Z3] V.R Zaharjuta, Spaces of harmonic functions, preprint.[ZS] V.R Zaharjuta and N.I. Skiba, Orthogonal harmonic polynomials and Bem-

stein-Walsh theorem i n R n, Doga-Tr. J. of Mathematics 16 (1992), 46-49.[Z] H.K. Zemukov, Expansion of the solutions of an elliptic equation in the deriva

tives of the fundamental solution, in Linear and Nonlinear Boundary Value Problems, ed. by Ju. O. Mitropol’skii and A.A. Berezovskii, Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev. (Russian), 1971, pp. 153-159.

Thomas Bagby Norman LevenbergRawles Hall University of AucklandIndiana University Private Bag 92019Bloomington IN 47405 AucklandU.S.A. NEW ZEALAND

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