Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory

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by  Len Bos

BERNSTEIN-WALSH THEORY ASSOCIATED TO
CONVEX BODIES AND APPLICATIONS TO
MULTIVARIATE APPROXIMATION THEORY
7
1
0
2 L. BOS AND N. LEVENBERG*

n
a
J

9 Abstract. We prove a version of the Bernstein-Walsh theorem
1
onuniformpolynomialapproximationofholomorphicfunctions on

] compact sets in several complex variables. Here we consider sub-
V
classes of the full polynomial space associated to a convex body
C P. As a consequence, we validate and clarify some observations of
. Trefethen in multivariate approximation theory.
h
t
a
m 1. Introduction.
[

A standard theorem in several complex variables, quantifying the
1
v classical Oka-Weil theorem on polynomial approximation – which itself
3 isthemultivariateversionoftheclassicalRungetheoremforpolynomial
1
approximation in the complex plane – is the Bernstein-Walsh theorem:
6
5
0 Theorem 1.1. Let K Cd be compact, nonpluripolar and polynomi-
⊂
. ally convex with V continuous. Let R > 1, and let Ω := z : V (z) <
1 K R K
{
0 logR . Let f be continuous on K. Then
7 }
1 limsupD (f,K)1/n 1/R
: n ≤
v n
→∞
i
X if and only if f is the restriction to K of a function holomorphic in Ω .
R
r
a Here for K Cd compact,
⊂
(1.1)
1
V (z) = max[0,sup log p(z) : p := max p(ζ) 1 ]
K K
{deg(p) | | || || ζ K | | ≤ }
∈
1991 Mathematics Subject Classification. 32U15, 32U20, 41A10.
Key words and phrases. convex body, Bernstein-Walsh, multivariate
approximation.
*Supported by Simons Foundation grant No. 354549.
1
2 L. BOS ANDN. LEVENBERG*
where p is a nonconstant holomorphic polynomial; and for a continuous
complex-valued function f on K,
D (f,K) := inf f p : p
n n K n n
{|| − || ∈ P }
where is the space of holomorphic polynomials of degree at most n.
n
P
See [1] for a survey and history of Theorem 1.1 in both one and several
complex variables.
In Trefethen [9], the author gives some evidence for why one might
consider non-traditional notions of “degree” of a polynomial in the set-
ting of multivariate approximation theory. More precisely, for certain
functions f on K = [ 1,1]d he compares the approximation num-
−
bers D (f,[ 1,1]d) where “degree” has three possible meanings: total
n
−
degree, Euclidean degree, or maximum degree. We clarify this dis-
tinction in a more general setting by describing generalizations of the
extremal functions V associated to subclasses of the full polynomial
K
space . Given a convex body P (R+)d = [0, )d, following
nPn ⊂ ∞
Bayraktar [2], we define a P extremal function V associated to K.
P,K
S −
In Section 2 we list and prove some basic properties of these functions.
We state and prove a generalization of Theorem 1.1 in this setting in
Section 3. Section 4 recovers the Trefethen cases for K = [ 1,1]d
−
by taking appropriate P and provides explicit examples of functions f
comparing rates of approximation.
2. Background: P extremal functions.
−
In what follows, we fix a convex body P (R+)d; i.e., a compact,
convex set in (R+)d with non-empty interio⊂r Po. Standard examples
include the case where
(1) P is a non-degenerate convex polytope, i.e., the convex hull of
a finite subset of (Z+)d in (R+)d with Po = ;
6 ∅
(2) P := (x ,...,x ) (R+)d : (xp + xp)1/p 1 is the (non-
nepgativ{e) 1portiond o∈f an lp ball in1 (R·+·)·d,d1 p≤ } .
≤ ≤ ∞
As a particular case of (2), with p = 1 we have P = Σ where
1
Σ := (x ,...,x ) Rd : x ,...,x 0, x + x 1 .
1 d 1 d 1 d
{ ∈ ≥ ··· ≤ }
We will consider convex bodies P (R+)d with the property that
⊂
(2.1) Σ kP for some k Z+.
⊂ ∈
POLYNOMIAL DEGREE DEFINED BY A CONVEX BODY 3
Associated with P, following [2], we consider the finite-dimensional
polynomial spaces
Poly(nP) := p(z) = c zJ : c C
J J
{ ∈ }
J nP (Z+)d
∈ X∩
for n = 1,2,.... Here J = (j ,...,j ). In the case P = Σ we have
1 d
Poly(nΣ) = , the usual space of holomorphic polynomials of degree
n
at most n inPCd. Clearly there exists a minimal positive integer A =
A(P) 1 such that P AΣ. Thus
≥ ⊂
(2.2) Poly(nP) A = for all n.
n An
⊂ P P
We let d =dim(Poly(nP)). From (2.2),
n
(2.3) d dimP = 0(nd).
n An
≤
Note it follows from convexity of P that
p Poly(nP), p Poly(mP) p p Poly((n+m)P).
n m n m
∈ ∈ ⇒ · ∈
ItsufficestoverifythisformonomialszA Poly(nP), zB Poly(mP).
∈ ∈
Then A nP, B mP so A = na, a P and B = mb, b P. Thus
∈ ∈ ∈ ∈
n m
A+B = na+mb = (n+m)[ a+ b] (n+m)P.
n+m n+m ∈
Recall the indicator function of a convex body P is
φ (x ,...,x ) := sup (x y + x y ).
P 1 d 1 1 d d
···
(y1,...,yd)∈P
For the P we consider, φ 0 on (R+)d with φ (0) = 0. Define the
P P
≥
logarithmic indicator function
H (z) := suplog zJ := φ (log z ,...,log z ).
P P 1 d
| | | | | |
J P
∈
Here zJ := z j1 z jd for J = (j ,...,j ) P (the components j
1 d 1 d k
| | | | ···| | ∈
need not be integers). From (2.1), we have
1 1
H (z) max log+ z = max [max(0,log z )].
P j j
≥ k j=1,...,d | | k j=1,...,d | |
WewilluseH todefinegeneralizationsoftheLelongclassesL(Cd), the
P
set of all plurisubharmonic (psh) functions u on Cd with the property
that u(z) log z = 0(1), z , and
− | | | | → ∞
L+(Cd) = u L(Cd) : u(z) log+ z +C
u
{ ∈ ≥ | | }
4 L. BOS ANDN. LEVENBERG*
where C is a constant depending on u. We remark that, a priori, for
u
a set E Cd, one defines the global extremal function
⊂
V (z) := sup u(z) : u L(Cd), u 0 on E .
E
{ ∈ ≤ }
It is a theorem, due to Siciak and to Zaharjuta (cf., Theorem 5.1.7 in
[5]), that for K Cd compact, V coincides with the function in (1.1).
K
⊂
Moreover,
V (z) := limsupV (ζ) L+(Cd)
K∗ K ∈
ζ z
→
precisely when K is nonpluripolar; i.e., for K such that u plurisubhar-
monic on a neighborhood of K with u = on K implies u .
−∞ ≡ −∞
Define
L = L (Cd) := u PSH(Cd) : u(z) H (z) = 0(1), z ,
P P P
{ ∈ − | | → ∞}
and
L = L (Cd) = u L (Cd) : u(z) H (z)+C .
P,+ P,+ P P u
{ ∈ ≥ }
Then L = L(Cd) and L = L+(Cd). Given E Cd, the P extremal
Σ Σ,+
⊂ −
function of E is given by V (z) := limsup V (ζ) where
P∗,E ζ z P,E
→
V (z) := sup u(z) : u L (Cd), u 0 on E .
P,E P
{ ∈ ≤ }
For P = Σ, we recover V = V . We will restrict to the case where
E Σ,E
E = K Cd is compact. In this case, Bayraktar [2] proved a Siciak-
⊂
Zaharjuta type theorem showing that V can be obtained using poly-
P,K
nomials. Note that 1 log p L for p Poly(nP).
n | n| ∈ P n ∈
Proposition 2.1. Let K Cd be compact and nonpluripolar. Then
⊂
1
V = lim logΦ
P,K n
n n
→∞
pointwise on Cd where
Φ (z) := sup p (z) : p Poly(nP), p 1 .
n n n n K
{| | ∈ || || ≤ }
If V is continuous, the convergence is locally uniform on Cd.
P,K
It follows that V = V where
P,K P,K
b
K = z : p(z) p , for all p
K n
{ | | ≤ || || ∈ P }
n
[
is the polynombial hull of K. Also, either V + , which occurs if
P∗,K ≡ ∞
and only if K is pluripolar, or V L .
P∗,K ∈ P,+
POLYNOMIAL DEGREE DEFINED BY A CONVEX BODY 5
Example 2.2. Let K = Td = (z ,...,z ) : z = = z = 1 , the
1 d 1 d
unit d torus in Cd. Then { | | ··· | | }
−
V (z) = H (z) = maxlog zJ .
P,Td P
J P | |
∈
This is Example 2.3 in [2]. There it is stated only for P a convex
polytope. We give an alternate proof in Proposition 2.4.
From Proposition 2.1 we have a Bernstein-Walsh inequality.
Proposition 2.3. Let K be nonpluripolar. Then for p Poly(nP),
n
∈
(2.4) p (z) p exp(nV (z)), z Cd.
n n K P,K
| | ≤ || || ∈
In particular, if V is continuous, for R > 1
P,K
Ω := z : V (z) < logR
R P,K
{ }
is an open neighborhood of K and for p Poly(nP),
n
∈
p (z) p Rn, z Ω .
n n K R
| | ≤ || || ∈
Many of the results in Chapter 5 of [5] remain valid for P extremal
−
functions. From the definition of V and Example 2.2, we obtain:
K,P
(1) If K are compact sets with K K and K := K ,
{ j} j+1 ⊂ j j j
then lim V = V ;
j→∞ P,Kj P,K T
(2) for K compact, V is lower semicontinuous;
P,K
(3) for K compact, if V = 0 then V is continuous (on Cd);
P∗,K|K P,K
(4) forK compact, lim V = V whereK := z : dist(z,K)
ǫ→0 P,Kǫ P,K ǫ { ≤
ǫ .
}
For P = Σ, we say a compact set K is L regular if V is contin-
K
−
uous on K; i.e., if V = V (equivalently, V = 0 on K). Given a
K K∗ K∗
convex body P (R+)d, we call a compact set PL regular if V is
P,K
⊂ −
continuous on K; i.e., if V = V . Since P AΣ, for each n we
P,K P∗,K ⊂
have
1
sup log p : p Poly(nP), p 1
n n n K
{n | | ∈ || || ≤ }
1
sup log p : p , p 1 .
n n An n K
≤ {n | | ∈ P || || ≤ }
Hence V (z) A V (z), z Cd. It follows that if K is L regular,
P,K K
then K is PL ≤regu·lar for any∈convex body P (R+)d. −
− ⊂
The proofs of Propositions 5.3.12, 5.3.14 and Corollary 5.3.13 in [5]
carry over in this setting to show that K = D is PL regular for D a
−
bounded open set with C1 boundary. Hence for any compact set K,
6 L. BOS ANDN. LEVENBERG*
one can find a decreasing sequence of bounded open sets D with
j
{ }
smooth boundaries (thus D is PL regular) such that K = D .
j j
−
We end this section with a result on P extremal functions for prod-
− T
uct sets. This will be useful in Section 4. Before proceeding we re-
quire a definition: we call a convex body P (R+)d a lower set
if for each n = 1,2,..., whenever (j ,...,j ) ⊂nP (Z+)d we have
1 d
(k ,...,k ) nP (Z+)d for all k j , l = 1,.∈..,d. ∩
1 d l l
∈ ∩ ≤
Proposition 2.4. Let P (R+)d be a lower set and let E ,...,E C
1 d
⊂ ⊂
be compact and nonpolar. Then
(2.5) V (z ,...,z ) = φ (V (z ),...,V (z )).
P∗,E1×···×Ed 1 d P E∗1 1 E∗d d
Proof. For simplicity, we do the case d = 2. Thus let E,F C be
⊂
compact sets and let φ = φ (x ,x ) be the support function of P.
P P 1 2
From properties (1) and (4) of P extremal functions, we can assume
−
that E,F are L regular in C with E = E, F = F and that E F is
PL regular in C−2. ×
−
To see that b b
φ (V (z),V (w)) V (z,w),
P E F P,E F
≤ ×
since φ (0,0) = 0, it suffices to show that φ (V (z),V (w)) L (C2).
P P E F P
∈
From the definition of φ ,
P
φ (V (z),V (w)) = sup [xV (z)+yV (w)]
P E F E F
(x,y) P
∈
which is an upper envelope of locally bounded above plurisubharmonic
functions. Asφ isconvexandV ,V arecontinuous, φ (V (z),V (w))
P E F P E F
is continuous. Finally, since V (z) = log z + 0(1) as z and
E
| | | | → ∞
V (w) = log w +0(1) as w , it follows that φ (V (z),V (w))
F P E F
L (C2). | | | | → ∞ ∈
P
To prove the reverse inequality, and hence (2.5), we modify the proof
of Theorem 5.1.8 in [5]. Let µ,ν be probability measures on E,F such
that (E,µ) and (F,ν) are Bernstein-Markov pairs: thus, given ǫ > 0,
there exists a positive constant M such that
(2.6) t M(1+ǫ)n t and t M(1+ǫ)n t
n E n L2(µ) n F n L2(ν)
|| || ≤ || || || || ≤ || ||
for all t (C). Any compact set B C admits a measure η so
n n
∈ P ⊂
that (B,η) is a Bernstein-Markov pair; cf., [4]. Let f = f(z,w)
∈
Poly(nP) with f 1. Given an orthonormal basis p = p (z)
E F j j
|| || × ≤ { }
POLYNOMIAL DEGREE DEFINED BY A CONVEX BODY 7
in L2(µ) and q = q (w) in L2(ν) for the univariate polynomials,
k k
{ }
where deg(p ) = j and deg(q ) = k, we can write
j k
f(z,w) = c p (z)q (w)
jk j k
(j,k) nP
X∈
where
c = < f,p q > 1
jk j k L2(µ ν)
| | | × | ≤
for all (j,k) nP. The lower set property of P implies that p q
j k
∈ ∈
Poly(nP) for (j,k) nP. Using (2.6),
∈
p M(1+ǫ)j, q M(1+ǫ)k
j E k F
|| || ≤ || || ≤
so that, using the univariate Bernstein-Walsh estimates, i.e., (2.4) for
E,F (d = 1),
p (z)q (w) M2(1+ǫ)j+kejVE(z)+kVF(w)
j k
| | ≤
for all (z,w) C2. Thus, using (2.2),
∈
f(z,w) d M2(1+ǫ)An max ejVE(z)+kVF(w).
n
| | ≤ ·(j,k) nP
∈
Now
max ejVE(z)+kVF(w) = max enjVE(z)+nkVF(w) n
(j,k) nP (j/n,k/n) P
∈ ∈
so that, since (cid:0) (cid:1)
j k
log max enjVE(z)+nkVF(w) = max [ VE(z)+ VF(w)],
(j/n,k/n) P (j/n,k/n) P n n
∈ ∈
(cid:0) (cid:1)
we have
1 1 j k
log f(z,w) log(d M2)+Alog(1+ǫ)+ max [ V (z)+ V (w)]
n E F
n | | ≤ n (j/n,k/n) P n n
∈
1
log(d M2)+Alog(1+ǫ)+φ (V (z),V (w)).
n P E F
≤ n
The result follows, using (2.3), upon letting n .
→ ∞ (cid:3)
In particular, this gives (another) proof of Example 2.2.
Remark 2.5. For x,y Rd, let x y = x y + x y . Let
1 1 d d
∈ · ···
Po := y Rd : sup x y 1
{ ∈ | · | ≤ }
x P
∈
8 L. BOS ANDN. LEVENBERG*
be the polar of P and let be the dual norm defined by Po; i.e.,
Po
||·||
(y ,...,y ) := sup x y .
1 d Po
|| || | · |
x P
∈
Thus Po is the unit ball in this norm. Then we can write (2.5) as
(2.7) V (z ,...,z ) = (V (z ),...,V (z )) .
P,E1×···×Ed 1 d || E1 1 Ed d ||Po
3. Bernstein-Walsh theorem.
As in the previous section, we fix a convex body P (R+)d. We
⊂
prove a Bernstein-Walsh theorem in this setting. Given a compact set
K, for a continuous complex-valued function f on K we define
D = D (f,K,P) inf f p : p Poly(nP) .
n n n K n
≡ {|| − || ∈ }
Theorem 3.1. Let K be compact and PL regular. Let R > 1, and let
−
Ω := z : V (z) < logR . Let f be continuous on K.
R P,K
{ }
(1) If P is a lower set and f is the restriction to K of a function
holomorphic in Ω , then
R
limsupD (f,P,K)1/n 1/R.
n
≤
n
→∞
(2) If K = K, limsup D (f,P,K)1/n 1/R implies f is the
restriction to K ofna→f∞unctnion holomorp≤hic in Ω .
R
b
Proof. (2). Suppose
limsupD1/n = 1/R
n
n
→∞
for some R > 1. We show that if p Poly(nP) satisfies D =
n n
∈
||f −pn||K, then the series p0+ ∞1 (pn−pn−1) converges uniformly on
compact subsets of Ω to a holomorphic function F which agrees with
R
P
f on K. To this end, choose R with 1 < R < R; by hypothesis the
′ ′
polynomials p satisfy
n
M
(3.1) f p , n = 0,1,2,...,
|| − n||K ≤ Rn
′
forsomeM > 0. Nowlet 1 < ρ < R, andapply(2.4) tothepolynomial
′
p p Poly(nP) to obtain
n n 1
− − ∈
sup p (z) p (z) ρn p p
n n 1 n n 1 K
| − − | ≤ || − − ||
Ωρ
M(1+R)
ρn( p f + f p ) ρn ′ .
≤ || n − ||K || − n−1||K ≤ R′n
POLYNOMIAL DEGREE DEFINED BY A CONVEX BODY 9
Since ρ and R were arbitrary numbers satisfying 1 < ρ < R < R, we
′ ′
ctoonaclhuodleomthoartphpi0c+func∞1tio(pnnF−. pFnr−o1m) c(o3n.1v)e,rFges=lofcaolnlyKu.niformly on Ω(cid:3)R
P
To verify (1) we will follow Bloom’s reasoning in [3]. Note that
Poly(nP) is a finite-dimensional complex vector space; we call its di-
mension d (see Remark 2.3). We begin with the key lemma. Fixn k
n
≥
where k is as in (2.1) and let Q ,...,Q be a basis for Poly(nP). For
1 dn
R > 0 define
D := z Cd : Q (z) < Rn, j = 1,...,d .
R j n
{ ∈ | | }
Lemma 3.2. Let P be a lower set and let f be holomorphic in a
neighborhood of D . Then for each positive integer m, there exists
R
G Poly(mP) such that for all ρ R,
m
∈ ≤
f G B(ρ/R)m
|| − m||Dρ ≤
where B is a constant independent of m.
Proof. We show for m = sn, an integer multiple of n, that there exists
G Poly(mP) such that for all ρ R,
m
∈ ≤
f G B(ρ/R)m+n
|| − m||Dρ ≤
where B is independent of m. To this end, let S : Cd Cdn via
→
S(z) := (Q (z),...,Q (z)).
1 dn
Then S(Cd) is a subvariety of Cdn and S(D ) is a subvariety of the
R
polydisk
∆ := ζ Cdn : ζ < Rn, j = 1,...,d .
R j n
{ ∈ | | }
ChooseR > Rsothatf isholomorphiconaneighborhoodofD . Let
1 R1
F be holomorphic in a neighborhood of ∆ Cdn such that F S = f
R1 ⊂ ◦
on D . That such an F exists follows from Theorem 8.2 in [6]; see
R1
Remark 3.3 below. Define
β := Rn, β := Rn, and α := ρn
1 1
Thus α β < β . Let
1
≤
F(ζ) := F ζI
I
I
X
be the Taylor series of F about 0 Cdn. By the Cauchy estimates on
∈
∆ , for each multiindex I we have
R1
I
|FI| ≤ ||F||∆R1β1−| |.
10 L. BOS ANDN. LEVENBERG*
Given a positive integer s, we let
E (ζ) := F ζI
s I
I s
|X|≤
be the Taylor polynomial of degree at most s of F at 0 Cdn. Then
∈
for m = sn, let
G (z) := E S(z) = F Q (z)i1 Q (z)idn
m s ◦ I 1 ··· dn
I s
|X|≤
where I = (i ,...,i ). It follows that G Poly(mP) because Q
1 dn m ∈ j ∈
Poly(nP) and P is a lower set. Since S(D ) ∆ , we have
ρ ρ
⊂
α
f G F E F ( )I .
|| − m||Dρ ≤ || − s||∆ρ ≤ || ||∆R1 β1 | |
I >s
|X|
To obtain the desired estimate, note first that
α ∞ α d +k 1
( )I = ( )k n − .
| |
β β k
I >s 1 k=s+1 1 (cid:18) (cid:19)
|X| X
Choose β with β < β < β and C > 0 so that
2 2 1
d +k 1
C(β /β )k n − , k = 1,2,3,...
1 2
≥ k
(cid:18) (cid:19)
Then
1
f G F C(α/β )s+1
|| − m||Dρ ≤ || ||∆R1 · 2 · 1 α/β2
−
B(α/β)s+1 B(α/β)sn
≤ ≤
C F
where B = || ||∆R1.
1 β/β2
− (cid:3)
Remark 3.3. A version of this lemma was proved in [8] using the
Oka extension theorem: instead of the mapping S : Cd Cdn via
S(z) := (Q (z),...,Q (z)), one considers S˜ : Cd Cd+dn v→ia S˜(z) :=
1 dn →
(z,Q (z),...,Q (z)); the Oka result provides the existence of F˜ holo-
1 dn
morphicona neighborhoodofS˜(D ) withF˜ S˜ = f ona neighborhood
R
◦
of D (cf., section 3.3 of [6]). We need to avoid using this Oka map
R
S˜ as not all powers of the coordinates of z Cd may be included in
∈
our Poly(nP) spaces. Here, to apply the result in [6], we need S to be
one-to-one on D . This follows since n k implies Σ nP so that
R
≥ ⊂