Table Of ContentDEGENERATING BEHAVIOR OF GREEN’S FUNCTION
F. PEHERSTORFER1
0
1
0 Abstract. Let the unions of real intervals I = ∪lj=1[a2j−1,a2j], a1 <
2 ... < a2l, and In = ∪mk=1[Bk,n,Ck,n] be such that ∩∞k=1[Bk,n,Ck,n] =
n {ck} for k = 1,...,m and dist(E,In) ≥ const > 0. We show how to
a express asymptotically the Green’s function φ(z,∞,E∪In) of E∪In
J at z = ∞ in terms of the Green’s function φ(z,∞,E) and φ(z,ck,E).
4 The formula yields immediately asymptotics for φn(z,∞,E∪In) with
respect to n which are important in many problems of approximation
] theory. Anotherconsequenceisanasymptoticrepresentationofcap(E∪
V
In) in terms of cap(E) and φ(z,ck,E) and of the harmonic measure
C ω(∞,Ej,E∪In).
.
h
t
a l
Let E = [a ,a ], < a < a < ... < a < , be a union of l
m j=1 2j−1 2j −∞ 1 2 2l ∞
disjoint intervals, put H(x) = 2l (x a ) and let φ(z,z ,E) be a so-called
[ S j=1 − j 0
complex Green’s function, that is, a mapping which maps C¯ E onto the
1 exterior of the unit circle, whQich has a simple pole at z = z \ C¯ E and
v 0
∈ \
5 satisfies φ(z,z ) 1 for z x E; or in other words log φ is the Green’s
0
| | → → ∈ | |
8 function. It is known, see e.g. [7], that
4
0 z dξ
. φ(z, ,E) := φ(z, ) = exp r (ξ)
01 ∞ ∞ Za2l ∞ H(ξ)!
0 where r (ξ) = ξl−1+... is the unique polynomial sucph that
1 ∞
: a2j+1 dξ
v (1) r (ξ) = 0 for j = 0,...,l 1
∞
Xi Za2j H(ξ) −
r and that for x R E p
a ∈ \
z r (ξ) dξ
(2) φ(z,x ,E) := φ(z,x ) = exp x0
0 0
Za2l ξ−x0 H(ξ)!
where rx0 ∈ Pl−1 is such that p
(3) r (x ) = H(x )
x0 0 − 0
and p
a2j+1 r (ξ) dξ
(4) x0 = 0 for j = 1,...,l 1.
Z−a2j ξ−x0 H(ξ) −
1The manuscript was prepared bpy the author in the two months preceding his passing
away in November 2009. The manuscript remained unsubmitted and is not published
elsewhere (submitted by P. Yuditskii and I. Moale).
1
2 F.PEHERSTORFER
The so-called capacity of E is given by
z
(5) cap(E) = lim
z→∞|φ(z, ,E)|
∞
By ω(z,K,E) we denote the harmonic measure of the set K E with
⊂
respect to E. Recall that
a2j dx
(6) ω( ,[a ,a ],E) = r (x)
2j−1 2j ∞
∞ Za2j−1| | |H(x)|
Theorem 0.1. (Asymptotic representation of the pGreen’s function in the
degenerate case) Let l,m N, E = l [a ,a ] and for a strictly
∈ j=1 2j−1 2j
monotone increasing subsequence (n) of N let I = m [B ,C ] with
∞ S n k=1 k,n k,n
dist(E,I ) const > 0 and [B ,C ] = c and let ω =
n ≥ n=1 k,n k,n { k} S k,n
ω( ,[B ,C ],E I ) with ω 0 for k = 1,...,m. Then uniformly
k,n k,n n k,n
∞ ∪ T n−→→∞
on compact subsets of C (E c ,...,c )
1 m
\ ∪{ }
φ(z, ,E)
(7) φ(z, ,E I ) = ∞ (1+O( max ω2 ))
∞ ∪ n m (φ(z,ck,E))ωk,n 1≤k≤m k,n
k=1
Q
Proof. For abbreviation we put
E˜ = E I , H (x) = H(x)H˜ (x)
n ∪ n E˜,n n
where
2l m
H(x) = (x a ), H˜ (x) = (x B )(x C )
j n k,n k,n
− − −
j=1 k=1
Y Y
By [7] we know that there exists a monic polynomial r (x) of degree l+
E˜,n
m 1 such that
−
nr (x)
φ(z, ,E˜ ) = exp E˜,n dx
n
∞
H (x)
Z E˜,n
q
Now let us represent r in the form
E˜,n
m
r (x) = s˜ (x) (x c )+t (x)
E˜,n n − k n
k=1
Y
m m
= r (x) (x c )+s (x) (x c )+t (x)
E k n k n
− −
k=1 k=1
Y Y
where r P is the monic polynomial associated with the Green’s func-
E l−1
tion φ(z, ∈,E), s (x) = s˜ (x) r (x) P and t P . Putting
n n E l−2 n m−1
∞ − ∈ ∈
L = max C B
n k,n k,n
1≤k≤n| − |
DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 3
we have on compact subsets of C c ,...,c
1 m
\{ }
1 1
= +O(L )
m n
HE˜,n(x) (x ck) H(x)
−
k=1
q
Q p
and thus
rE˜,n(x) rE(x) sn(x)
dx = dx+ dx
H (x) H(x) H(x)
E˜,n
(8)
q p t (x) p dx
n
+ +O(L )
m n
H(x)
(x c )
k
−
k=1
p
Q
Partial fraction expansion gives, recall t P ,
n m−1
∈
m
t (x) λ
n k,n
(9) =
m x c
(x−ck) Xk=1 − k
k=1
Q
To determine the λ ’s we integrate both sides in (8) counterclockwise
k,n
around a circle with center c and fixed small radius ε. Taking into con-
k
sideration the facts, that the limiting values H (x) := lim 1
± E˜,n z→x± √H(z)
q x∈E˜,n
from the upper- and lower half plane satisfy + H (x) = H (x)
E˜,n − E˜,n
and that q q
+ 1 = sgn rE˜,n(x) for x E˜,n,
∈
H (x) i H (x)
E˜,n | E˜,n |
q q
weobtainfortheLHSbyshrinkingthecircletotheinterval[B ,C ]that
k,n k,n
(10) rE˜,n(ξ) dξ = 2 Bk,n |rE˜,n(x)| dx = 2iπω
k,n
i
I HE˜,n(ξ) ZCk,n |HE˜,n(x)|
q q
On the other hand the RHS from (8) becomes, note that all the terms at
the RHS are analytic on the circle up to 1/(x c ),
k
−
rE˜,n(x) 1 dx
dx = λ +O(L )
k,n n
H (x) x ck H(x)
I E˜,n I −
q λ 2πi p
k,n
= +O(L )
n
H(c )
k
that is, p
(11) λ = ω H(c )+O(L )
k,n k,n k n
p
4 F.PEHERSTORFER
Summarizing, on compact subsets of C c ,...,c we have
1 m
\{ }
rE˜,n(x) rE(x) dx
dx = dx+s (x)
n
H (x) H(x) H(x)
E˜,n
(12)
q p p
H(c ) dx
k
+ ω +O(L )
k,n n
px−ck H(x)
X
Finally let us determine sn(x) asymptoticallpy. We have by (8), (9) and
(11) that
Cν,n rE˜,n(x) a2ν+1 rE˜,n(x)
iπω = dx = dx
ν,n
−
ZBν,n HE˜,n(x) Za2ν HE˜,n(x)
a2ν+1 qm ω H(c ) dxq
k,n k
= s (x)+ +O(L )
n n
Za2ν k=1 xp−ck ! H(x)
X
Observing that p
a2ν+1 r (ξ) dξ a2ν+1 r (x) r (c ) dx
ck = ck − ck k
Z−a2ν x−ck H(ξ) Za2ν x−ck H(x)
a2ν+1 1 dξ
p H(c ) p
k
− Z−a2ν ξ−ck H(ξ)
p
and that by (3) p
a2ν+1 1 dξ
(13) = iπδ
kν
Z−a2ν ξ−ck H(ξ)
it follows that for ν = 1,...,l 1
p
−
a2ν+1 m dξ
(14) s (ξ) ω q (ξ) dξ = O(L )
Z−a2ν n −k=1 k,n ck H(ξ)! n
X
Recall that sn(ξ) − mk=1ωk,nqck(ξ) ∈ Pl−p2. Since a polynomial q ∈ Pl−2
satisfying
P
a2k+1 dx
(15) q(x) = 0 for k = 1,...,l 1
Za2k H(x) −
is identically the zero polynomial it follows with the help of Cramer’s rule
p
that
m
s (x) ω q (x) = O(L )
n − k,n ck,n n
k=1
X
which gives by (12) the assertion, if we are able to show that
(16) L = O( max ω2 )
n k,n
1≤k≤n
If c (a ,a ) let us consider
k 1 2l
∈
= [a ,C ] [a ,a ], respectively, = [a ,c ] [a ,a ]
k,n 1 k,n 2k+1 2l k,∞ 1 k 2k+1 2l
E ∪ E ∪
DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 5
andletA andA bethecriticalpointoflnφ(x, , )andlnφ(x, , ),
k,n k,∞ k,n k,∞
∞ E ∞ E
respectively. Then it follows that
A A (c ,a ).
k,n k,∞ k 2k+1
→ ∈
Thus we obtain for n n
0
≥
ω( ,[B ,C ],E˜ ) ω( ,[B ,C ], )
k,n k,n n k,n k,n k,n
∞ ≥ ∞ E
Ck,n x Ak,n
= − dx
(x a )(x a )(x C )(x a )
ZBk,n − 1 − 2l − k,n − 2k+1
pCk,n dx
const = const C B ,
k,n k,n
≥ C x −
ZBk,n k,n−
p
where const > 0 is indepenpdent of n, and this proves (16).
If c R [a ,a ] then we consider := [a ,C ] E˜ , respectively,
k 1 2l k,n 1 k,n n
[B ,a∈] an\d the assertion follows eveEn simpler as abov⊃e. (cid:3)
k,n 2l
Remark 0.2. By conformal transplantation and the invariance property of
the harmonic measure it follows that the Theorem holds for much more
general sets as circular slits in the plane, etc. .
Corollary 0.3. As above let ω = ω( ,[B ,C ],E m [B ,C ]).
k,n ∞ k,n k,n ∪ k=1 k,n k,n
Then the following statements hold:
S
m
a) cap(E∪ mk=1[Bk,n,Ck,n])= (capE) |φ(∞,ck,E)|ωk,n(capE)(1+
k=1
O(maxωS2 )) Q
k,n
k
m
m
b) ω( ,E ,E [B ,C ])= ω( ,E ,E) ω ω(c ,E ,E)+
∞ j ∪ k=1 k,n k,n ∞ j − k,n k j
k=1
O(maxω2 )S P
k,n
k
Proof. a) Follows immediately by the fact that for a compact set K
lim z = cap (K)
z→∞|φ(z,∞,K)|
b) follows by taking the log in (7) and using (6). (cid:3)
Notation 0.4. a)Letp (x) = a xn+...;bylc(p ) = a wedenotetheleading
n n n n
coefficient of p .
n
b) x ,...,x E with x < x < ... < x are called alternation points of
1 m 1 2 m
∈
f C[a,b] if f(x ) = ( 1)i f for i = 1,...,m. By ♯A(f;K) we denote
i ∞,E
∈ ± − || ||
the number of alternation points of f on K.
Theorem0.5. Letl,m N,E = l [a ,a ]andI = m [B ,C ]
∈ j=1 2j−1 2j n k=1 k,n k,n
with dist(E,I ) const > 0 for a strictly monotone increasing subsequence
n
of N and ∞ [B≥ ,C ] = c .S S
Suppose tkh=a1t thke,nre iks,na seq{uekn}ce of polynomials (p ) such that p P
n n n
has n reaTl zeros, min p (y ) : p′ (y ) = 0 1, p 1∈and
{| n i | n i } ≥ || n||E∪In ≤
6 F.PEHERSTORFER
♯A(p ;I ) = ν + 1 for k = 1,...,m and n N. Then on compact sub-
n k,n k
sets of C (E c ,...,c ) p has an asymptot∈ic representation of the form
1 m n
\ ∪{ }
1
(17) 2p (x) = ψ (x)+ (1+o(1))
n n
ψ (x)
n
(cid:18) (cid:19)
where
φn(x, ,E)
ψ (x) = ∞
n m
φνk(x,ck,E)
k=1
Q
Furthermore the norm of the monic polynomial pˆ (x) = p (x)/lc(p ) is
n n n
given by
m
(18) pˆ = 1/lc(p ) = 2(cap E)n φνk(c ; ,E)(1+o(1))
n E n j
|| || ∞
j=1
Y
where 0 < r < 1, and concerning the alternation points on E we have
ν
l−1
1
♯A(p ;E ) = nω( ,E ,E)+ ω(c ,E ,E)+O( ) for ν = 1,...,l
n ν ν k ν
∞ n
k=1
X
Proof. By the properties of p it follows that p−1([ 1,1]) = E I which
n n n
− ∪
implies, see e.g. [3, 4] that
1
φ(z, ,E I ) = p (z) p2(z) 1 n
n n n
∞ ∪ − −
(cid:16) 1 pz p′ (x)(cid:17)
= exp n dx
n Za2l p2n(x)−1 !
Thus p
1
2p (z) = φn(z, ,E I )+
n ∞ ∪ n φn(z, ,E I )
n
∞ ∪
and the representation (17) follows by Theorem 0.1.
Concerning relation (18) let us observe that, with the help of (17) and
φ(z, ,E) > 1 on C E,
| ∞ | \
n
p (z) 1 φ(z, ,E) 1
n
lc(p ) = lim = lim ∞ (1+o(1))
n z→∞ zn z→∞(cid:18)2 (cid:18) z (cid:19) mk=1φνk(z,ck,E)(cid:19)
(cid:3)
Q
BytheAlternationTheoremitfollowsimmediatelythatmanyL -minimal
∞
polynomials without or with restrictions as minimal polynomials on several
intervals satisfy the assumptions of Theorem 0.5 with ν = 1, k = 1,...,m,
k
and thus have a representation of the form (17). One of the remaining chal-
lenging problems is to describe when c ’s appear between two consecutive
k
E ’s. For a solution of this problem in the case of minimal polynomials on
ν
several intervals, see [2].
DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 7
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[3] F. Peherstorfer, R. Steinbauer, Orthogonal and Lq- extremal polynomials on inverse
images of polynomial mappings, J. Comput. Appl. Math. 127 (2001), 297-315.
[4] T. Ransford, Potential theory in the complex plane, Cambr. Univ. Press., 1995.
[5] V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math. (Scan-
dinavian) 187 (2001), 139-160.
[6] V. Totik, Chebyshev constants and the inheritance problem, to appear in J. Approx.
Theory.
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