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Spectral transforms of measures and orthogonal polynomials on regions
J. Math. Anal. Appl. 407 (2013) 290–304
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Spectral transforms of measures and orthogonal polynomials on regions Erwin Miña-Díaz a,∗ , Brian Simanek b a
University of Mississippi, Department of Mathematics, Hume Hall 305, P.O. Box 1848, University, MS 38677-1848, USA
b
Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, Nashville, TN 37240, USA
article
abstract
info
Article history: Received 25 January 2013 Available online 21 May 2013 Submitted by Paul Nevai
We discuss Szegő-type strong asymptotics for polynomials orthogonal over a region bounded by an analytic curve, and find necessary conditions for such asymptotics to hold on an interior neighborhood of the curve. We also prove that this property is not destroyed when the measure is perturbed by multiplication by a rational function and the addition of point masses. © 2013 Elsevier Inc. All rights reserved.
Keywords: Orthogonal polynomials Strong asymptotics Geronimus transform Christoffel transform Uvarov transform
1. Introduction and main results Let µ be a finite Borel measure with compact and infinite support in the complex plane. Given such a measure, one can perform Gram–Schmidt orthogonalization on the sequence {1, z , z 2 , z 3 , . . .} in the space L2 (C, µ) to arrive at a sequence of polynomials {φn (z ; µ)}n≥0 satisfying
φn (z ; µ)φm (z ; µ)dµ(z ) = δmn , C
and normalized so that φn has positive leading coefficient. The leading coefficient of φn (z ; µ) will be denoted by κn (µ), and the monic orthogonal polynomial by
Φn (z ; µ) := κn−1 (µ)φn (z ; µ),
n ≥ 0.
This paper concerns the behavior that Φn (z ; µ) displays as n → ∞ when the measure µ is supported on the closure of an analytic domain. Some of the basic geometric properties of analytic domains that are essential in our discussion are briefly summarized in what follows. Hereafter, G will always denote a bounded region in the complex plane whose boundary is an analytic Jordan curve. Let D := {w : |w| < 1} be the unit disk. Since G is simply connected, there is a unique conformal bijection ψ that maps C \ D onto C \ G with a Laurent expansion at ∞ of the form
ψ(w) = dw + d0 + d−1 w −1 + d−2 w −2 + · · · ,
∗
d > 0.
Corresponding author. E-mail addresses: [email protected] (E. Miña-Díaz), [email protected] (B. Simanek).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.05.035
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Let ρ ∈ [0, 1] denote the smallest number with the property that ψ admits an analytic and univalent continuation to |w| > ρ . It is well-known that since G has an analytic boundary, ρ < 1. The set
Ωρ := {z = ψ(w) : |w| > ρ} is a simply-connected neighborhood of ∞ containing all of ∂ G. We can then define
ϕ : Ωρ → {w : |w| > ρ} to be the inverse of ψ , so that for every r > ρ , the level set
ℓr = {z : |ϕ(z )| = r } is an analytic Jordan curve. We shall let Ωr and Gr denote, respectively, the exterior and interior domains of the curve ℓr . Notice that with this notation we have G1 = G, ℓ1 = ∂ G, and Ω1 = C \ G. By a measure µ supported on G we shall always mean a measure whose support is an infinite subset of G and that satisfies
µ(Ωs ) > 0,
s ∈ [ρ, 1).
(A1)
This last condition is really more of a convenience than a necessity, since any measure µ on G can be made to satisfy (A1) by an appropriate rescaling. However, if we are to expect the geometry of G to have some influence on the behavior of Φn (z ; µ), the condition (A1) is the least we can impose on µ. There are many ways to study the asymptotics of the polynomial sequence {Φn }n≥0 . The asymptotics we will be interested in are called strong or Szegő asymptotics, and are devoted to finding conditions on µ guaranteeing the existence of some analytic function Dµ (z ) such that lim
n→∞
Φn (z ; µ) = Dµ (z ) dn ϕ(z )n
(2)
on some region of the complex plane. Here d is the first coefficient of the Laurent expansion of ψ in (1), which coincides with the so-called logarithmic capacity of ℓ1 = ∂ G. If ℓ1 is the unit circle and supp(µ) ⊂ ℓ1 , then the well-known Szegő–Geronimus theorem [8, Chapter I, II] states that (2) holds for all |z | > 1, if and only if, the Radon–Nikodym derivative µ′ (z ) of µ with respect to the arc-length measure |dz | satisfies the Szegő condition
ℓ1
log µ′ (z )|dz | > −∞.
In fact, a similar theorem was obtained by Geronimus [6, Theorem 7.1] for a measure µ supported on an arbitrary analytic curve. Szegő-type asymptotic formulas have also been established for planar type measures such as those given by Lipschitz continuous weights [18] and perturbations of product measures [14] on analytic regions, as well as for area measure on regions with corners [17]. Generally speaking, if the measure µ is sufficiently irregular, Szegő asymptotics can only be derived for z ∈ Ω1 . However, for nicer measures it is possible to have (2) holding across the boundary of G. For instance, Carleman proved in [1] that if µ = dA is the area measure on G, then (2) holds locally uniformly for z ∈ Ωρ with Dµ (z ) = ϕ ′ (z ) (for an extension of this theorem, see [2]). Korovkin [11] and Smirnov [16] generalized this result to measures µ whose restriction to Ωs (for some ρ ≤ s < 1) is of the form |g (z )|2 dA(z ), where g (z ) is such that for some integer m, g (z )ϕ(z )m is analytic and nonvanishing on Ωs . In the present work, we derive several relations that must be satisfied in order to have Szegő asymptotics holding on some interior neighborhood of ℓ1 = ∂ G. With this knowledge in hand, we will then be able to explore certain perturbations of the orthogonality measure that do not destroy the property of having Szegő asymptotics across ∂ G. Here, the so-called functions of the second kind Qn (z ; µ) :=
Φn (t ; µ) dµ(t ), z−t
qn (z ; µ) := κn (µ)Qn (z ; µ),
z ̸∈ supp(µ),
will play an important role as well. The question of whether (2) holds on a domain larger than Ω1 is completely settled for measures µ supported on the unit circle. In such a case, it is a well-known result that for (2) to hold on the exterior of some circle of radius less than 1, it is necessary1 and sufficient2 that µ be of the form |w(z )|2 |dz |, with w analytic and nonvanishing on the unit circle. Theorem 1.1 recovers, in particular, the ‘‘necessary’’ half of this result. We will be interested in measures µ that are supported on some G, and that admit the Szegő asymptotics (2) in the region Ωr , for some r ∈ [ρ, 1). For such measures, we will also impose a very natural additional condition, namely, that for
1 See (b) ⇒ (c) in the corollary of [12]. 2 The sufficiency is even true for a general analytic curve [19, Theorem 16.5].
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every s ∈ (r , 1),
lim sup
|ϕ(z )|2n |Dµ (z )|2 dµ(z ) > 0. |ϕ(z )|2n dµ(z ) Ωs
Ωs
n→∞
(A2)
It will follow from Lemma 3.2 below that the above lim sup, whether positive or not, has the same value for every s ∈ (r , 1). It is entirely possible for a measure µ to satisfy (2) and not (A2). For instance, take the measure µ supported on the closed N unit disk D and given by µ = dA1/2 + j=1 δzj , where dA1/2 is the area measure restricted to |z | < 1/2, and the zj ’s are arbitrary points located on the unit circle |z | = 1. It follows from Theorem 1.2 below that
Φn (z ; µ)
lim
zn
n→∞
=
N z − zj , z − 4z1 j=1 j
|z | > 1/4.
This can happen because the measure µ is too thin near the boundary of the domain, namely, a sum of point masses. As a matter of fact, at least in the following weak sense this will always be the case. Helly’s selection theorem tells us that through any subsequence of the natural numbers, we may take a further subsequence through which the measures3
|ϕ|2n dµ|Ωρ |ϕ|2n dµ|Ωρ converge weakly to a measure γ that, by condition (A1), has for support a subset of ∂ G. Therefore, if the condition (A2) is not satisfied, every such weak limit point γ must satisfy
∂G
|Dµ (z )|2 dγ (z ) = 0.
Since Dµ is an analytic function in some neighborhood of ∂ G, this last equality forces γ to be a sum of masses located at the finitely many zeros that Dµ necessarily has on ∂ G. For simplicity, hereafter we make the assumption that the capacity of ∂ G is 1. There is no loss of generality in doing so since this can always be attained by rescaling the domain G (see the third remark following Theorem 1.2 in [14]). We can now state our result giving necessary conditions in order to have Szegő asymptotics holding on some interior neighborhood of ℓ1 = ∂ G. Theorem 1.1. Let µ be a measure supported on G such that lim
n→∞
Φn (z ; µ) = Dµ (z ) ϕ(z )n
(3)
uniformly on closed subsets of Ωr , for some r ∈ [ρ, 1), and suppose also that µ satisfies the condition (A2). Then, the following statements hold true: (a) Dµ has no zeros on Ω 1 ; (b) for every s ∈ (r , 1),
κ −2 (µ) = lim lim n 2n n→∞ |ϕ| dµ n→∞ Ωρ
|ϕ|2n |Dµ |2 dµ 2π = ; 2n −2 ′ |ϕ| dµ Ωs ℓ1 |Dµ | |ϕ | |dz |
Ωs
(4)
(c)
|ϕ|2n dµ|Ωρ |Dµ |−2 |ϕ ′ | |dz | = ; 2n −2 ′ |ϕ| dµ|Ωρ ℓ1 |Dµ | |ϕ | |dz |
w-lim n→∞
(5)
(d) w-lim |φn |2 dµ = n→∞
|ϕ ′ | |dz | ; 2π
(6)
(e) lim κn2 (µ)ϕ(z )n+1 Qn (z ; µ) =
n→∞
ϕ ′ (z ) Dµ (z )
uniformly on closed subsets of Ω1 . 3 As usual, µ| stands for the restriction of the measure µ to the measurable set A. A
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Since the measures in (5) ‘‘suppress’’ the properties of µ away from ∂ G, Theorem 1.1 tells us that in our setting, the measure µ must resemble a positive analytic weight near the boundary of G. Observe also that (4) is telling us that, indeed, the full limit in (A2) exists. We now turn our attention to perturbations of the measure µ of the form K
|z − zk |2
k=1
dµ( ˆ z) =
L
dµ(z ) +
J
mj δaj ,
mj > 0, 1 ≤ j ≤ J ,
(7)
j =1
2
|z − bl |
l=1
with the aj ’s pairwise distinct, each pole bl ∈ Ω1 , and no zero zk or mass point aj lying on ∂ G. We shall see in our next theorem that this new measure µ ˆ retains the property of having Szegő asymptotics across ∂ G. But first, we need a new piece of notation. Corresponding to a domain G and its associated number ρ (the univalency radius of ψ ), we define the quantity ρ(r , a) for every r ∈ (ρ, 1) and a ∈ C by
r , ρ(r , a) := |ϕ(a)|−1 , |ϕ(a)|,
a ̸∈ Ωr ∩ G1/r , 1 < |ϕ(a)| < 1/r , r < |ϕ(a)| ≤ 1,
(8)
while for a set E ⊂ C, we put
ρ(r , E ) := sup{ρ(r , a) : a ∈ E }. Let us also define F (z , t ) :=
ϕ(z ) − ϕ(t ) ϕ(t )ϕ(z ) − 1
.
Theorem 1.2. Let µ be a measure satisfying the conditions of Theorem 1.1. Then, for a measure µ ˆ of the form (7), we have lim
n→∞
Φn (z ; µ) ˆ = Dµˆ (z ) ϕ(z )n
(9)
uniformly on closed subsets of Ωρ(r ,E ) , where E = {z1 , . . . , zK , a1 , . . . , aJ } and
Dµˆ (z ) =
L
Dµ (z )
(z − bl )
l=1
L ϕ(z )L−K
ϕ(bl )F (z , bl )
l =1
zk ∈Ω1
ϕ(zk )F (z , zk )
K
(z − zk )
ϕ(aj )F (z , aj ).
aj ∈Ω1
k=1
Moreover, 2π
|ϕ(aj )|2 |ϕ(zk )|2 aj ∈Ω1 ˆ κn−2 (µ) zk ∈Ω1 lim · L = . −2 ′ n→∞ |ϕ|2n dµ Ωρ ℓ1 |Dµ | |ϕ | |dz | 2 |ϕ(bl )|
(10)
l=1
The measure µ ˆ in (7) results from the repeated application of three simpler types of spectral transforms on a measure µ, namely, the addition of a pole |z − a|−2 µ, of a zero |z − a|2 µ, and of a point mass µ + mδa . In Section 2, we will consider each of these transformations individually, stating in Propositions 2.2, 2.4 and 2.5, respectively, the simplified version of Theorem 1.2 that corresponds to each transform. The reader will have no difficulty in seeing that Theorem 1.2 follows from these propositions by a trivial mathematical induction argument. The proof of all the results, as well as some auxiliary lemmas are given in Section 3. Our exposition will include some examples that are of particular interest. 2. Spectral transforms 2.1. The Geronimus transform The Geronimus spectral transform of a measure µ, corresponding to a point a ̸∈ supp(µ), is the measure µa defined by d µa ( z ) =
1
|z − a|2
dµ(z ).
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Such transformations have been studied previously in the context of general orthogonal polynomials in [9,10]. In particular, there is a concise formula describing the polynomial Φn+1 (z ; µa ) in terms of the orthogonal polynomials and the functions of the second kind for µ. More precisely, it was proven in [9, Proposition 3] (see also [10, Proposition 3]) that for all n ≥ 0,
Qn (a; µ) (z − a)
n−1
qj (a; µ)φj (z ; µ) + 1
j =0
Φn+1 (z ; µa ) = (z − a)Φn (z ; µ) +
n −1
∥µa ∥ −
.
(12)
|qj (a; µ)|
2
j=0
Eq. (12) will be the key to our calculations, though it will require the following reformulation before we can effectively apply it. Proposition 2.1. Let µ be a measure supported on G such that lim |Φn (z ; µ)|1/n = |ϕ(z )|
n→∞
uniformly on closed subsets of Ω1 . Then, for every point a ∈ Ω1 and n ≥ 0, the equality
(z − a)Qn (a; µ) Φn+1 (z ; µa ) = (z − a)Φn (z ; µ) −
∞
qj (a; µ)φj (z ; µ)
j =n
∞
(13)
|qj (a; µ)|2
j =n
holds true locally uniformly for z ∈ G|ϕ(a)| . We remark that the sum on the right-hand side of (13) does not converge for z ̸∈ G|ϕ(a)| . Combining Proposition 2.1 and Theorem 1.1 we will be able to establish the following proposition. Proposition 2.2. Let µ be a measure satisfying the conditions of Theorem 1.1. If µa is the measure given by (11), then for every a ∈ Ω1 , lim
n→∞
Φn (z ; µa ) Dµ (z )(ϕ(a)ϕ(z ) − 1)(z − a) = ϕ(z )n ϕ(a)ϕ(z )(ϕ(z ) − ϕ(a))
(14)
locally uniformly on Ωr . We have already mentioned in the introduction several classes of measures µ known to satisfy the conditions of Theorem 1.1. The simplest example would be the arc-length measure µ = |dz | on the unit circle, for which the monic orthogonal polynomials are just the monomials Φn (z ; µ) = z n , n ≥ 0, and the leading coefficients κn (µ) = (2π )−1/2 are constant. Repeated applications of the Geronimus transform to this measure give rise to the so-called Bernstein–Szegő polynomials. These are the polynomials orthogonal over the unit circle with respect to a measure of the form dµ( ˆ z ) = |(z − b1 ) · · · (z − bL )|−2 |dz |,
|bl | > 1,
1 ≤ l ≤ L.
In this case, it is well-known and easy to verify that [19, Theorem 11.2]
Φn+L (z ; µ) ˆ = z n (z − b∗1 ) · · · (z − b∗L ),
n ≥ 0,
where, as it is customary, z ∗ := 1/z denotes the reflection of z about the unit circle. More generally, if µ is a measure supported on D having the monomials z n for monic orthogonal polynomials, then it follows straightforwardly from Proposition 2.1 that Φn+1 (z ; µa ) is divisible by z n for all n ≥ 0. From this observation and Proposition 2.2, we can use a clear induction argument to arrive at the following corollary. Corollary 2.3. Let µ be a measure supported on D such that Φn (z ; µ) = z n , n ≥ 0. For every measure µ ˆ of the form dµ( ˆ z ) = |(z − b1 ) · · · (z − bL )|−2 dµ(z ),
|bl | > 1,
1 ≤ l ≤ L,
we have
Φn+L (z ; µ) ˆ = z n (z − b∗1 ) · · · (z − b∗L ) + εn (z ) ,
n ≥ 0,
with εn (z ) a polynomial such that limn→∞ εn (z ) = 0 uniformly on compact subsets of the complex plane.
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A well-known class of measures having the monomials z n for monic orthogonal polynomials is that of the radially symmetric measures, that is, measures µ that are the image by the map (r , θ ) → reiθ of a product measure σ × dθ on [0, 1]× [0, 2π ]. For such a measure µ, we have
1
z z dµ(z ) = n m
r
n+m
dσ (r )
ei(n−m)θ dθ ,
0
0
D
2π
and after easy computations one finds
Φn (z ; µ) = z n ,
1
κn−2 (µ) = 2π
r 2n dσ (r ),
Qn (a; µ) =
κn−2 (µ) an + 1
0
∞
qj (a; µ)φj (z ; µ) =
j=n
(z /a)n , a−z
∞
2π
|qj (a; µ)|2 =
2n
| a|
j =n
0
1
r 2n 2
|a| − r 2
, dσ (r ).
Inserting these expressions in (13) one arrives at the explicit representation
Φn+1 (z ; µa ) = z n
1
z−
+
a
r 2n (1−r 2 ) d 0 |a|2 −r 2
1
1
r 2n d 0 |a|2 −r 2
1
a
σ
σ
,
n ≥ 0.
(16)
Notice that if µ is supported on D, or equivalently, if 1 ∈ supp(σ ), the second fraction in the right-hand side of (16) converges to 0 as n → ∞. We can also prove (16) in a direct way that will allow us to extend this explicit representation to polynomials Φn (z ; µ) ˆ of sufficiently large degree corresponding to a measure µ ˆ constructed in the form (15) out of a radially symmetric µ = σ × dθ supported on D. We know from the last corollary that for all n ≥ 0, the (n + L)th monic orthogonal polynomial has the form
Φn+L (z ; µ) ˆ = z n (αn,0 + αn,1 z + · · · + αn,L z L ),
αn,L = 1.
(17)
For every pair of integers n, m, we have
1
z n z m dµ( ˆ z) =
i
D
1
d σ (r )
|ζ |=r
0
ζ n−1 ζ m dζ . L 2 |ζ − bj |
(18)
j=1
Using that ζ = r /ζ on |ζ | = r, followed by the change of variables ζ = r 2 z and a deformation of the contour of integration from |z | = 1/r to |z | = 1, we get 2
|ζ |=r
ζ n−1 ζ m dζ = L 2 |ζ − bj |
ζ n+L−m−1 r 2m
|ζ |=r
j =1
L
(
r2
dζ
− bj ζ )(ζ − bj )
j =1
z n+L−m−1 r 2n
= |z |=1
L
dz .
(1 − bj z )(r 2 z − bj )
j =1
Combining this last equality with (18) and (17), we arrive at the following identity, valid for every integer m:
Φn+L (z ; µ) ˆ z m dµ( ˆ z) =
1
D
i
z n+L−m−1
L
αn,l z l gn,l (z ) dz
l =0
|z |=1
L
,
(19)
(1 − bj z )
j =1
where gn,l (z ) :=
1
0
r 2n+2l dσ (r ) L
(
r 2z
,
0 ≤ l ≤ L.
− bj )
j =1
Each function gn,l is analytic on the closed unit disk, so that for every 0 ≤ m ≤ n + L − 1, the integral in (19) will reduce to zero L L if constants αn,l ’s are found such that l=0 αn,l z l gn,l (z ) is divisible by j=1 (z − b∗j ). Thus, we can prescribe the conditions L−1 l =0
αn,l (b∗j )l gn,l (b∗j ) + (b∗j )L gn,L (b∗j ) = 0,
1 ≤ j ≤ L,
(20)
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where in case bj has multiplicity βj > 1, βj − 1 of these equations must be replaced by L−1
αn,l [z l gn,l (z )](v) + [z L gn,L (z )](v) = 0,
z = b∗j , 1 ≤ v ≤ βj − 1.
l =0
Observe that this system remains constant (independent of n) when σ = δ1 , that is, when µ is the arc-length measure on the unit circle. 1 If we divide each equation in (20) by 0 r 2n dσ (r ) and take the limit as n → ∞, we obtain a limiting system (precisely, the one corresponding to σ = δ1 ) with a unique solution given by the coefficients of the polynomial j=1 (z − b∗j ). Hence, for all n sufficiently large, the system (20) also has a unique solution that can be explicitly computed using Cramer’s formula.
L
2.2. The Christoffel and Uvarov transforms For a point a ∈ C, the Christoffel transform µca of the measure µ is defined by dµca (z ) = |z − a|2 dµ(z ).
(21)
Using the orthogonality relations, it is easy to verify that
Φn+1 (a; µ)Kn (z , a; µ) Φn+1 (z ; µ) − , Φn (z ; µ ) = z−a Kn (a, a; µ) 1
c a
n ≥ 0,
(22)
where Kn (z , t ; µ) :=
n
φj (t ; µ)φj (z ; µ).
j=0
The relation (22) appears in several publications such as [5,9]. We will use it to establish the following proposition. Proposition 2.4. Let µ be a measure satisfying the conditions of Theorem 1.1, and let µca be the measure given by (21). Then, with ρ(r , a) as defined by (8), we have
Dµ (z )ϕ(a)ϕ(z ) ϕ(z ) − ϕ(a) · , Φn (z ; µca ) z−a ϕ(a)ϕ(z ) − 1 lim = n→∞ ϕ(z )n Dµ (z )ϕ(z ) , z−a
a ∈ Ω 1, (23) a ∈ G,
uniformly on closed subsets of Ωρ(r ,a) . We now consider for a point a ∈ C and a number m > 0, the measure µua defined by
µua = µ + mδa , where δa is the point mass Dirac measure at a. This measure µua is often called the Uvarov transform of µ. Using the orthogonality relations, a rather simple calculation reveals that
Φn (z ; µua ) = Φn (z ; µ) −
mΦn (a; µ)Kn−1 (z , a; µ) 1 + mKn−1 (a, a; µ)
,
n ≥ 0.
(24)
This formula is due to Geronimus [7, Formula (29.12)], who obtained it not just for one but for any finite number of point masses. It will allow us to prove the following result. Proposition 2.5. Let µ be a measure supported on G and satisfying the conditions of Theorem 1.1. Let µ ˜ be a measure of the form
µ+
J
mj δaj ,
mj > 0 , 1 ≤ j ≤ J ,
j =1
with the aj ’s pairwise distinct and contained in Ω1 . Suppose that lim
n→∞
J Φn (z ; µ) ˜ ∗ = D ( z ) = D ( z ) (z − aj ) µ ˜ µ ˜ ϕ(z )n j =1
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297
uniformly on closed subsets of Ωr , with D∗µ˜ analytic and nonvanishing on Ω 1 . Then, lim
n→∞
2 ′ ˜ /Dµ | |ϕ | |dz | κn−2 (µ) ˜ ℓ1 |Dµ , = −2 ′ |ϕ|2n dµ Ωρ ℓ1 |Dµ | |ϕ | |dz |
(26)
and for the measure µ ˜ ua = µ ˜ + mδa , m > 0, a ̸∈ {a1 , . . . , aJ }, we have
ϕ(z ) − ϕ(a) , Φn (z ; µ ˜ ) ϕ(a)Dµ˜ (z ) · lim = ϕ( a)ϕ(z ) − 1 n→∞ ϕ(z )n Dµ˜ (z ), u a
a ∈ Ω 1,
(27)
a ∈ G,
uniformly on closed subsets of Ωρ(r ,a) . In conclusion, we see that if a ̸∈ ∂ G, the polynomials Φn (z ; µca ) and Φn (z ; µua ) also satisfy a strong asymptotic formula on a domain containing Ω 1 , namely, on Ωρ(r ,a) . It is easy to see that, in general, this is no longer true if a ∈ ∂ G. For instance, take G = D and let µ = |dz |/2π be the normalized arc-length measure on the unit circle ∂ D. For this measure, we have Φn (z ; µ) = φn (z ; µ) = z n , and we can compute from (22) and (24) very simple explicit formulas for the orthogonal polynomials corresponding to µca and µua . From these formulas, we find that if |a| = 1, then lim (n + 1)a−n−2 Φn (z ; µca ) = (z − a)−2 ,
|z | < 1,
lim (n + 1)a−n−1 Φn (z ; µua ) = (z − a)−1 ,
|z | < 1,
n→∞ n→∞
and so limn→∞ z −n Φn (z ; µca ) = limn→∞ z −n Φn (z ; µua ) = ∞ on 0 < |z | < 1. 3. Proofs The proofs will be given following the same order in which the corresponding results were presented. To abbreviate the notation, we will drop µ whenever there is no cause for confusion. Thus, we will write, for instance, Φn (z ) = Φn (z ; µ), qn (a) = qn (a; µ), and κn = κn (µ). Lemma 3.1. Let µ be a measure supported on G and let E ⊂ Ω1 be a set consisting of isolated points. Suppose that lim |Φn (z ; µ)|1/n = |ϕ(z )|
(28)
n→∞
locally uniformly on Ω1 \ E. Then, for every a ∈ Ω1 , 1 a−z
=
∞
qn (a; µ)φn (z ; µ)
(29)
n=0
uniformly on compact subsets of G|ϕ(a)| , and lim sup |κn (µ)qn (a; µ)|1/n = |ϕ(a)|−1 , n→∞
a ∈ Ω1 .
Proof. By orthogonality, we have
G
Φn (z ) − Φn (a) Φn (z )dµ(z ) = 0, a−z
or equivalently,
κn Φn (a)qn (a) =
G
|φn (z )|2 dµ(z ) , a−z
a ∈ Ω1 , n ≥ 0 ,
(30)
so that
|κn Φn (a)qn (a)| ≤
1 d(a, ℓ1 )
,
where d(a, ℓ1 ) = min |a − z | > 0. z ∈ℓ1
a ∈ Ω1 , n ≥ 0 ,
(31)
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Taking nth roots in (31) and using (28) gives lim sup |κn qn (a)|1/n ≤ |ϕ(a)|−1
(32)
n→∞
uniformly on compacts of Ω1 \ E. Indeed, this last inequality also holds uniformly on compacts of Ω1 . To see this, suppose a0 ∈ E and that D ⊂ Ω1 is a small closed disk centered at a0 such that D ∩ E = {a0 }. We then infer from (32) and the maximum modulus principle that
1/n
lim sup |ϕ(a)| |κn qn (a)|1/n ≤ lim sup max κn ϕ(t )n qn (t )
n→∞
n→∞
t ∈∂ D
≤ 1,
a ∈ D.
A similar argument shows that if (28) holds, then lim sup |Φn (z ; µ)|1/n ≤ |ϕ(z )|
(33)
n→∞
locally uniformly on Ω1 . Now, fix a number η such that 1 < η < |ϕ(a)|, and choose a corresponding ϵ > 0 satisfying that η(1 + ϵ) < |ϕ(a)|. Then, by (33) we can find a constant Mη such that
|Φn (z )| ≤ Mη |ϕ(z )|n (1 + ϵ)n ,
z ∈ ℓη , n ≥ 0.
Hence, ∞
∞
|qn (a)φn (z )| ≤ Mη
n=0
|κn qn (a)| |ϕ(z )|n (1 + ϵ)n
n =0
∞
≤ Mη
|κn qn (a)|ηn (1 + ϵ)n ,
z ∈ ℓη .
n =0
This and (32) show that the series in the right-hand side of (29) converges normally on G|ϕ(a)| (hence, in the L2 (µ)-norm as well) to an analytic function, say g (z ). By Runge’s Theorem [4, Theorem 2.3.2], (a − z )−1 can be arbitrarily approximated by polynomials in the uniform norm over G, and so it is in the L2 (µ)-closure of the space of the polynomials. Therefore, (a − z )−1 coincides, as an element of L2 (µ), with its Fourier series, which is precisely the series in the right-hand side of (29). Thus, the analytic function (a − z )−1 − g (z ) vanishes at every point of supp(µ), which is infinite, and so by the uniqueness theorem for analytic functions, we must have (a − z )−1 ≡ g (z ). Arguing similarly we can see that if lim supn→∞ |κn qn (a)|1/n = L < |ϕ(a)|−1 , then the series in the right-hand side of (29) would converge normally to an analytic function on GL , which is impossible since (a − z )−1 has a pole at a. From now on, we shall employ the notation
cn,s :=
Ωs
Dµ,n (z ) :=
|ϕ|2n dµ,
ρ ≤ s < 1, n ≥ 0,
Φn (z ; µ) , ϕ(z )n
n ≥ 0.
Our next lemma is essentially telling us that the weak convergence of the measures cn−,s1 |ϕ|2n dµ|Ωs is independent of
s ∈ [ρ, 1). In particular, when applied with the choice F = |Dµ |2 , it shows that the value of the lim sup in the condition (A2) is the same for every value of s. Lemma 3.2. Let µ be a measure supported on G and let F be a function defined and bounded on Ωs ∩ G, for some s ∈ [ρ, 1). Then,
Ωs
lim
n→∞
|ϕ|2n Fdµ
|ϕ|2n dµ Ωs
Ωs′
−
Ωs′
|ϕ|2n Fdµ |ϕ|2n dµ
= 0,
s < s′ < 1.
(34)
Proof. Since µ satisfies condition (A1), we have that for all ρ < s < s′ < 1, 0≤
cn,ρ cn,s
−1≤
Ωρ ∩Gs
|ϕ|2n dµ
cn,s′
≤
(s/s′ )2n ∥µ∥ −→ 0. µ(Ωs′ ) n→∞
Hence, lim cn,s′ /cn,s = 1,
n→∞
ρ ≤ s < s′ < 1 .
(35)
E. Miña-Díaz, B. Simanek / J. Math. Anal. Appl. 407 (2013) 290–304
299
Let us now denote the difference under the limit in (34) by In (s, s′ ), and let us assume that |F (z )| ≤ M for all z ∈ Ωs ∩ G. Then, the conclusion of the lemma follows by observing that with s′′ ∈ (s′ , 1), we have
Ωs ∩Gs′
In (s, s′ ) ≤
|ϕ|2n |F |dµ cn,s′′
+
Ωs′
|ϕ|2n |F |dµ cn,s′
1 − cn,s′ /cn,s
(s /s ) M ∥µ∥ + M 1 − cn,s′ /cn,s −→ 0. n→∞ µ(Ωs′′ ) ′
′′ 2n
≤
We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Since Dµ,n → Dµ uniformly on compacts of Ωr , an application of the maximum modulus principle produces the following estimate, valid for every function F continuous in the complex plane, every j ∈ Z and nonnegative integer n with n + j ≥ 0, and all r < s < s′ < 1:
Φ Φ Fdµ |Φn Φn+j | |F |dµ |ϕ|2n Dµ,n Dµ,n+j ϕ j Fdµ n n+j G Ωs − ≤ s cn,s cn,s cn,s′ s 2n sj ∥µ∥ −→ 0. max |Dµ,n Dµ,n+j | max |F | ≤ ′ z ∈ℓs s µ(Ωs′ ) n→∞ z ∈Gs
(36)
Since µ satisfies the condition (A2), there exists a subsequence N ⊆ N such that for every s ∈ (r , 1),
|ϕ|2n |Dµ |2 dµ > 0. |ϕ|2n dµ Ωs
Ωs
∆ := nlim →∞ n∈N
(37)
Using Helly’s selection theorem [13, Theorem 1.3], we can find a subsequence M ⊂ N such that the measures cn−,ρ1 |ϕ|2n dµ|Ωρ converge to some probability measure γ on ℓ1 = ∂ G as n → ∞ through M . Then, from Lemma 3.2 and (37), we obtain w-lim cn−,s1 |ϕ|2n dµ|Ωs = dγ ,
s ∈ (ρ, 1),
n→∞ n∈M
∆=
ℓ1
(38)
|Dµ |2 dγ > 0.
Applying (36) with F ≡ 1, in conjunction with (38), we deduce that for every j ∈ Z \ {0}, lim c −1 n→∞ n,s
Φn Φn+j dµ =
ℓ1
n∈M
|Dµ | ϕ dγ = 2
j
|Dµ (ψ(w))|2 w j d(ϕ∗ γ )(w),
(39)
|w|=1
where ϕ∗ γ is the push-forward via ϕ of the measure γ to {w : |w| = 1} (similarly, for any measure ν on {w : |w| = 1}, ψ∗ ν is the push-forward via ψ to ℓ1 ). For j = 0, (39) implies that lim
n→∞ n∈M
κn−2 cn,s
= ∆ = nlim →∞ n∈M
|ϕ|2n |Dµ |2 dµ , |ϕ|2n dµ Ωs
Ωs
s ∈ (r , 1),
(40)
while by the orthogonality property of the polynomials, (39) also yields that the measure |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) satisfies 2π
eijθ |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) = 0,
j ∈ Z \ {0}.
0
Hence, it follows that4
|Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) =
∆ 2π
dθ ,
or equivalently,
|Dµ (t )| dγ (t ) = ψ∗ |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) = ψ∗ 2
∆ dθ 2π
4 See, for instance, the last paragraph of Section 1.1 of [3], or Section 1.3.4 of [15].
=
∆ 2π
|ϕ ′ (t )| |dt |.
(41)
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Multiplying by |Dµ |−2 and integrating over E := {z ∈ ℓ1 : Dµ (z ) ̸= 0}, we get
∆ 2π
γ (E ) =
|Dµ (t )|−2 |ϕ ′ (t )| |dt |,
(42)
E
which is only possible if E = ℓ1 (for otherwise the right-hand side would be ∞). Being γ a probability measure, we conclude from (42) and (41) that
|Dµ |−2 |ϕ ′ | |dt | , −2 ′ ℓ1 |Dµ | |ϕ | |dt |
dγ =
∆=
2π
ℓ1
|Dµ |−2 |ϕ ′ | |dt |
.
(43)
Having Dµ not vanishing on ℓ1 implies that for some s ∈ (r , 1), the infimum of |Dµ | on Ωs ∩ G is strictly positive, and so by Lemma 3.2,
|ϕ|2n |Dµ |2 dµ > 0, |ϕ|2n dµ Ωs
lim inf
Ωs
n→∞
s ∈ (r , 1).
Hence, we can reapply the exact same arguments to realize that every subsequence of the natural numbers has in turn a subsequence, say M , through which the limits (38) and (40) hold true, with γ and ∆ given by (43). Since these limits are unique, that is, independent of the subsequences, the conclusions of Theorem 1.1(b)–(c) follow. As for Part (d), this is now a direct consequence of relations(36), (4), and (35). ∞ Finally, we see from (31) that κn ϕ n+1 Dµ,n qn n=0 is uniformly bounded on closed subsets of Ω1 . Since Dµ ̸≡ 0, the zeros of Dµ are isolated, and by Hurwitz’s theorem, attract those of Dµ,n . Then, a standard argument using the maximum modulus principle establishes the same boundedness, and by Montel’s theorem, the normality of the family
∞ κn ϕ n+1 qn n=0
(44)
on the domain Ω1 . Then, from (30) and (6), we get lim κn ϕ(z )n+1 qn (z ) =
n→∞
ϕ(z ) 2π Dµ (z ) ϕ (z )
ℓ1
|ϕ ′ (t )| |dt | ϕ(z ) = z−t 2π iDµ (z )
ℓ1
ϕ ′ (t )dt ϕ(t )(z − t )
′
=
Dµ (z )
,
(45)
whenever Dµ (z ) ̸= 0. But this and the normality of (44) imply that Dµ (z ) ̸= 0 for all z ∈ Ω1 , and that the limit in (45) holds uniformly on closed subsets of Ω1 . We now prove Proposition 2.1, which is actually a straightforward consequence of Lemma 3.1. Proof of Proposition 2.1. From the remarks following the proof of [9, Proposition 3], we obtain
∥µa ∥ =
∞
|qj (a)|2 ,
j =0
so that
∥µa ∥ −
n −1
|qj (a)|2 =
∞
|qj (a)|2 .
j =n
j =0
The proposition now follows by substituting Eq. (29) into Eq. (12).
In order to prove the theorems dealing with the spectral transforms, it is convenient to establish first the following lemma. Lemma 3.3. Let {λn }n≥0 be a sequence of positive numbers with the property that for every s < 1, there exists another sequence {cs,n }n≥0 of positive terms such that sj ≤
cs,n+j cs,n
≤ 1,
j, n ≥ 0,
lim λn /cs,n > 0.
n→∞
(46)
Then, lim
n→∞
∞ λn+j j =0
λn
z j = lim
n→∞
n λn j =0
λj
z n−j =
1 1−z
(47)
E. Miña-Díaz, B. Simanek / J. Math. Anal. Appl. 407 (2013) 290–304
301
locally uniformly on |z | < 1, and lim
n λn
n→∞
λj
j =0
= ∞.
(48)
Proof. For every n ≥ 0, let us denote by fn (z ) the sum under the first limit in (47). By writing
λn+j = λn
λn+j
cs,n
cs,n+j
λn
cs,n+j
cs,n
,
(49)
we see that the conditions (46) imply that λn+j /λn is uniformly bounded in n and j, so that the sequence {fn }∞ n=0 is a normal family of well-defined analytic functions on |z | < 1, and it suffices to prove that limn→∞ fn (z ) = (1 − z )−1 for z = |z | ∈ [0, 1). Again, from (49) and (46) we deduce that for every ϵ > 0, we can find and index N such that for all n ≥ N, 1−ϵ 1 − s|z |
≤ fn (|z |) ≤
1+ϵ 1 − |z |
,
so that 1−ϵ 1 − s|z |
1+ϵ
≤ lim inf fn (|z |) ≤ lim sup fn (|z |) ≤ n→∞
1 − |z |
n→∞
,
and the first limit in (47) follows after letting ϵ → 0 and s → 1. Let us now denote by gn (z ) the sum under the second limit in (47). Writing
λn = λj
λn cs,n
cs,j
λj
cs,n cs,j
,
we see that the sequence {gn }∞ n=0 is a normal family on |z | < 1, and that moreover, for every ϵ > 0, we can find an index N such that for all n ≥ N, n λ n n −j 1+ϵ (1 − ϵ)(1 − |zs|n−N +1 ) ≤ |z | ≤ . 1 − |zs| λ 1 − |z | j j =N
(50)
Taking the limit as n → ∞ we get 1−ϵ 1 − s|z |
≤ lim inf gn (|z |) ≤ lim sup gn (|z |) ≤ n→∞
n→∞
1+ϵ 1 − |z |
,
|z | < 1,
(51)
and the second limit in (47) follows after letting ϵ → 0 and s → 1. Notice that, the lower bound for the lim inf in (51) is equally valid for |z | = 1, whence (48) follows.
Later, we will also use the fact that under the assumptions of Lemma 3.3, and for any sequence of functions {νn (z )}n≥0 , we have lim νn (z ) = 0 ⇒ lim
n→∞
n→∞
n λn j =0
λj
z n−j νj (z ) = 0,
(52)
with both limits understood as holding locally uniformly on |z | < 1. This follows quite trivially from the inequalities (50). Proof of Proposition 2.2. By the maximum modulus principle, it suffices to show that (14) holds true uniformly on any contour ℓs where r < s < 1. By Proposition 2.1, we have that for all z ∈ ℓs ,
Φn (z ; µ) Φn+1 (z ; µa ) = − (z − a)ϕ(z )n ϕ(z )n
Qn (a; µ)ϕ(z )−n
∞
qn+j (a; µ)φn+j (z ; µ)
j =0
∞
|qn+j (a; µ)|2
j =0
One of our assumptions is that
Φn (z ; µ) = ϕ(z )n [Dµ (z ) + ϵn (z )],
ϵn (z ) → 0, z ∈ Ωr ,
and according to Theorem 1.1(e), we can write
κn (µ)ϕ(a)n+1 qn (a; µ) = h(a) + en ,
n ≥ 0,
.
(53)
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where h(a) = ϕ ′ (a)/Dµ (a) and limn→∞ en = 0. Therefore, the second fraction in the right-hand side of (53) can be rewritten in the form (we drop µ from the notation)
ϕ(a)n+1 κn qn (a)
∞ κn+j ϕ(a)n+j+1 qn+j (a) ϕ(a)−j ϕ(z )−n Φn+j (z ) j =0
∞ j =0
(h(a) + en )
κn2 |ϕ(a)|−2j |ϕ(a)n+1+j κn+j qn+j (a)|2 κn2+j ∞
(h(a) + en+j )(ϕ(z )/ϕ(a))j [Dµ (z ) + ϵn+j (z )]
j =0
=
∞ j =0
.
κn2 (µ)|ϕ(a)|−2j |h(a)+en+j |2 κn2+j (µ)
(54)
Since by Theorem 1.1(b) and (35), we have that for every s ∈ [ρ, 1)
κn−2 (µ) > 0, |ϕ|2n dµ Ωs
lim
n→∞
we can apply Lemma 3.3 with the choice λn = κn−2 (µ) and cs,n = Ω |ϕ|2n dµ to get from the first limit in (47) that this last s expression (54) converges uniformly on ℓs as n → ∞, and computing its limit yields
lim
n→∞
Φn+1 (z ; µa ) (z − a)Dµ (z ) Dµ (z )(|ϕ(a)|2 − 1)(z − a) = + , ϕ(z )n+1 ϕ(z ) ϕ(z ) ϕ(a) (ϕ(z ) − ϕ(a))
which is the desired expression.
Proof of Proposition 2.4. Suppose first that a ∈ Ω 1 , and let us rewrite (22) in the form n
κj2 (µ)
[ϕ(a)ϕ(z )]j−n Dµ,j (a)Dµ,j (z ) 2 (z − a)Φn (z ; µ ) Φn+1 (z ; µ) Φn+1 (a; µ) j=0 κn (µ) = − . n κ 2 (µ) ϕ(z )n ϕ(z )n ϕ(a)n j 2(j−n) |D 2 |ϕ( a )| ( a )| µ, j κ 2 (µ) c a
j =0
(55)
n
We can now apply Lemma 3.3, with the same choice as above λn = κn−2 (µ) and cs,n = (55) to get from (48), (52), and the second limit in (47) that
Ωs
|ϕ|2n dµ, and take the limit in
ϕ(z )Dµ (z )(|ϕ(a)|2 − 1) (z − a)Φn (z ; µca ) = ϕ(z )Dµ (z ) − , n n→∞ ϕ(z ) ϕ(a)ϕ(z ) − 1
(56)
lim
uniformly on closed subsets of |ϕ(z )| > max{r , |ϕ(a)|−1 }. Notice that, this last formula is equally valid for a ∈ ℓ1 , since in that case the fraction in the right-hand side of (56) is identically zero. Suppose now that a ∈ G. Again, by the maximum modulus principle, it suffices to show that (23) holds true for z ∈ ℓs , ρ(r , a) < s < 1. Fix s′ and η with ρ(r , a) < s′ < s < η < 1. Since s′ > r and a lies interior to the level curve ℓs′ , our assumption (3) and the behavior of κn (µ) given by (4) imply that there exist constants Ms , Ms′ and Mη such that
|Φn (z ; µ)| ≤ Ms sn ,
|Φn (a; µ)| ≤ Ms′ (s′ )n ,
κn2 (µ) ≤ Mη η−n ,
z ∈ ℓs , n ≥ 0.
Then from (22) we obtain that for every z ∈ ℓs and n ≥ 0, n κj2 (µ)|Φj (a; µ)| |Φj (z ; µ)| s′ Ms′ (s′ /s)n (z − a)Φn (z ; µca ) Φn+1 (z ; µ) j =0 ≤ − n ϕ(z )n ϕ(z )n κj2 (µ)|Φj (a; µ)|2 j =0
≤
s′ Ms Mη Ms2′
κ02 (µ)
(s′ /s)n
n ′ j ss j=0
η
,
and (23) for the case a ∈ G follows since the right-hand side of this inequality converges to 0 as n → ∞.
Proof of Proposition 2.5. From our assumption (25) and Hurwitz’s theorem, we see that for all n sufficiently large, Φn (z ; µ) ˜ has exactly J zeros on Ω 1 , say zn,1 , . . . , zn,J , which are simple and can be enumerated in such a way that zn,j → aj as n → ∞.
E. Miña-Díaz, B. Simanek / J. Math. Anal. Appl. 407 (2013) 290–304
J
303
J
Set pJ (z ) := j=1 (z − aj ), pn,J (z ) := j=1 (z − zn,j ). Then, by the extremal property of the monic orthogonal polynomials, and since µ ≤ µ ˜ , we have that for all n large,
|Φn (z ; µ)| ˜ 2 dµ ˜ ≤
|Φn (z ; µ) ˜ pJ (z )/pn,J (z )|2 dµ ˜
≤ max |pJ (z )/pn,J (z )|2
|Φn (z ; µ)| ˜ 2 dµ.
|Φn (z ; µ)| ˜ 2 dµ ≤
z ∈G
(57)
Since the maximum in (57) converges to 1 as n → ∞, these inequalities combine with (25), (5), and (35), to yield that for every s ∈ (r , 1), lim
n→∞
|Dµ˜ |2 |ϕ|2n dµ |Φn (z ; µ)| ˜ 2 dµ κn−2 (µ) ˜ Ωs = lim = lim n→∞ n→∞ |ϕ|2n dµ |ϕ|2n dµ |ϕ|2n dµ Ωs Ωs Ωs 2 ′ ˜ /Dµ | |ϕ | |dz | ℓ |Dµ > 0. = 1 −2 ′ ℓ1 |Dµ | |ϕ | |dz |
Thus, Lemma 3.3 can and will be applied with the choice λn = κn−2 (µ) ˜ and cs,n = −2 κn (µ) ˜ exists and is finite. Then, suppose first that a ∈ Ω 1 , and let us write (24) in the form
Φn+1 (z ; µ ˜ ) Φn+1 (z ; µ) ˜ Φn+1 (a; µ) ˜ = − · ϕ(z )n+1 ϕ(z )n+1 ϕ(a)n ϕ(z ) u a
m
n j =0
κj2 (µ) ˜ κn2 (µ) ˜
κn−2 (µ) ˜ |ϕ(a)|2n
Ωs
|ϕ|2n dµ. Observe that limn→∞
[ϕ(a)ϕ(z )]j−n Dµ, ˜ j (a)Dµ, ˜ j (z )
+m
n
κj2 (µ)|ϕ( ˜ a)|2(j−n) κn2 (µ) ˜
j =0
. 2 |Dµ, ˜ j (a)|
Using again (48), (52), and the second limit in (47) during the process of taking the limit in this last equality yields lim
n→∞
Φn (z ; µ ˜ ua ) Dµ˜ (z )(|ϕ(a)|2 − 1) = D ( z ) − µ ˜ ϕ(z )n (ϕ(a)ϕ(z ) − 1)
uniformly on closed subsets of |ϕ(z )| > max{r , |ϕ(a)|−1 }. We omit the proof of (27) for the case a ∈ G since it follows from (24) very much in the same way we proved (23) for a ∈ G in the second half of the proof of Proposition 2.4. Proof of Theorem 1.2. The limit (9) follows directly from Propositions 2.2, 2.4 and 2.5 by a trivial mathematical induction argument. If we set K
h(z ) :=
(z − zk )
k=1 L
,
(z − bl )
l =1
then (9) tells us that the measure |h|2 µ satisfies the conditions of Theorem 1.1, so that we can apply the relation (26) of Proposition 2.5 to get
κn (µ) ˆ lim = 2n |h|2 dµ n→∞ |ϕ| Ωρ
2π
−2
L
aj ∈Ω1
|ϕ(bl )|2
l=1
zk ∈Ω1
|ϕ(zk )|2
ℓ1
|ϕ(aj )|2 ,
|h/Dµ |2 |ϕ ′ | |dz |
and (10) follows at once by combining this last equality with (5).
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[7] Ya.L. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. 1954 (104) (1954) 79. [8] Ya.L. Geronimus, Ortogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau, New York, 1961. [9] E. Godoy, F. Marcellán, Orthogonal polynomials and rational modifications of measures, Canad. J. Math. 45 (5) (1993) 930–943. [10] J. Hernández, F. Marcellán, Geronimus spectral transforms and measures on the complex plane, J. Comput. Appl. Math. 219 (2008) 441–456. [11] P.P. Korovkin, The asymptotic representation of polynomials orthogonal over a region, Dokl. Akad. Nauk SSSR 58 (1947) 1883–1885. [12] P. Nevai, V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. 53 (1989) 99–104. [13] E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997. [14] B. Simanek, Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions, J. Approx. Theory 170 (2013) 172–197. [15] B. Simon, Orthogonal Polynomials on the Unit Circle, Part One: Classical Theory, American Mathematical Society, Providence, RI, 2005. [16] V.I. Smirnov, N.A. Lebedev, Functions of a Complex Variable: Constructive Theory, MIT Press, Cambridge, MA, 1968. [17] N. Stylianopoulos, Strong asymptotics for Bergman orthogonal polynomials over domains with corners and applications, Constr. Approx., in press (http://dx.doi.org/10.1007/s00365-012-9174-y). [18] P.K. Suetin, Polynomials Orthogonal Over a Region and Bieberbach Polynomials, American Mathematical Society, Providence, RI, 1974. [19] G. Szegő, Orthogonal Polynomials, fourth ed., Amer. Math. Soc., Providence, RI, 1975.
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Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Spectral transforms of measures and orthogonal polynomials on regions Erwin Miña-Díaz a,∗ , Brian Simanek b a
University of Mississippi, Department of Mathematics, Hume Hall 305, P.O. Box 1848, University, MS 38677-1848, USA
b
Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, Nashville, TN 37240, USA
article
abstract
info
Article history: Received 25 January 2013 Available online 21 May 2013 Submitted by Paul Nevai
We discuss Szegő-type strong asymptotics for polynomials orthogonal over a region bounded by an analytic curve, and find necessary conditions for such asymptotics to hold on an interior neighborhood of the curve. We also prove that this property is not destroyed when the measure is perturbed by multiplication by a rational function and the addition of point masses. © 2013 Elsevier Inc. All rights reserved.
Keywords: Orthogonal polynomials Strong asymptotics Geronimus transform Christoffel transform Uvarov transform
1. Introduction and main results Let µ be a finite Borel measure with compact and infinite support in the complex plane. Given such a measure, one can perform Gram–Schmidt orthogonalization on the sequence {1, z , z 2 , z 3 , . . .} in the space L2 (C, µ) to arrive at a sequence of polynomials {φn (z ; µ)}n≥0 satisfying
φn (z ; µ)φm (z ; µ)dµ(z ) = δmn , C
and normalized so that φn has positive leading coefficient. The leading coefficient of φn (z ; µ) will be denoted by κn (µ), and the monic orthogonal polynomial by
Φn (z ; µ) := κn−1 (µ)φn (z ; µ),
n ≥ 0.
This paper concerns the behavior that Φn (z ; µ) displays as n → ∞ when the measure µ is supported on the closure of an analytic domain. Some of the basic geometric properties of analytic domains that are essential in our discussion are briefly summarized in what follows. Hereafter, G will always denote a bounded region in the complex plane whose boundary is an analytic Jordan curve. Let D := {w : |w| < 1} be the unit disk. Since G is simply connected, there is a unique conformal bijection ψ that maps C \ D onto C \ G with a Laurent expansion at ∞ of the form
ψ(w) = dw + d0 + d−1 w −1 + d−2 w −2 + · · · ,
∗
d > 0.
Corresponding author. E-mail addresses: [email protected] (E. Miña-Díaz), [email protected] (B. Simanek).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.05.035
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Let ρ ∈ [0, 1] denote the smallest number with the property that ψ admits an analytic and univalent continuation to |w| > ρ . It is well-known that since G has an analytic boundary, ρ < 1. The set
Ωρ := {z = ψ(w) : |w| > ρ} is a simply-connected neighborhood of ∞ containing all of ∂ G. We can then define
ϕ : Ωρ → {w : |w| > ρ} to be the inverse of ψ , so that for every r > ρ , the level set
ℓr = {z : |ϕ(z )| = r } is an analytic Jordan curve. We shall let Ωr and Gr denote, respectively, the exterior and interior domains of the curve ℓr . Notice that with this notation we have G1 = G, ℓ1 = ∂ G, and Ω1 = C \ G. By a measure µ supported on G we shall always mean a measure whose support is an infinite subset of G and that satisfies
µ(Ωs ) > 0,
s ∈ [ρ, 1).
(A1)
This last condition is really more of a convenience than a necessity, since any measure µ on G can be made to satisfy (A1) by an appropriate rescaling. However, if we are to expect the geometry of G to have some influence on the behavior of Φn (z ; µ), the condition (A1) is the least we can impose on µ. There are many ways to study the asymptotics of the polynomial sequence {Φn }n≥0 . The asymptotics we will be interested in are called strong or Szegő asymptotics, and are devoted to finding conditions on µ guaranteeing the existence of some analytic function Dµ (z ) such that lim
n→∞
Φn (z ; µ) = Dµ (z ) dn ϕ(z )n
(2)
on some region of the complex plane. Here d is the first coefficient of the Laurent expansion of ψ in (1), which coincides with the so-called logarithmic capacity of ℓ1 = ∂ G. If ℓ1 is the unit circle and supp(µ) ⊂ ℓ1 , then the well-known Szegő–Geronimus theorem [8, Chapter I, II] states that (2) holds for all |z | > 1, if and only if, the Radon–Nikodym derivative µ′ (z ) of µ with respect to the arc-length measure |dz | satisfies the Szegő condition
ℓ1
log µ′ (z )|dz | > −∞.
In fact, a similar theorem was obtained by Geronimus [6, Theorem 7.1] for a measure µ supported on an arbitrary analytic curve. Szegő-type asymptotic formulas have also been established for planar type measures such as those given by Lipschitz continuous weights [18] and perturbations of product measures [14] on analytic regions, as well as for area measure on regions with corners [17]. Generally speaking, if the measure µ is sufficiently irregular, Szegő asymptotics can only be derived for z ∈ Ω1 . However, for nicer measures it is possible to have (2) holding across the boundary of G. For instance, Carleman proved in [1] that if µ = dA is the area measure on G, then (2) holds locally uniformly for z ∈ Ωρ with Dµ (z ) = ϕ ′ (z ) (for an extension of this theorem, see [2]). Korovkin [11] and Smirnov [16] generalized this result to measures µ whose restriction to Ωs (for some ρ ≤ s < 1) is of the form |g (z )|2 dA(z ), where g (z ) is such that for some integer m, g (z )ϕ(z )m is analytic and nonvanishing on Ωs . In the present work, we derive several relations that must be satisfied in order to have Szegő asymptotics holding on some interior neighborhood of ℓ1 = ∂ G. With this knowledge in hand, we will then be able to explore certain perturbations of the orthogonality measure that do not destroy the property of having Szegő asymptotics across ∂ G. Here, the so-called functions of the second kind Qn (z ; µ) :=
Φn (t ; µ) dµ(t ), z−t
qn (z ; µ) := κn (µ)Qn (z ; µ),
z ̸∈ supp(µ),
will play an important role as well. The question of whether (2) holds on a domain larger than Ω1 is completely settled for measures µ supported on the unit circle. In such a case, it is a well-known result that for (2) to hold on the exterior of some circle of radius less than 1, it is necessary1 and sufficient2 that µ be of the form |w(z )|2 |dz |, with w analytic and nonvanishing on the unit circle. Theorem 1.1 recovers, in particular, the ‘‘necessary’’ half of this result. We will be interested in measures µ that are supported on some G, and that admit the Szegő asymptotics (2) in the region Ωr , for some r ∈ [ρ, 1). For such measures, we will also impose a very natural additional condition, namely, that for
1 See (b) ⇒ (c) in the corollary of [12]. 2 The sufficiency is even true for a general analytic curve [19, Theorem 16.5].
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every s ∈ (r , 1),
lim sup
|ϕ(z )|2n |Dµ (z )|2 dµ(z ) > 0. |ϕ(z )|2n dµ(z ) Ωs
Ωs
n→∞
(A2)
It will follow from Lemma 3.2 below that the above lim sup, whether positive or not, has the same value for every s ∈ (r , 1). It is entirely possible for a measure µ to satisfy (2) and not (A2). For instance, take the measure µ supported on the closed N unit disk D and given by µ = dA1/2 + j=1 δzj , where dA1/2 is the area measure restricted to |z | < 1/2, and the zj ’s are arbitrary points located on the unit circle |z | = 1. It follows from Theorem 1.2 below that
Φn (z ; µ)
lim
zn
n→∞
=
N z − zj , z − 4z1 j=1 j
|z | > 1/4.
This can happen because the measure µ is too thin near the boundary of the domain, namely, a sum of point masses. As a matter of fact, at least in the following weak sense this will always be the case. Helly’s selection theorem tells us that through any subsequence of the natural numbers, we may take a further subsequence through which the measures3
|ϕ|2n dµ|Ωρ |ϕ|2n dµ|Ωρ converge weakly to a measure γ that, by condition (A1), has for support a subset of ∂ G. Therefore, if the condition (A2) is not satisfied, every such weak limit point γ must satisfy
∂G
|Dµ (z )|2 dγ (z ) = 0.
Since Dµ is an analytic function in some neighborhood of ∂ G, this last equality forces γ to be a sum of masses located at the finitely many zeros that Dµ necessarily has on ∂ G. For simplicity, hereafter we make the assumption that the capacity of ∂ G is 1. There is no loss of generality in doing so since this can always be attained by rescaling the domain G (see the third remark following Theorem 1.2 in [14]). We can now state our result giving necessary conditions in order to have Szegő asymptotics holding on some interior neighborhood of ℓ1 = ∂ G. Theorem 1.1. Let µ be a measure supported on G such that lim
n→∞
Φn (z ; µ) = Dµ (z ) ϕ(z )n
(3)
uniformly on closed subsets of Ωr , for some r ∈ [ρ, 1), and suppose also that µ satisfies the condition (A2). Then, the following statements hold true: (a) Dµ has no zeros on Ω 1 ; (b) for every s ∈ (r , 1),
κ −2 (µ) = lim lim n 2n n→∞ |ϕ| dµ n→∞ Ωρ
|ϕ|2n |Dµ |2 dµ 2π = ; 2n −2 ′ |ϕ| dµ Ωs ℓ1 |Dµ | |ϕ | |dz |
Ωs
(4)
(c)
|ϕ|2n dµ|Ωρ |Dµ |−2 |ϕ ′ | |dz | = ; 2n −2 ′ |ϕ| dµ|Ωρ ℓ1 |Dµ | |ϕ | |dz |
w-lim n→∞
(5)
(d) w-lim |φn |2 dµ = n→∞
|ϕ ′ | |dz | ; 2π
(6)
(e) lim κn2 (µ)ϕ(z )n+1 Qn (z ; µ) =
n→∞
ϕ ′ (z ) Dµ (z )
uniformly on closed subsets of Ω1 . 3 As usual, µ| stands for the restriction of the measure µ to the measurable set A. A
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293
Since the measures in (5) ‘‘suppress’’ the properties of µ away from ∂ G, Theorem 1.1 tells us that in our setting, the measure µ must resemble a positive analytic weight near the boundary of G. Observe also that (4) is telling us that, indeed, the full limit in (A2) exists. We now turn our attention to perturbations of the measure µ of the form K
|z − zk |2
k=1
dµ( ˆ z) =
L
dµ(z ) +
J
mj δaj ,
mj > 0, 1 ≤ j ≤ J ,
(7)
j =1
2
|z − bl |
l=1
with the aj ’s pairwise distinct, each pole bl ∈ Ω1 , and no zero zk or mass point aj lying on ∂ G. We shall see in our next theorem that this new measure µ ˆ retains the property of having Szegő asymptotics across ∂ G. But first, we need a new piece of notation. Corresponding to a domain G and its associated number ρ (the univalency radius of ψ ), we define the quantity ρ(r , a) for every r ∈ (ρ, 1) and a ∈ C by
r , ρ(r , a) := |ϕ(a)|−1 , |ϕ(a)|,
a ̸∈ Ωr ∩ G1/r , 1 < |ϕ(a)| < 1/r , r < |ϕ(a)| ≤ 1,
(8)
while for a set E ⊂ C, we put
ρ(r , E ) := sup{ρ(r , a) : a ∈ E }. Let us also define F (z , t ) :=
ϕ(z ) − ϕ(t ) ϕ(t )ϕ(z ) − 1
.
Theorem 1.2. Let µ be a measure satisfying the conditions of Theorem 1.1. Then, for a measure µ ˆ of the form (7), we have lim
n→∞
Φn (z ; µ) ˆ = Dµˆ (z ) ϕ(z )n
(9)
uniformly on closed subsets of Ωρ(r ,E ) , where E = {z1 , . . . , zK , a1 , . . . , aJ } and
Dµˆ (z ) =
L
Dµ (z )
(z − bl )
l=1
L ϕ(z )L−K
ϕ(bl )F (z , bl )
l =1
zk ∈Ω1
ϕ(zk )F (z , zk )
K
(z − zk )
ϕ(aj )F (z , aj ).
aj ∈Ω1
k=1
Moreover, 2π
|ϕ(aj )|2 |ϕ(zk )|2 aj ∈Ω1 ˆ κn−2 (µ) zk ∈Ω1 lim · L = . −2 ′ n→∞ |ϕ|2n dµ Ωρ ℓ1 |Dµ | |ϕ | |dz | 2 |ϕ(bl )|
(10)
l=1
The measure µ ˆ in (7) results from the repeated application of three simpler types of spectral transforms on a measure µ, namely, the addition of a pole |z − a|−2 µ, of a zero |z − a|2 µ, and of a point mass µ + mδa . In Section 2, we will consider each of these transformations individually, stating in Propositions 2.2, 2.4 and 2.5, respectively, the simplified version of Theorem 1.2 that corresponds to each transform. The reader will have no difficulty in seeing that Theorem 1.2 follows from these propositions by a trivial mathematical induction argument. The proof of all the results, as well as some auxiliary lemmas are given in Section 3. Our exposition will include some examples that are of particular interest. 2. Spectral transforms 2.1. The Geronimus transform The Geronimus spectral transform of a measure µ, corresponding to a point a ̸∈ supp(µ), is the measure µa defined by d µa ( z ) =
1
|z − a|2
dµ(z ).
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Such transformations have been studied previously in the context of general orthogonal polynomials in [9,10]. In particular, there is a concise formula describing the polynomial Φn+1 (z ; µa ) in terms of the orthogonal polynomials and the functions of the second kind for µ. More precisely, it was proven in [9, Proposition 3] (see also [10, Proposition 3]) that for all n ≥ 0,
Qn (a; µ) (z − a)
n−1
qj (a; µ)φj (z ; µ) + 1
j =0
Φn+1 (z ; µa ) = (z − a)Φn (z ; µ) +
n −1
∥µa ∥ −
.
(12)
|qj (a; µ)|
2
j=0
Eq. (12) will be the key to our calculations, though it will require the following reformulation before we can effectively apply it. Proposition 2.1. Let µ be a measure supported on G such that lim |Φn (z ; µ)|1/n = |ϕ(z )|
n→∞
uniformly on closed subsets of Ω1 . Then, for every point a ∈ Ω1 and n ≥ 0, the equality
(z − a)Qn (a; µ) Φn+1 (z ; µa ) = (z − a)Φn (z ; µ) −
∞
qj (a; µ)φj (z ; µ)
j =n
∞
(13)
|qj (a; µ)|2
j =n
holds true locally uniformly for z ∈ G|ϕ(a)| . We remark that the sum on the right-hand side of (13) does not converge for z ̸∈ G|ϕ(a)| . Combining Proposition 2.1 and Theorem 1.1 we will be able to establish the following proposition. Proposition 2.2. Let µ be a measure satisfying the conditions of Theorem 1.1. If µa is the measure given by (11), then for every a ∈ Ω1 , lim
n→∞
Φn (z ; µa ) Dµ (z )(ϕ(a)ϕ(z ) − 1)(z − a) = ϕ(z )n ϕ(a)ϕ(z )(ϕ(z ) − ϕ(a))
(14)
locally uniformly on Ωr . We have already mentioned in the introduction several classes of measures µ known to satisfy the conditions of Theorem 1.1. The simplest example would be the arc-length measure µ = |dz | on the unit circle, for which the monic orthogonal polynomials are just the monomials Φn (z ; µ) = z n , n ≥ 0, and the leading coefficients κn (µ) = (2π )−1/2 are constant. Repeated applications of the Geronimus transform to this measure give rise to the so-called Bernstein–Szegő polynomials. These are the polynomials orthogonal over the unit circle with respect to a measure of the form dµ( ˆ z ) = |(z − b1 ) · · · (z − bL )|−2 |dz |,
|bl | > 1,
1 ≤ l ≤ L.
In this case, it is well-known and easy to verify that [19, Theorem 11.2]
Φn+L (z ; µ) ˆ = z n (z − b∗1 ) · · · (z − b∗L ),
n ≥ 0,
where, as it is customary, z ∗ := 1/z denotes the reflection of z about the unit circle. More generally, if µ is a measure supported on D having the monomials z n for monic orthogonal polynomials, then it follows straightforwardly from Proposition 2.1 that Φn+1 (z ; µa ) is divisible by z n for all n ≥ 0. From this observation and Proposition 2.2, we can use a clear induction argument to arrive at the following corollary. Corollary 2.3. Let µ be a measure supported on D such that Φn (z ; µ) = z n , n ≥ 0. For every measure µ ˆ of the form dµ( ˆ z ) = |(z − b1 ) · · · (z − bL )|−2 dµ(z ),
|bl | > 1,
1 ≤ l ≤ L,
we have
Φn+L (z ; µ) ˆ = z n (z − b∗1 ) · · · (z − b∗L ) + εn (z ) ,
n ≥ 0,
with εn (z ) a polynomial such that limn→∞ εn (z ) = 0 uniformly on compact subsets of the complex plane.
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295
A well-known class of measures having the monomials z n for monic orthogonal polynomials is that of the radially symmetric measures, that is, measures µ that are the image by the map (r , θ ) → reiθ of a product measure σ × dθ on [0, 1]× [0, 2π ]. For such a measure µ, we have
1
z z dµ(z ) = n m
r
n+m
dσ (r )
ei(n−m)θ dθ ,
0
0
D
2π
and after easy computations one finds
Φn (z ; µ) = z n ,
1
κn−2 (µ) = 2π
r 2n dσ (r ),
Qn (a; µ) =
κn−2 (µ) an + 1
0
∞
qj (a; µ)φj (z ; µ) =
j=n
(z /a)n , a−z
∞
2π
|qj (a; µ)|2 =
2n
| a|
j =n
0
1
r 2n 2
|a| − r 2
, dσ (r ).
Inserting these expressions in (13) one arrives at the explicit representation
Φn+1 (z ; µa ) = z n
1
z−
+
a
r 2n (1−r 2 ) d 0 |a|2 −r 2
1
1
r 2n d 0 |a|2 −r 2
1
a
σ
σ
,
n ≥ 0.
(16)
Notice that if µ is supported on D, or equivalently, if 1 ∈ supp(σ ), the second fraction in the right-hand side of (16) converges to 0 as n → ∞. We can also prove (16) in a direct way that will allow us to extend this explicit representation to polynomials Φn (z ; µ) ˆ of sufficiently large degree corresponding to a measure µ ˆ constructed in the form (15) out of a radially symmetric µ = σ × dθ supported on D. We know from the last corollary that for all n ≥ 0, the (n + L)th monic orthogonal polynomial has the form
Φn+L (z ; µ) ˆ = z n (αn,0 + αn,1 z + · · · + αn,L z L ),
αn,L = 1.
(17)
For every pair of integers n, m, we have
1
z n z m dµ( ˆ z) =
i
D
1
d σ (r )
|ζ |=r
0
ζ n−1 ζ m dζ . L 2 |ζ − bj |
(18)
j=1
Using that ζ = r /ζ on |ζ | = r, followed by the change of variables ζ = r 2 z and a deformation of the contour of integration from |z | = 1/r to |z | = 1, we get 2
|ζ |=r
ζ n−1 ζ m dζ = L 2 |ζ − bj |
ζ n+L−m−1 r 2m
|ζ |=r
j =1
L
(
r2
dζ
− bj ζ )(ζ − bj )
j =1
z n+L−m−1 r 2n
= |z |=1
L
dz .
(1 − bj z )(r 2 z − bj )
j =1
Combining this last equality with (18) and (17), we arrive at the following identity, valid for every integer m:
Φn+L (z ; µ) ˆ z m dµ( ˆ z) =
1
D
i
z n+L−m−1
L
αn,l z l gn,l (z ) dz
l =0
|z |=1
L
,
(19)
(1 − bj z )
j =1
where gn,l (z ) :=
1
0
r 2n+2l dσ (r ) L
(
r 2z
,
0 ≤ l ≤ L.
− bj )
j =1
Each function gn,l is analytic on the closed unit disk, so that for every 0 ≤ m ≤ n + L − 1, the integral in (19) will reduce to zero L L if constants αn,l ’s are found such that l=0 αn,l z l gn,l (z ) is divisible by j=1 (z − b∗j ). Thus, we can prescribe the conditions L−1 l =0
αn,l (b∗j )l gn,l (b∗j ) + (b∗j )L gn,L (b∗j ) = 0,
1 ≤ j ≤ L,
(20)
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where in case bj has multiplicity βj > 1, βj − 1 of these equations must be replaced by L−1
αn,l [z l gn,l (z )](v) + [z L gn,L (z )](v) = 0,
z = b∗j , 1 ≤ v ≤ βj − 1.
l =0
Observe that this system remains constant (independent of n) when σ = δ1 , that is, when µ is the arc-length measure on the unit circle. 1 If we divide each equation in (20) by 0 r 2n dσ (r ) and take the limit as n → ∞, we obtain a limiting system (precisely, the one corresponding to σ = δ1 ) with a unique solution given by the coefficients of the polynomial j=1 (z − b∗j ). Hence, for all n sufficiently large, the system (20) also has a unique solution that can be explicitly computed using Cramer’s formula.
L
2.2. The Christoffel and Uvarov transforms For a point a ∈ C, the Christoffel transform µca of the measure µ is defined by dµca (z ) = |z − a|2 dµ(z ).
(21)
Using the orthogonality relations, it is easy to verify that
Φn+1 (a; µ)Kn (z , a; µ) Φn+1 (z ; µ) − , Φn (z ; µ ) = z−a Kn (a, a; µ) 1
c a
n ≥ 0,
(22)
where Kn (z , t ; µ) :=
n
φj (t ; µ)φj (z ; µ).
j=0
The relation (22) appears in several publications such as [5,9]. We will use it to establish the following proposition. Proposition 2.4. Let µ be a measure satisfying the conditions of Theorem 1.1, and let µca be the measure given by (21). Then, with ρ(r , a) as defined by (8), we have
Dµ (z )ϕ(a)ϕ(z ) ϕ(z ) − ϕ(a) · , Φn (z ; µca ) z−a ϕ(a)ϕ(z ) − 1 lim = n→∞ ϕ(z )n Dµ (z )ϕ(z ) , z−a
a ∈ Ω 1, (23) a ∈ G,
uniformly on closed subsets of Ωρ(r ,a) . We now consider for a point a ∈ C and a number m > 0, the measure µua defined by
µua = µ + mδa , where δa is the point mass Dirac measure at a. This measure µua is often called the Uvarov transform of µ. Using the orthogonality relations, a rather simple calculation reveals that
Φn (z ; µua ) = Φn (z ; µ) −
mΦn (a; µ)Kn−1 (z , a; µ) 1 + mKn−1 (a, a; µ)
,
n ≥ 0.
(24)
This formula is due to Geronimus [7, Formula (29.12)], who obtained it not just for one but for any finite number of point masses. It will allow us to prove the following result. Proposition 2.5. Let µ be a measure supported on G and satisfying the conditions of Theorem 1.1. Let µ ˜ be a measure of the form
µ+
J
mj δaj ,
mj > 0 , 1 ≤ j ≤ J ,
j =1
with the aj ’s pairwise distinct and contained in Ω1 . Suppose that lim
n→∞
J Φn (z ; µ) ˜ ∗ = D ( z ) = D ( z ) (z − aj ) µ ˜ µ ˜ ϕ(z )n j =1
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297
uniformly on closed subsets of Ωr , with D∗µ˜ analytic and nonvanishing on Ω 1 . Then, lim
n→∞
2 ′ ˜ /Dµ | |ϕ | |dz | κn−2 (µ) ˜ ℓ1 |Dµ , = −2 ′ |ϕ|2n dµ Ωρ ℓ1 |Dµ | |ϕ | |dz |
(26)
and for the measure µ ˜ ua = µ ˜ + mδa , m > 0, a ̸∈ {a1 , . . . , aJ }, we have
ϕ(z ) − ϕ(a) , Φn (z ; µ ˜ ) ϕ(a)Dµ˜ (z ) · lim = ϕ( a)ϕ(z ) − 1 n→∞ ϕ(z )n Dµ˜ (z ), u a
a ∈ Ω 1,
(27)
a ∈ G,
uniformly on closed subsets of Ωρ(r ,a) . In conclusion, we see that if a ̸∈ ∂ G, the polynomials Φn (z ; µca ) and Φn (z ; µua ) also satisfy a strong asymptotic formula on a domain containing Ω 1 , namely, on Ωρ(r ,a) . It is easy to see that, in general, this is no longer true if a ∈ ∂ G. For instance, take G = D and let µ = |dz |/2π be the normalized arc-length measure on the unit circle ∂ D. For this measure, we have Φn (z ; µ) = φn (z ; µ) = z n , and we can compute from (22) and (24) very simple explicit formulas for the orthogonal polynomials corresponding to µca and µua . From these formulas, we find that if |a| = 1, then lim (n + 1)a−n−2 Φn (z ; µca ) = (z − a)−2 ,
|z | < 1,
lim (n + 1)a−n−1 Φn (z ; µua ) = (z − a)−1 ,
|z | < 1,
n→∞ n→∞
and so limn→∞ z −n Φn (z ; µca ) = limn→∞ z −n Φn (z ; µua ) = ∞ on 0 < |z | < 1. 3. Proofs The proofs will be given following the same order in which the corresponding results were presented. To abbreviate the notation, we will drop µ whenever there is no cause for confusion. Thus, we will write, for instance, Φn (z ) = Φn (z ; µ), qn (a) = qn (a; µ), and κn = κn (µ). Lemma 3.1. Let µ be a measure supported on G and let E ⊂ Ω1 be a set consisting of isolated points. Suppose that lim |Φn (z ; µ)|1/n = |ϕ(z )|
(28)
n→∞
locally uniformly on Ω1 \ E. Then, for every a ∈ Ω1 , 1 a−z
=
∞
qn (a; µ)φn (z ; µ)
(29)
n=0
uniformly on compact subsets of G|ϕ(a)| , and lim sup |κn (µ)qn (a; µ)|1/n = |ϕ(a)|−1 , n→∞
a ∈ Ω1 .
Proof. By orthogonality, we have
G
Φn (z ) − Φn (a) Φn (z )dµ(z ) = 0, a−z
or equivalently,
κn Φn (a)qn (a) =
G
|φn (z )|2 dµ(z ) , a−z
a ∈ Ω1 , n ≥ 0 ,
(30)
so that
|κn Φn (a)qn (a)| ≤
1 d(a, ℓ1 )
,
where d(a, ℓ1 ) = min |a − z | > 0. z ∈ℓ1
a ∈ Ω1 , n ≥ 0 ,
(31)
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Taking nth roots in (31) and using (28) gives lim sup |κn qn (a)|1/n ≤ |ϕ(a)|−1
(32)
n→∞
uniformly on compacts of Ω1 \ E. Indeed, this last inequality also holds uniformly on compacts of Ω1 . To see this, suppose a0 ∈ E and that D ⊂ Ω1 is a small closed disk centered at a0 such that D ∩ E = {a0 }. We then infer from (32) and the maximum modulus principle that
1/n
lim sup |ϕ(a)| |κn qn (a)|1/n ≤ lim sup max κn ϕ(t )n qn (t )
n→∞
n→∞
t ∈∂ D
≤ 1,
a ∈ D.
A similar argument shows that if (28) holds, then lim sup |Φn (z ; µ)|1/n ≤ |ϕ(z )|
(33)
n→∞
locally uniformly on Ω1 . Now, fix a number η such that 1 < η < |ϕ(a)|, and choose a corresponding ϵ > 0 satisfying that η(1 + ϵ) < |ϕ(a)|. Then, by (33) we can find a constant Mη such that
|Φn (z )| ≤ Mη |ϕ(z )|n (1 + ϵ)n ,
z ∈ ℓη , n ≥ 0.
Hence, ∞
∞
|qn (a)φn (z )| ≤ Mη
n=0
|κn qn (a)| |ϕ(z )|n (1 + ϵ)n
n =0
∞
≤ Mη
|κn qn (a)|ηn (1 + ϵ)n ,
z ∈ ℓη .
n =0
This and (32) show that the series in the right-hand side of (29) converges normally on G|ϕ(a)| (hence, in the L2 (µ)-norm as well) to an analytic function, say g (z ). By Runge’s Theorem [4, Theorem 2.3.2], (a − z )−1 can be arbitrarily approximated by polynomials in the uniform norm over G, and so it is in the L2 (µ)-closure of the space of the polynomials. Therefore, (a − z )−1 coincides, as an element of L2 (µ), with its Fourier series, which is precisely the series in the right-hand side of (29). Thus, the analytic function (a − z )−1 − g (z ) vanishes at every point of supp(µ), which is infinite, and so by the uniqueness theorem for analytic functions, we must have (a − z )−1 ≡ g (z ). Arguing similarly we can see that if lim supn→∞ |κn qn (a)|1/n = L < |ϕ(a)|−1 , then the series in the right-hand side of (29) would converge normally to an analytic function on GL , which is impossible since (a − z )−1 has a pole at a. From now on, we shall employ the notation
cn,s :=
Ωs
Dµ,n (z ) :=
|ϕ|2n dµ,
ρ ≤ s < 1, n ≥ 0,
Φn (z ; µ) , ϕ(z )n
n ≥ 0.
Our next lemma is essentially telling us that the weak convergence of the measures cn−,s1 |ϕ|2n dµ|Ωs is independent of
s ∈ [ρ, 1). In particular, when applied with the choice F = |Dµ |2 , it shows that the value of the lim sup in the condition (A2) is the same for every value of s. Lemma 3.2. Let µ be a measure supported on G and let F be a function defined and bounded on Ωs ∩ G, for some s ∈ [ρ, 1). Then,
Ωs
lim
n→∞
|ϕ|2n Fdµ
|ϕ|2n dµ Ωs
Ωs′
−
Ωs′
|ϕ|2n Fdµ |ϕ|2n dµ
= 0,
s < s′ < 1.
(34)
Proof. Since µ satisfies condition (A1), we have that for all ρ < s < s′ < 1, 0≤
cn,ρ cn,s
−1≤
Ωρ ∩Gs
|ϕ|2n dµ
cn,s′
≤
(s/s′ )2n ∥µ∥ −→ 0. µ(Ωs′ ) n→∞
Hence, lim cn,s′ /cn,s = 1,
n→∞
ρ ≤ s < s′ < 1 .
(35)
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299
Let us now denote the difference under the limit in (34) by In (s, s′ ), and let us assume that |F (z )| ≤ M for all z ∈ Ωs ∩ G. Then, the conclusion of the lemma follows by observing that with s′′ ∈ (s′ , 1), we have
Ωs ∩Gs′
In (s, s′ ) ≤
|ϕ|2n |F |dµ cn,s′′
+
Ωs′
|ϕ|2n |F |dµ cn,s′
1 − cn,s′ /cn,s
(s /s ) M ∥µ∥ + M 1 − cn,s′ /cn,s −→ 0. n→∞ µ(Ωs′′ ) ′
′′ 2n
≤
We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Since Dµ,n → Dµ uniformly on compacts of Ωr , an application of the maximum modulus principle produces the following estimate, valid for every function F continuous in the complex plane, every j ∈ Z and nonnegative integer n with n + j ≥ 0, and all r < s < s′ < 1:
Φ Φ Fdµ |Φn Φn+j | |F |dµ |ϕ|2n Dµ,n Dµ,n+j ϕ j Fdµ n n+j G Ωs − ≤ s cn,s cn,s cn,s′ s 2n sj ∥µ∥ −→ 0. max |Dµ,n Dµ,n+j | max |F | ≤ ′ z ∈ℓs s µ(Ωs′ ) n→∞ z ∈Gs
(36)
Since µ satisfies the condition (A2), there exists a subsequence N ⊆ N such that for every s ∈ (r , 1),
|ϕ|2n |Dµ |2 dµ > 0. |ϕ|2n dµ Ωs
Ωs
∆ := nlim →∞ n∈N
(37)
Using Helly’s selection theorem [13, Theorem 1.3], we can find a subsequence M ⊂ N such that the measures cn−,ρ1 |ϕ|2n dµ|Ωρ converge to some probability measure γ on ℓ1 = ∂ G as n → ∞ through M . Then, from Lemma 3.2 and (37), we obtain w-lim cn−,s1 |ϕ|2n dµ|Ωs = dγ ,
s ∈ (ρ, 1),
n→∞ n∈M
∆=
ℓ1
(38)
|Dµ |2 dγ > 0.
Applying (36) with F ≡ 1, in conjunction with (38), we deduce that for every j ∈ Z \ {0}, lim c −1 n→∞ n,s
Φn Φn+j dµ =
ℓ1
n∈M
|Dµ | ϕ dγ = 2
j
|Dµ (ψ(w))|2 w j d(ϕ∗ γ )(w),
(39)
|w|=1
where ϕ∗ γ is the push-forward via ϕ of the measure γ to {w : |w| = 1} (similarly, for any measure ν on {w : |w| = 1}, ψ∗ ν is the push-forward via ψ to ℓ1 ). For j = 0, (39) implies that lim
n→∞ n∈M
κn−2 cn,s
= ∆ = nlim →∞ n∈M
|ϕ|2n |Dµ |2 dµ , |ϕ|2n dµ Ωs
Ωs
s ∈ (r , 1),
(40)
while by the orthogonality property of the polynomials, (39) also yields that the measure |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) satisfies 2π
eijθ |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) = 0,
j ∈ Z \ {0}.
0
Hence, it follows that4
|Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) =
∆ 2π
dθ ,
or equivalently,
|Dµ (t )| dγ (t ) = ψ∗ |Dµ (ψ(eiθ ))|2 d(ϕ∗ γ )(θ ) = ψ∗ 2
∆ dθ 2π
4 See, for instance, the last paragraph of Section 1.1 of [3], or Section 1.3.4 of [15].
=
∆ 2π
|ϕ ′ (t )| |dt |.
(41)
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Multiplying by |Dµ |−2 and integrating over E := {z ∈ ℓ1 : Dµ (z ) ̸= 0}, we get
∆ 2π
γ (E ) =
|Dµ (t )|−2 |ϕ ′ (t )| |dt |,
(42)
E
which is only possible if E = ℓ1 (for otherwise the right-hand side would be ∞). Being γ a probability measure, we conclude from (42) and (41) that
|Dµ |−2 |ϕ ′ | |dt | , −2 ′ ℓ1 |Dµ | |ϕ | |dt |
dγ =
∆=
2π
ℓ1
|Dµ |−2 |ϕ ′ | |dt |
.
(43)
Having Dµ not vanishing on ℓ1 implies that for some s ∈ (r , 1), the infimum of |Dµ | on Ωs ∩ G is strictly positive, and so by Lemma 3.2,
|ϕ|2n |Dµ |2 dµ > 0, |ϕ|2n dµ Ωs
lim inf
Ωs
n→∞
s ∈ (r , 1).
Hence, we can reapply the exact same arguments to realize that every subsequence of the natural numbers has in turn a subsequence, say M , through which the limits (38) and (40) hold true, with γ and ∆ given by (43). Since these limits are unique, that is, independent of the subsequences, the conclusions of Theorem 1.1(b)–(c) follow. As for Part (d), this is now a direct consequence of relations(36), (4), and (35). ∞ Finally, we see from (31) that κn ϕ n+1 Dµ,n qn n=0 is uniformly bounded on closed subsets of Ω1 . Since Dµ ̸≡ 0, the zeros of Dµ are isolated, and by Hurwitz’s theorem, attract those of Dµ,n . Then, a standard argument using the maximum modulus principle establishes the same boundedness, and by Montel’s theorem, the normality of the family
∞ κn ϕ n+1 qn n=0
(44)
on the domain Ω1 . Then, from (30) and (6), we get lim κn ϕ(z )n+1 qn (z ) =
n→∞
ϕ(z ) 2π Dµ (z ) ϕ (z )
ℓ1
|ϕ ′ (t )| |dt | ϕ(z ) = z−t 2π iDµ (z )
ℓ1
ϕ ′ (t )dt ϕ(t )(z − t )
′
=
Dµ (z )
,
(45)
whenever Dµ (z ) ̸= 0. But this and the normality of (44) imply that Dµ (z ) ̸= 0 for all z ∈ Ω1 , and that the limit in (45) holds uniformly on closed subsets of Ω1 . We now prove Proposition 2.1, which is actually a straightforward consequence of Lemma 3.1. Proof of Proposition 2.1. From the remarks following the proof of [9, Proposition 3], we obtain
∥µa ∥ =
∞
|qj (a)|2 ,
j =0
so that
∥µa ∥ −
n −1
|qj (a)|2 =
∞
|qj (a)|2 .
j =n
j =0
The proposition now follows by substituting Eq. (29) into Eq. (12).
In order to prove the theorems dealing with the spectral transforms, it is convenient to establish first the following lemma. Lemma 3.3. Let {λn }n≥0 be a sequence of positive numbers with the property that for every s < 1, there exists another sequence {cs,n }n≥0 of positive terms such that sj ≤
cs,n+j cs,n
≤ 1,
j, n ≥ 0,
lim λn /cs,n > 0.
n→∞
(46)
Then, lim
n→∞
∞ λn+j j =0
λn
z j = lim
n→∞
n λn j =0
λj
z n−j =
1 1−z
(47)
E. Miña-Díaz, B. Simanek / J. Math. Anal. Appl. 407 (2013) 290–304
301
locally uniformly on |z | < 1, and lim
n λn
n→∞
λj
j =0
= ∞.
(48)
Proof. For every n ≥ 0, let us denote by fn (z ) the sum under the first limit in (47). By writing
λn+j = λn
λn+j
cs,n
cs,n+j
λn
cs,n+j
cs,n
,
(49)
we see that the conditions (46) imply that λn+j /λn is uniformly bounded in n and j, so that the sequence {fn }∞ n=0 is a normal family of well-defined analytic functions on |z | < 1, and it suffices to prove that limn→∞ fn (z ) = (1 − z )−1 for z = |z | ∈ [0, 1). Again, from (49) and (46) we deduce that for every ϵ > 0, we can find and index N such that for all n ≥ N, 1−ϵ 1 − s|z |
≤ fn (|z |) ≤
1+ϵ 1 − |z |
,
so that 1−ϵ 1 − s|z |
1+ϵ
≤ lim inf fn (|z |) ≤ lim sup fn (|z |) ≤ n→∞
1 − |z |
n→∞
,
and the first limit in (47) follows after letting ϵ → 0 and s → 1. Let us now denote by gn (z ) the sum under the second limit in (47). Writing
λn = λj
λn cs,n
cs,j
λj
cs,n cs,j
,
we see that the sequence {gn }∞ n=0 is a normal family on |z | < 1, and that moreover, for every ϵ > 0, we can find an index N such that for all n ≥ N, n λ n n −j 1+ϵ (1 − ϵ)(1 − |zs|n−N +1 ) ≤ |z | ≤ . 1 − |zs| λ 1 − |z | j j =N
(50)
Taking the limit as n → ∞ we get 1−ϵ 1 − s|z |
≤ lim inf gn (|z |) ≤ lim sup gn (|z |) ≤ n→∞
n→∞
1+ϵ 1 − |z |
,
|z | < 1,
(51)
and the second limit in (47) follows after letting ϵ → 0 and s → 1. Notice that, the lower bound for the lim inf in (51) is equally valid for |z | = 1, whence (48) follows.
Later, we will also use the fact that under the assumptions of Lemma 3.3, and for any sequence of functions {νn (z )}n≥0 , we have lim νn (z ) = 0 ⇒ lim
n→∞
n→∞
n λn j =0
λj
z n−j νj (z ) = 0,
(52)
with both limits understood as holding locally uniformly on |z | < 1. This follows quite trivially from the inequalities (50). Proof of Proposition 2.2. By the maximum modulus principle, it suffices to show that (14) holds true uniformly on any contour ℓs where r < s < 1. By Proposition 2.1, we have that for all z ∈ ℓs ,
Φn (z ; µ) Φn+1 (z ; µa ) = − (z − a)ϕ(z )n ϕ(z )n
Qn (a; µ)ϕ(z )−n
∞
qn+j (a; µ)φn+j (z ; µ)
j =0
∞
|qn+j (a; µ)|2
j =0
One of our assumptions is that
Φn (z ; µ) = ϕ(z )n [Dµ (z ) + ϵn (z )],
ϵn (z ) → 0, z ∈ Ωr ,
and according to Theorem 1.1(e), we can write
κn (µ)ϕ(a)n+1 qn (a; µ) = h(a) + en ,
n ≥ 0,
.
(53)
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where h(a) = ϕ ′ (a)/Dµ (a) and limn→∞ en = 0. Therefore, the second fraction in the right-hand side of (53) can be rewritten in the form (we drop µ from the notation)
ϕ(a)n+1 κn qn (a)
∞ κn+j ϕ(a)n+j+1 qn+j (a) ϕ(a)−j ϕ(z )−n Φn+j (z ) j =0
∞ j =0
(h(a) + en )
κn2 |ϕ(a)|−2j |ϕ(a)n+1+j κn+j qn+j (a)|2 κn2+j ∞
(h(a) + en+j )(ϕ(z )/ϕ(a))j [Dµ (z ) + ϵn+j (z )]
j =0
=
∞ j =0
.
κn2 (µ)|ϕ(a)|−2j |h(a)+en+j |2 κn2+j (µ)
(54)
Since by Theorem 1.1(b) and (35), we have that for every s ∈ [ρ, 1)
κn−2 (µ) > 0, |ϕ|2n dµ Ωs
lim
n→∞
we can apply Lemma 3.3 with the choice λn = κn−2 (µ) and cs,n = Ω |ϕ|2n dµ to get from the first limit in (47) that this last s expression (54) converges uniformly on ℓs as n → ∞, and computing its limit yields
lim
n→∞
Φn+1 (z ; µa ) (z − a)Dµ (z ) Dµ (z )(|ϕ(a)|2 − 1)(z − a) = + , ϕ(z )n+1 ϕ(z ) ϕ(z ) ϕ(a) (ϕ(z ) − ϕ(a))
which is the desired expression.
Proof of Proposition 2.4. Suppose first that a ∈ Ω 1 , and let us rewrite (22) in the form n
κj2 (µ)
[ϕ(a)ϕ(z )]j−n Dµ,j (a)Dµ,j (z ) 2 (z − a)Φn (z ; µ ) Φn+1 (z ; µ) Φn+1 (a; µ) j=0 κn (µ) = − . n κ 2 (µ) ϕ(z )n ϕ(z )n ϕ(a)n j 2(j−n) |D 2 |ϕ( a )| ( a )| µ, j κ 2 (µ) c a
j =0
(55)
n
We can now apply Lemma 3.3, with the same choice as above λn = κn−2 (µ) and cs,n = (55) to get from (48), (52), and the second limit in (47) that
Ωs
|ϕ|2n dµ, and take the limit in
ϕ(z )Dµ (z )(|ϕ(a)|2 − 1) (z − a)Φn (z ; µca ) = ϕ(z )Dµ (z ) − , n n→∞ ϕ(z ) ϕ(a)ϕ(z ) − 1
(56)
lim
uniformly on closed subsets of |ϕ(z )| > max{r , |ϕ(a)|−1 }. Notice that, this last formula is equally valid for a ∈ ℓ1 , since in that case the fraction in the right-hand side of (56) is identically zero. Suppose now that a ∈ G. Again, by the maximum modulus principle, it suffices to show that (23) holds true for z ∈ ℓs , ρ(r , a) < s < 1. Fix s′ and η with ρ(r , a) < s′ < s < η < 1. Since s′ > r and a lies interior to the level curve ℓs′ , our assumption (3) and the behavior of κn (µ) given by (4) imply that there exist constants Ms , Ms′ and Mη such that
|Φn (z ; µ)| ≤ Ms sn ,
|Φn (a; µ)| ≤ Ms′ (s′ )n ,
κn2 (µ) ≤ Mη η−n ,
z ∈ ℓs , n ≥ 0.
Then from (22) we obtain that for every z ∈ ℓs and n ≥ 0, n κj2 (µ)|Φj (a; µ)| |Φj (z ; µ)| s′ Ms′ (s′ /s)n (z − a)Φn (z ; µca ) Φn+1 (z ; µ) j =0 ≤ − n ϕ(z )n ϕ(z )n κj2 (µ)|Φj (a; µ)|2 j =0
≤
s′ Ms Mη Ms2′
κ02 (µ)
(s′ /s)n
n ′ j ss j=0
η
,
and (23) for the case a ∈ G follows since the right-hand side of this inequality converges to 0 as n → ∞.
Proof of Proposition 2.5. From our assumption (25) and Hurwitz’s theorem, we see that for all n sufficiently large, Φn (z ; µ) ˜ has exactly J zeros on Ω 1 , say zn,1 , . . . , zn,J , which are simple and can be enumerated in such a way that zn,j → aj as n → ∞.
E. Miña-Díaz, B. Simanek / J. Math. Anal. Appl. 407 (2013) 290–304
J
303
J
Set pJ (z ) := j=1 (z − aj ), pn,J (z ) := j=1 (z − zn,j ). Then, by the extremal property of the monic orthogonal polynomials, and since µ ≤ µ ˜ , we have that for all n large,
|Φn (z ; µ)| ˜ 2 dµ ˜ ≤
|Φn (z ; µ) ˜ pJ (z )/pn,J (z )|2 dµ ˜
≤ max |pJ (z )/pn,J (z )|2
|Φn (z ; µ)| ˜ 2 dµ.
|Φn (z ; µ)| ˜ 2 dµ ≤
z ∈G
(57)
Since the maximum in (57) converges to 1 as n → ∞, these inequalities combine with (25), (5), and (35), to yield that for every s ∈ (r , 1), lim
n→∞
|Dµ˜ |2 |ϕ|2n dµ |Φn (z ; µ)| ˜ 2 dµ κn−2 (µ) ˜ Ωs = lim = lim n→∞ n→∞ |ϕ|2n dµ |ϕ|2n dµ |ϕ|2n dµ Ωs Ωs Ωs 2 ′ ˜ /Dµ | |ϕ | |dz | ℓ |Dµ > 0. = 1 −2 ′ ℓ1 |Dµ | |ϕ | |dz |
Thus, Lemma 3.3 can and will be applied with the choice λn = κn−2 (µ) ˜ and cs,n = −2 κn (µ) ˜ exists and is finite. Then, suppose first that a ∈ Ω 1 , and let us write (24) in the form
Φn+1 (z ; µ ˜ ) Φn+1 (z ; µ) ˜ Φn+1 (a; µ) ˜ = − · ϕ(z )n+1 ϕ(z )n+1 ϕ(a)n ϕ(z ) u a
m
n j =0
κj2 (µ) ˜ κn2 (µ) ˜
κn−2 (µ) ˜ |ϕ(a)|2n
Ωs
|ϕ|2n dµ. Observe that limn→∞
[ϕ(a)ϕ(z )]j−n Dµ, ˜ j (a)Dµ, ˜ j (z )
+m
n
κj2 (µ)|ϕ( ˜ a)|2(j−n) κn2 (µ) ˜
j =0
. 2 |Dµ, ˜ j (a)|
Using again (48), (52), and the second limit in (47) during the process of taking the limit in this last equality yields lim
n→∞
Φn (z ; µ ˜ ua ) Dµ˜ (z )(|ϕ(a)|2 − 1) = D ( z ) − µ ˜ ϕ(z )n (ϕ(a)ϕ(z ) − 1)
uniformly on closed subsets of |ϕ(z )| > max{r , |ϕ(a)|−1 }. We omit the proof of (27) for the case a ∈ G since it follows from (24) very much in the same way we proved (23) for a ∈ G in the second half of the proof of Proposition 2.4. Proof of Theorem 1.2. The limit (9) follows directly from Propositions 2.2, 2.4 and 2.5 by a trivial mathematical induction argument. If we set K
h(z ) :=
(z − zk )
k=1 L
,
(z − bl )
l =1
then (9) tells us that the measure |h|2 µ satisfies the conditions of Theorem 1.1, so that we can apply the relation (26) of Proposition 2.5 to get
κn (µ) ˆ lim = 2n |h|2 dµ n→∞ |ϕ| Ωρ
2π
−2
L
aj ∈Ω1
|ϕ(bl )|2
l=1
zk ∈Ω1
|ϕ(zk )|2
ℓ1
|ϕ(aj )|2 ,
|h/Dµ |2 |ϕ ′ | |dz |
and (10) follows at once by combining this last equality with (5).
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