KERNEL ESTIMATES
YACIN AMEUR AND JOAQUIM ORTEGA-CERD `A
Abstract. LetQbe a suitable real valued function onCwhich increases sufficiently rapidly asz→ ∞. Ann-Fekete set corresponding toQis a subset{zn1, . . . ,znn}ofCwhich maximizes the weighted Vandermonde determinantQn
i<j
zni−zn j
2e−n(Q(zn1)+···+Q(znn)). It is well known that there exists a compact setSknown as the “droplet” such that the sequence of measuresµn=n−1(δzn1+· · ·+δznn) converges to the equilibrium measure∆Q·1SdAasn→ ∞. In this note we consider a related topic, proving that Fekete sets are in a sense maximally spread out with respect to the equilibrium measure.
In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the Ginibre potentialQ=|z|2we obtain results which hold globally; we conjecture that such global results are true for a wide range of potentials.
In this paper we discuss equidistribution results for weighted Fekete sets in subsets of the plane. More precisely, we show that Fekete sets are maximally spread out relative to a rescaled version of the Beurling–Landau density, in the “droplet” corresponding to the given weight. Our method combines Landau’s idea to relate the density of a family of discrete sets to properties of the spectrum of the concentration operator, with estimates for the correlation kernel of the corresponding random normal matrix ensemble.
1. Fekete sets
1.1. Potentials and droplets. We recapture some notions and results from weighted potential theory. Proofs and further results can be found in [23]. Cf. also [2] and [17] where the setting is more tuned to fit the present discussion.
LetQ:C→R∪ {+∞}be a suitable function (the“potential” or “external field”) satisfying lim inf
z→∞
Q(z)
log|z|2 = +∞.
(In detail: we require in addition that the functionw:=e−Q/2satisfy the mild condition of being an “admissible weight” in the sense of [23], p. 26. This means thatwis upper semi-continuous and the set{w>0}has positive logarithmic capacity.)
We associate toQthe “equilibrium potential”Qbin the following way: Let SHQbe the set of all subharmonic functionsu:C→Rsuch thatu(z)≤log+|z|2+const. andu≤QonC. One defines
2010Mathematics Subject Classification. 31C20; 82B20; 30E05; 94A20.
Key words and phrases. Weighted Fekete set; droplet; equidistribution; concentration operator; correlation kernel.
This work is a contribution to the research program on “Complex Analysis and Spectral Problems” which was conducted at the CRM in Barcelona during the spring semester of 2011. The first author was supported by grants from Magnussons fond, Vetenskapsrådet, SveFum, and the European Science Foundation. The second author was supported by grants MTM2008-05561-C02-01 and 2009 SGR 1303.
1
the equilibrium potential asQ(z)b =sup{u(z) ; u∈SHQ}.The droplet associated toQis the set S={z∈C;Q(z)=Q(z)b }.
This is a compact set; one has that∆Q≥0 onSand that theequilibrium measure
(1.1) dσ(z)=1S(z)∆Q(z)dA(z)
is a probability measure onC. Here we agree that dAis normalized area measure dA= π1dxdy, while∆ = ∂∂ = 14(∂2/∂x2+∂2/∂y2) is the normalized Laplacian; ∂ = 12(∂/∂x−i∂/∂y) and∂ =
1
2(∂/∂x+i∂/∂y) are the complex derivatives.
We will make the standing assumption thatQbeC3-smooth and strictly subharmonic in some neighbourhoodΛofS. In other words, we assume that theconformal metricds2(z)= ∆Q(z)|dz|2is comparable to the Euclidean metric onΛ.
1.2. Fekete sets. Consider the weighted Vandermonde determinant Vn(z1, . . . ,zn)=Y
i<j
zi−zj
2e−n(Q(z1)+···+Q(zn)), z1, . . . ,zn∈C.
A setFn={zn1, . . . ,znn}which maximizesVnis called ann-Fekete setcorresponding toQ. Notice that Fekete sets are not unique.
Equivalently, the setFnminimizes the weighted energy (1.2) Hn(z1, . . . ,zn)=X
i,j
log zi−zj
−1
+n
n
X
j=1
Q(zj) over all configurations{zj}n
j=1 ⊂C. If we think of the pointszjas giving locations fornidentical repelling point charges with total charge 1 confined toC under the influence of the external magnetic fieldnQ, thenHncan be regarded as the the energy of the system.
The following classical result displays some fundamental and well-known properties of Fekete sets.
Theorem 1.1. For any Fekete setFn={zn1, . . . ,znn}holds:
(1) Fn⊂S
(2) Letσbe the equilibrium measure(1.1). We then have convergence in the sense of measures 1
n
n
X
j=1
δzn j→σ, as n→ ∞.
A proof can be found in [23], theorems III.1.2 and III.1.3. (Notice that our assumptions onQ imply thatS=S∗in the notation of [23].) The theorem 1.1 was generalized to line bundles over complex manifolds in [9], [10].
We remark that the property (1) is essential to the analysis in this paper, and that the standard proof of (1) (e.g. in [23]) depends on the “maximum principle for weighted polynomials”, which is reproduced in Lemma 2.8 below.
We will consider related questions concerning the distribution of Fekete points. In a sense, we will prove that these points are maximally spread out with respect to the conformal metric. To quantify this assertion, we introduce some definitions.
Definition 1.2. LetF ={Fn}∞
n=1be a family ofn-Fekete sets. Also letζ=(zn)∞1 be a sequence of points inS. We define thelower Beurling–Landau’s densityofF with respect toζby
D−(F;ζ)=lim inf
R→∞ lim inf
n→∞
#
Fn∩D zn;R/√
n R2∆Q(zn) , and we define the corresponding upper density by
D+(F;ζ)=lim sup
R→∞
lim sup
n→∞
#
Fn∩D zn;R/√
n R2∆Q(zn) . We also put
dn(ζ)=dist(zn,C\S).
Here “dist” denotes the Euclidean distance in the plane, andD(ζ;r) is the open disk with center ζand radiusr.
We have the following theorem.
Theorem 1.3. Putδn=log2n/√
n,and suppose that dn(ζ)≥3δnfor all n. Then
(1.3) D−(F;ζ)=D+(F;ζ)=1.
A proof is given in§2.3.
Remark1.4. The function%n(z)−2defined bynσ(D(z;%n(z)))=1 can be considered as a regularized version of the Laplacian ∆Q(z). Replacing ∆Q(zn) by %n(zn)−2 in our definition of Beurling–
Landau’s densities, it becomes possible to extend our results to cover some situations in which
∆Q=0 at isolated points of the droplet.
1.3. The Ginibre case. The potentialQ(z)=|z|2is known as theGinibre potential. It is easy to see that for this potential, the droplet isS=D, i.e. the closed unit disk with center 0.
Theorem 1.5. Suppose that Q(z) = |z|2. Letζ = (zn) be a sequence inDand assume that the limit L=limn→∞
√
n(1− |zn|)exists. Then (1) If L= +∞, then(1.3)holds (2) If L<+∞, then
(1.4) D−(F;ζ)=D+(F;ζ)= 1
2. A proof is given in§2.4.
Remark 1.6. The condition that the limitLexists is really superfluous and is made merely for technical convenience. Indeed, we can assert that lim infn→∞
√
n(1− |zn|)= +∞then (1.3) holds while if lim supn→∞
√
n(1− |zn|)<+∞then (1.4) holds. These somewhat more general statements can be proved without difficulty by using the arguments below.
1.4. A conjecture. The boundary of a droplet corresponding to a smooth potential is in general a quite complicated set. However, owing to Sakai’s theory [24], it is known that the situation is more manageable for potentialsQwhich arereal-analyticin a neighbourhood of the droplet.
Namely, for a real analytic potentialQ, the boundary ofSis a finite union of real analytic arcs and possibly a finite number of isolated points. The boundary ofSmay also have finitely many singularities which can be either cusps or double-points. This result can easily be proved using arguments from [18], Section 4.
Suppose thatQis real-analytic and strictly subharmonic in a neighbourhood ofS, and assume that∂Shas no singularities. LetS∗denote the setSwith eventual singularities and isolated points removed. Also letζ=(zn)∞1 be a sequence of points inS∗and assume for simplicity that the limit L = limn→∞
√
ndn(ζ) exists, wheredn(ζ) is the distance ofzn to∂S. We conjecture that for any sequenceF ={Fn}of weighted Fekete sets, we have (i) ifL= +∞, thenD−(F, ζ)=D+(F, ζ)=1 and (ii) ifL<+∞, thenD−(F, ζ)=D+(F, ζ)=1/2.
The conjecture is supported by the results of the forthcoming paper [5].
1.5. Earlier work and related topics. The topics considered in this note, as well as our basic strategy, were inspired by the paper [19] by Landau, which concerns questions about interpo- lation and sampling for functions in Paley–Wiener spaces. In particular, our “Beurling–Landau densities” can be seen as straightforward adaptations of the densities defined in [19], and our re- sults below are parallel to those of Landau. The historically interested reader should also consult Beurling’s lecture notes (see the references in [19]), where some of the basic concepts appeared earlier; in fact Landau’s exposition depends in an essential way on Beurling’s earlier work.
In the one-component plasma (or “OCP”) setting, one introduces a temperature 1/β, where β > 0. The probability measure dPβn(z) =(Zβn)−1e−βHn(z)dVn(z) onCnis known as the density of states at the temperature 1/β. Here dVnis Lebesgue measure onCn,Hnis the Hamiltonian (1.2), andZβnis a normalizing constant. One then considers configurationsΨβn={zi}n
1picked randomly with respect toPβn.
Intuitively, Fekete sets should correspond to particle configurations at temperature zero, or rather, the “limiting configurations” as 1/β → 0, although the latter “limit” so far has been understood mostly on a physical level. In this interpretation, the methods of the present note prove that the Beurling–Landau density of temperature zero configurations is in fact completely determined by properties atβ=1. (More precisely: it is determined by the one- and two-point functions ofP1n.)
A more subtle problem is to characterise Fekete sets amongst all configurations of Beurling–
Landau density one. It is believed that a certain crystalline structure will manifest itself (known as the “Abrikosov lattice”). We refer to [14], [25] and the references therein for further details on this topic. A survey of related questions for minimum energy points on manifolds is found in [16].
2. Weighted polynomials and triangular lattices
Our approach combines the method for characterizing Fekete sets and triangular lattices from the papers [21] and [22] with correlation kernel estimates of the type found in [6], [1], [2], [4].
In the Ginibre case, we use the explicit representation of the correlation kernel available for that potential, as well as estimates from the papers [27], [15], [12], and [11].
2.1. Weighted polynomials. LetHnbe the space of polynomialspof degree at mostn−1, normed by
p
2 nQ := R
C
p(z)
2e−nQ(z)dA(z). The reproducing kernel for Hn is Kn(z,w) = Pn−1
j=0ej(z)ej(w), where{ej}n−1
0 is an orthonormal basis forHn.
For our purposes, it is advantageous to work with spaces ˜Hnof weighted polynomials f = p·e−nQ/2, wherepis a polynomial of degree≤n−1, and one defines the norm in ˜Hnas the usual L2(dA)-norm. The reproducing kernel for ˜Hnis given by
Kn(z,w)=Kn(z,w)e−nQ(z)/2−nQ(w)/2.
The functionKnis known as thecorrelation kernelcorresponding to the potentialQ; the reproduc- ing property means that
f(z)=hf,Kn,zi, f ∈H˜n, z∈C,
whereKn,z(ζ)=Kn(ζ,z), and the inner product is the usual one inL2=L2(C,dA).
When ρn is not an integer, we interpret ˜Hρn as the space Hk wherek is the largest integer satisfying k < ρn. All statements below shall be understood in terms of this convention; in particular,Kρn(z,w) :=Kk(z,w)e−k(Q(z)+Q(w))/2.
2.2. Triangular lattices. LetZ=n Zjo∞
j=1be a triangular lattice of points inC. We write Zn={zn1,zn2, . . . ,znmn}.
It will be convenient to introduce some classes of lattices.
Letρ >0. A familyZis said to beρ-interpolatingif there is some constantCsuch that, for all families of valuesc={cn}∞
1 ,cn={cn j}mn
j=1, such that sup
n
1 nρ
mn
X
j=1
cn j
2<∞,
there exists a sequence fn∈H˜ρnsuch that fn(zn j)=cn j, 1≤ j≤mn, and
fn
2≤C 1 nρ
mn
X
j=1
cn j
2.
We say that a familyZisuniformly separatedif there is a numbers>0 such that for any two distinct pointsz,w∈ Znwe have|z−w|>s/√
n. The following simple lemma holds.
Lemma 2.1. Any interpolating family which is contained in S is uniformly separated.
A proof is given in§3.2.
Intuitively, an interpolating family should be “sparse”. We will also need a notion which implies the “density” of a family contained inS. For this purpose, the following classes have turned out to be convenient.
Definition 2.2. WriteS+ =S+D(0;s/√
n),wheresis some fixed positive number. LetZ ⊂Sbe a triangular family. We hay thatZis of classMS,ρifZis uniformly 2s-separated and
Z
S+
f
2 ≤C 1 nρ
X
zn j∈Zn
f(zn j)
2, f ∈H˜nρ
for all largen.
Definition 2.3. Letδn=log2n/√
nand putSn={z∈S; dist(z, ∂S)≥2δn}.We say that a triangular familyZ ⊂Sis of classMSn,ρifZis uniformly separated and
Z
Sn
f
2≤C 1 nρ
X
zn j∈Zn
f(zn j)
2, f ∈H˜nρ
for all largen.
2.3. Results in the interior of the droplet. We have the following lemma.
Lemma 2.4. Letζ=(zn)be a convergent sequence in S withdist(zn, ∂S)≥3δnfor all n. Then (i) IfZis of class MSn,ρ, then D−(Z;ζ)≥ρ,
(ii) IfZisρ-interpolating, then D+(Z;ζ)≤ρ.
A proof is given in Section 5.
WhenFnis a Fekete set, we writeF0
n=Fn∩SnandF0={F0
n}. Lemma 2.5. One has that
(1) F is uniformly separated,
(2) F0isρ-interpolating for anyρ >1, (3) F is of class MSn,ρwheneverρ <1.
A proof is given in Section 6.
Using lemmas 2.4 and 2.5, we infer that forζ=(zn) with dist(zn, ∂S) ≥3δn, we have for any ε >0 thatD−(F;ζ) ≥1−εandD+(F;ζ) ≤1+ε. This finishes the proof of Theorem 1.3, since evidentlyD−≤D+. q.e.d.
2.4. The Ginibre case. Now letQ=|z|2so thatS=D, and fix a convergent sequenceζ=(zj) in Dsuch that the limitL=limn→∞
√
n(1− |zn|) exists.
Lemma 2.6. Suppose that Q=|z|2, and letZbe a triangular family contained inD.
(1) IfZis of class MD,ρ, then
D−(Z;ζ)≥
ρ if L= +∞, ρ/2 if L<+∞. (2) IfZisρ-interpolating, then
D+(Z;ζ)≤
ρ if L= +∞, ρ/2 if L<+∞. A proof is given in§8.3.
Lemma 2.7. LetF ={Fn}be a family of Fekete sets with respect to the potential Q=|z|2. ThenF is of class MD,ρfor anyρ <1andρ-interpolating for anyρ >1.
A proof is given in§8.5.
To finish the proof of Theorem 1.5 it suffices to combine Lemma 2.6 and Lemma 2.7. q.e.d.
2.5. Auxiliary lemmas. We state a couple of known facts which are used frequently in the following. The following uniform estimate is well-known (see e.g. [23]).
Lemma 2.8. Let f ∈H˜nand z∈C\S. Assume that f
≤1on S. Then f(z)
≤e−n
Q(z)−bQ(z)
/2, z∈C.
(Proof: Let f =p·e−nQ/2. The assumption gives that 1nlog p
2is a subharmonic minorant ofQ which grows no faster than log|z|2+const. asz→ ∞. Thus 1nlog
p
2≤Q.)b We will also use the following well-known pointwise-L2estimate.
Lemma 2.9. Let f = ue−nQ/2 where u is holomorphic and bounded in D(z0;c/√
n) for some c > 0.
Suppose that∆Q(z)≤K for all z∈D(z0;c/√
n). Then
(2.1)
f(z0)
2≤n·eKc2c−2 Z
D(z0;c/√ n)
f
2dA.
In particular, ifZis2s-separated, then for all f ∈H˜n
(2.2) 1
n X
zn j∈Ω
f(zn j)
2 ≤Cs−2 Z
Ω+
f(ζ)
2dA(ζ),
where C depends only on the upper bound of∆Q on S+andΩ+=n
ζ∈C; dist(ζ,Ω)≤s/√ no
. A proof of (2.1) can be found e.g. in [2], Section 3. The estimate (2.2) is immediate from this.
We will also need the following lemma on uniform estimates and “off-diagonal damping” for correlation kernels.
Lemma 2.10. (i) There is a constant C such that for all z,w∈C,
|Kn(z,w)| ≤Cne−n(Q(z)−bQ(z))/2e−n(Q(w)−bQ(w))/2.
(ii) Suppose that z∈S and letδ=dist(z, ∂S). There are then positive constants C and c such that
|Kn(z,w)| ≤Cnexp
−c
√
nmin{|z−w|, δ}
·e−n(Q(w)−bQ(w))/2, w∈C.
Part (i) is standard, see e.g. [2], Sect. 3. For a proof of (ii) we refer to [2], Corollary 8.2 (which also shows that the constantccan be taken proportional to inf{p
∆Q(z); z∈S}).
2.6. Notation. We use the same letterKto denote a kernelK(z,w) and its corresponding integral operator K(f)(z) = R
C f(w)K(z,w)dA(w). We will denote by the same symbol “C” a constant independent ofn, which can change meaning as we go along. The notation “An -Bn” means thatAn≤CBn. We shall write
(2.3) An(z)=D(z;R/√
n) , A+n(z)=D(z; (R+s)/√
n) , A−n(z)=D(z; (R−s)/√ n).
3. Preliminary estimates
In this section, we discuss gradient estimates for weighted polynomials; these will be useful in the following. In particular they imply that interpolating families are uniformly separated.
3.1. Inequalities of Bernstein type. The following lemma is analogous to Lemma 18 in [20].
Lemma 3.1. Let p be a polynomial of degree at most n. Fix a point z such that p(z),0and|∆Q(z)|<K.
Then (3.1)
∇ p
e−nQ/2
(z)
≤C
√ n
pe−nQ/2 L∞, and
(3.2)
∇ p
e−nQ/2
(z)
≤Cn
pe−nQ/2 L2, where the constant C depends only on K.
Proof. LetHz(ζ)=Q(z)+2∂Q(z)·(ζ−z)+∂2Q(z)(ζ−z)2andhz(ζ)=ReHz(ζ),so that Q(ζ)=hz(ζ)+ ∆Q(z)|ζ−z|2+O(|z−ζ|3).
In particular, there is a constantCsuch that
(3.3) n|Q(ζ)−hz(ζ)| ≤C when |ζ−z| ≤1/√ n, whereCdepends only onK.
Now observe that, (3.4)
∇ p
e−nQ/2
(ζ) =
p0(ζ)−n·∂Q(ζ)·p(ζ)
e−nQ(ζ)/2, and
∇ p
e−nhz/2
(ζ) =
p0(ζ)−n·∂hz(ζ)·p(ζ)
e−nhz(ζ)/2=
=
d
dζ(pe−nHz/2)(ζ) . (3.5)
The expressions (3.4) and (3.5) are identical whenζ=z.
By Cauchy’s estimate applied to the circleC1/√n(z) with centerzand radius 1/√ n, (3.6)
d
dζ(pe−nHz/2)(z) = 1
2π Z
C1/√n(z)
p(ζ)e−nHz(ζ)/2 (z−ζ)2 dζ
≤ n 2π
Z
C1/√n(z)
p(ζ)
e−nhz(ζ)/2|dζ|. In view of (3.3), the right side can be estimated by a constant depending only onK, times
(3.7) n
Z
C1/√n(z)
p(ζ)
e−nQ(ζ)/2|dζ|.
To prove (3.1), it suffices to notice that (3.7) can be estimated by 2π√ n
pe−nQ/2 L∞. Next notice that, by Lemma 2.9,
p(ζ)
2e−nQ(ζ) ≤C0n Z
D(ζ;1/√ n)
p(ξ)
2e−nQ(ξ)dA(ξ)≤C0n
pe−nQ/2
2 L2
with another constantC0depending only onK. We conclude that
∇ p
e−nQ/2
(ζ) -n
√ n
pe−nQ/2 L2
Z
C1/√n(z)
|dζ|-n
pe−nQ/2 L2,
with a constant depending only onK. This proves (3.2).
3.2. Proof of Lemma 2.1. LetZbe an interpolating family contained inS. (W.l.o.g. putρ=1.) Fix an index j, 1≤ j≤mn. SinceZis interpolating, we can find a function f = fn ∈H˜nsuch that f(zn j0) = δj j0 andkfk2 ≤ C/n. Letδ > 0 be small enough thatD(zn j;δ) ⊂ Λ. Also assume w.l.o.g. that a pointzn j0satisfies
zn j−zn j0
< δ; if there is no such j0there is nothing to prove.
Evidently,
1= f(zn j)
−
f(zn j0)
≤
∇
f
L∞(Λ)
zn j−zn j0
. Thus Lemma 3.1 gives
1≤C1n f
zn j−zn j0
≤CC1
√ n
zn j−zn j0
.
This proves thatZiss-separated withs=1/(CC1).
4. The spectrum of the concentration operator
LetΩbe a measurable subset of the plane. Theconcentration operatorKΩnρis defined by KΩnρ(f)(z)=
Z
Ωf(w)Knρ(z,w)dA(w)=Knρ(1Ω·f)(z).
This is a positive contraction on ˜Hnρ.
In this section, we apply a technique which relates the spectrum of the concentration operator to the number of points inΩ∩ ZnwhenZis either an interpolating family or an M-family; the technique essentially goes back to Landau’s paper [19]. We here follow the strategy in [22], in a suitably modified form.
We first turn toM-families. We will consider the cases ofMSand ofMSnfamilies separately.
4.1. MS,ρ-families. Fix a pointz∈Sand letλnρj =λnρj (z) denote the eigenvalues ofKAnρn(z): ˜Hnρ→ H˜nρ, taken in decreasing order. Letφnjρbe corresponding normalized eigenvectors. We write
N+nρ=N+nρ(z)=# Zn∩A+n(z). (See (2.3) for the definitions of the setsAnandA+n.)
Lemma 4.1. Suppose thatZ ⊂S is of class MS,ρ. There is then a constantγ <1and a number n0such that for all z∈S and n≥n0, we haveλnρN+
nρ(z)+1 < γ.
Proof. W.l.o.g. put ρ = 1. Fix z ∈ S and suppose that f ∈ H˜n is such that f(zn j) = 0 when zn j∈A+n(z). SinceZis 2s-separated,
(4.1)
Z
S+
f
2≤C1 n
X
zn j∈S\A+n(z)
f(zn j)
2≤Cs−2 Z
S+\An(z)
f
2.
Now define f =PNn++1
j=1 cnjφnj,where the numberscnj (not all zero) are chosen so that f(zn j)=0 for allzn j∈A+n(z). This is possible for all largen, because the separation ofZensures thatN+n ≤C for some constantC=C(R,s).
Since the operatorKnis the orthogonal projection ofL2onto ˜Hn, we have
Nn++1
X
j=1
λnj cnj
2=D
KAnn(z)f,fE
=D
1An(z)·Kn(f),Kn(f)E
= Z
An(z)
f
2dA.
(4.2)
We infer by means of (4.1) and (4.2) that λnNn++1
N+n+1
X
j=1
cnj
2
≤
N+n+1
X
j=1
λnj cnj
2= Z
S+
− Z
S+\An(z)
! f
2dA
≤ 1−s2 C
! Z
S+
f
2≤ 1−s2 C
! f
2≤ 1−s2 C
!Nn++1
X
j=1
cnj
2,
which proves thatλN+n+1≤1−s2/C.
Notice that the separability ofZimplies that #(Zn∩(D(z; (R+s)/√
n)\D(z;R/√
n))) ≤ CR.
Therefore Lemma 4.1 implies the estimate
(4.3) #(Zn∩D(z;R/√
n))≥#{j; λnρj ≥γ}+O(R), where theO-constant is independent ofn.
4.2. MSn,ρ-families. We now modify the construction in the previous subsection.
Recall that Sn = {z ∈ S; dist(z, ∂S) ≥ 2δn} where δn = log2n/√
n. Fix a sequence zn ∈ Sn
satisfying
dist(zn, ∂S)≥3δn.
As before, we consider the eigenvaluesλnρj (decreasing order) and corresponding eigenfunctions φnρj of the concentration operatorKAnρn(zn). We will use the following lemma.
Lemma 4.2. For any positive integer K there is a constant CKand a number n0 =n0(R)such that for all j λnjρ
Z
C\Sn
φnjρ
2
≤CKn−K, n≥n0.
Proof. W.l.o.g. letρ=1. Choosen0such that dist(An(zn),C\Sn)≥δn/2 whenn≥n0. By Lemma 2.10, we then have an estimate
|Kn(ζ,w)| ≤Cne−clog2ne−n(Q(w)−bQ(w))/2, w∈C\Sn, ζ∈An(zn), wherecandCare positive constants. This gives
Z
C\Sn
Z
An(zn)
φj(w)Kn(ζ,w)φj(ζ)dA(ζ)dA(w)
≤Cne−clog2n Z
φj(z)
e−n(Q(z)−bQ(z))/2
!2
≤
≤CKn−Kkφjk2=CKn−K, where we have used the Cauchy-Schwarz inequality and thatR
e−n(Q−Q)b =1+o(1).
Lemma 4.3. Suppose thatZ ⊂ S is of class MSn,ρ. There is then a constant γ < 1 and a number n0 =n0(R)such that for all z∈S satisfyingdist(z, ∂S)≥3δnand all n≥n0, we haveλnρNn+ρ(z)+1< γ.
Proof. W.l.o.g. putρ=1. Assume that f ∈H˜nis such thatf(zn j)=0 for allzn j∈ Zn∩A+n(z). Then sinceZis 2s-separated for a sufficiently smalls,
Z
Sn
f
2≤C1 n
X
zn j∈S\A+n(z)
f(zn j)
2 ≤Cs−2 Z
S+\An(z)
f
2dA.
(4.4)
We again define f =PN+n+1
j=1 cnjφnj,where the numberscnj (not all zero) are chosen so thatf(zn j)=0 for allzn j∈ Zn∩A+n(z).
This time, observe that Lemma 4.2 and the Cauchy-Schwarz inequality implies λnN+
n+1
Z
C\Sn
f
2≤
N+n+1
X
j,k=1
qλnjλnk Z
C\Sn
cnjφnj·cnkφnk
≤
≤CKn−K
N+n+1
X
j,k=1
cnjcnk
≤CKn−K(Nn++1)
Nn++1
X
j=1
cnj
2
≤C0n−k
N+n+1
X
j=1
cnj
2.
In view of (4.4), we now conclude that λnN+
n+1 Nn++1
X
j=1
cnj
2
≤
Nn++1
X
j=1
λnj cnj
2 = Z
S+
− Z
S+\An(z)
! f
2dA≤
≤ 1−s2 C
! Z
Sn
f
2+ Z
S+\Sn
f
2≤ 1−s2 C
! f
2+ Z
S+\Sn
f
2≤
≤
1−s2
C + CKn−K λnN+
n+1
Nn++1
X
j=1
cnj
2,
whereCKdepends only onK,R, ands.
Withα=1−s2/C, this impliesλnN+ n+1 < p
α2+4CKn−K.Thus if we defineγas any number in the interval (α,1), we obtainλnN+
n+1≤γfor allnlarge enough.
As a corollary, we obtain the following estimate: Let Zbe as in Lemma 4.3. Then for all n≥n0(R)
(4.5) #(Zn∩D(z;R/√
n))≥#{j; λnjρ≥γ}+O(R), where theO-constant is independent ofn.
4.3. Interpolating families. Assume thatZbe aρ-interpolating sequence contained inS, and let 2sbe a separation constant forZ. We can w.l.o.g. assume thatρ=1.
Fixz∈ S. We defineInas the set of indicesjsuch thatzn j ∈A−n(z) and letNn−=N−n(z) be the cardinality ofIn. By the separation we have a uniform boundNn−≤C=C(R,s).
Now let{cj}mn
1 be a sequence withcj=0 whenj<In. SinceZis interpolating we can choose fn j ∈ H˜n such that fn j(zn j0)= δj j0 and
fn j
2 ≤C/nfor allnand j. The functions fn j, j∈ Inare linearly independent and span anN−n-dimensional subspace of ˜Hn. We denote this subspace by
F=spann
fn j; j∈ Ino . Note that an arbitrary f =P
j∈Incjfn j∈Fsatisfies
f
2≤CNn−1 n
X
j∈In
cj
2 ≤C01 n
X
j∈In
f(zn j)
2.
Applying (2.2) now gives f
2≤C Z
An(z)
f
2=CD
KAnn(z)f,fE .
Withδ=1/C, we have shown that
hKAnn(z)f,fi
hf,fi ≥δ, f ∈F, f .0.
Letλnj be the eigenvalues of the operatorKAnn(z) on ˜Hnarranged in decreasing order. By the Weyl–Courant lemma (see [13], p. 908) we have
λnj−1≥ inf
g∈Ej
hKAnn(z)g,gi hg,gi ,
whereEjranges over all j-dimensional subspaces of ˜Hn. Since dimF=N−n, we obtainλnN−
n−1 ≥δ.
The construction can obviously be carried out forρ ,1 as well. We have proved the following lemma.
Lemma 4.4. Suppose thatZisρ-interpolating, and letλnjρbe the eigenvalues of the operatorKAnρn(z)on H˜nρ, where z ∈ S. Also let N−nρbe the number of points inZn∩A−n(z). Then there is a numberδ >0 independent of n and z such that
#{j; λnρj ≥δ} ≥Nnρ− −1.
Next notice that sinceZis 2s-separated (Lemma 2.1), there is a constantCsuch that
#(Zn∩D(z;R/√
n))−Nnρ− ≤C(R2−(R−s)2)/s2. Using Lemma 4.4, we conclude that
(4.6) #(Zn∩D(z;R/√
n))≤O(R)+#{j; λnρj ≥δ}, as R→ ∞, where theO-constant is independent ofn.
5. Beurling–Landau densities ofMfamilies and of interpolating families
In this section, we prove Lemma 2.4. Our proof depends partly on trace estimates for the concentration operator, which are proved in Section 7.
5.1. Proof of Lemma 2.4(i). Let Z ⊂ Sbe of class MSn,ρ, and letζ = (zn) be a sequence with dist(zn,C\S)≥3δn.
Consider the eigenvalues λnjρ = λnjρ(zn) of the concentration operator KAnρn(zn), and putµn = Pmn
j=1δλnρ
j whereδzis the Dirac measure atz. We then have trace
KAnρn(zn)
= Z 1
0
xdµn(x) , trace
KAnn(zn)◦KAnn(zn)
= Z 1
0
x2dµn(x).
Letγandn0be given by Lemma 4.3. We then have, for alln≥n0(R),
#{j;λnjρ> γ}= Z 1
γ dµn(x)≥ Z 1
0
xdµn(x)− 1 1−γ
Z 1
0
x(1−x)dµn(x)=
=trace KAnρn(zn)
− 1 1−γ
htrace KAnρn(zn)
−trace
KAnρn(zn)◦KAnρn(zn)
i.
By the estimate (4.5) followed by the trace estimates in lemmas 7.1 and 7.3, we now get lim inf
n→∞
#(Zn∩D(zn;R/√ n))
R2∆Q(zn) ≥lim inf
n→∞
#{j;λnρj > γ}+O(R) R2∆Q(zn) ≥
≥lim inf
n→∞
trace
KAnρn(zn)
R2∆Q(zn) − 1 1−γ
trace KAnρn(zn)
−trace
KAnρn(zn)◦KAnρn(zn)
R2∆Q(zn)
+O(1/R)=
=ρ(R+s)2/R2+O(1/R).
SendingR→ ∞, we obtainD−(Z;ζ)≥ρ, and the proof of Lemma 2.4(i) is finished.
5.2. Proof of Lemma 2.4(ii). LetZbe aρ-interpolating family and letζ=(zn) be a sequence with zn∈Snfor alln. Again letλnjρbe the eigenvalues of the concentration operatorKAnρn(zn).
Letµnbe the measureµn=Pmn
j=1δλnρ
j . Then for anyδ∈(0,1) {j;λnρj ≥δ}=
Z 1
δ dµn(x)≤ Z 1
0
xdµn(x)+1 δ
Z 1
0
x(1−x)dµn(x).
In view of the estimate (4.6), we can pickδ=δ(R,s)>0 so that
#(Zn∩D(z;R/√
n))≤O(R)+trace KAnρn(zn)
+1 δ
htrace KAnρn(zn)
−trace
KAnρn(zn)◦KAnρn(zn)i . Forzn∈Sn, the trace estimates in lemmas 7.3 and 7.1 now imply
lim sup
n→∞
#(Zn∩D(zn;R/√ n)) R2∆Q(zn) ≤
≤lim sup
n→∞
trace(KAnρn(zn)) R2∆Q(zn) +1
δlim sup
n→∞
trace KAnρn(zn)
−trace
KAnρn(zn)◦KAnρn(zn)
R2∆Q(zn) +O(1/R)=
=ρ+O(1/R).
LettingR→ ∞now shows thatD+(Z;ζ)≤ρ, which finishes the proof of Lemma 2.4(ii).
6. Equidistribution of the bulk part of aFekete set
In this section we prove Lemma 2.5. The proof is given modulo some estimates for the correlation kernel, which are postponed to the next section.
6.1. Proof of Lemma 2.5(1). LetFn = {zn1, . . . ,znn} be a Fekete set and consider the Lagrange interpolation polynomials
ln j(z)=Y
i,j
(z−zni)/Y
i,j
(zn j−zni).
To avoid bulky notation, from now on writezj:=zn jetc.
Now consider theLeja–Siciak functioncorresponding toFn, Φn(z)=max
lj(z)
2enQ(zj); j=1, . . . ,n . It is known that for allz∈C
(6.1) Φn(z)1/n≤eQ(z)b and Φn(z)1/n→eQ(z)b , as n→ ∞.
We refer to [23],§III.5, notably eq. (5.3) and Corollary 5.3, for proofs of these statements.
Let us write
(6.2) `j(z)=lj(z)e−n(Q(z)−Q(zj))/2, and notice that (6.1) implies that
(6.3)
`j(z)
≤e−n(Q(z)−Q(z))/2b .
The following lemma concludes our proof for part (1) of Lemma 2.5.
Lemma 6.1. LetF ={Fn}be a family of Fekete sets. ThenF is uniformly separated.
Proof. By (6.3) we havek`jkL∞≤1 for allj. Hence Lemma 3.1 implies that there is a neighbourhood ΛofSsuch that
∇ `j
L∞(Λ)
≤C
√ n for some constantCindependent ofnand j.
Fixzn j∈ Fnand assume that a pointznk ∈ Fnis sufficiently close tozn j. Then 1=
`j(zn j)
−
`j(znk) ≤
∇
`j
L∞(Λ)
zn j−znk
≤C
√ n
zn j−znk
.
We have shown thatF is uniformly separated with best separation constant≥1/C.
6.2. Proof of Lemma 2.5(2). We now modify the weighted Lagrangian polynomials`j(6.2), by multiplying by certain “peak polynomials”, to localize to a small neighbourhood ofzj.
Takeε >0 small; consider the corresponding kernelKεn(z,w), and put
(6.4) Lj(z)= Kεn(z,zj)
Kεn(zj,zj)
!2
·`j(z).
These are weighted polynomials of degree (1+2ε)n; evidentlyLj(zk)=δjk.We have the following lemma.
Lemma 6.2. There is a constant C depending onεbut not on n such that for all zj∈ Fn∩Sn, we have
Lj
L1≤ C
n. Proof. Fixzj∈ Fn∩Sn. By (6.1) we have
Lj(z)
≤
Kεn(z,zj)
2
Kεn(zj,zj)2e−n(Q(z)−bQ(z))/2.
Using the asymptotics in Lemma 7.4 and the fact thatQb≤Qeverywhere, we conclude that
Lj
L1≤ C (εn)2
Z
C
Knε(z,zj)
2dA(z)= C
(εn)2Knε(zj,zj)≤ C0
εn.
Lemma 6.3. Let
Fn(z)= X
zj∈Sn
Lj(z)
.
There are then constants C=C(ε,s)and n0=n0(ε)such thatkFnkL∞≤C when n≥n0.
Proof. Forzj∈Snwe haveKn(zj,zj)≥cnwherec>0, by Lemma 7.4. Using Lemma 2.10 we find that
Lj(z)
≤CVj(z), zj∈Sn, z∈C, where
(6.5) Vj(z)=exp
−c
√nεminn z−zj
, δn
o, wherecis a positive constant.
Observe thatVj(z)≤e−c
√εlog2n≤1/nwhen z−zj
≥δnandnis large enough. This gives that there isn0=n0(ε) such that
Fn(z)≤C X
zj∈D(z;δn)
Vj(z)+1, n≥n0.
Now whenzj ∈ D(z;δn) we haveVj(z) = e−c
√nε|z−zj|. Hence when w−zj
≤ s/√
n we have Vj(z)≤Ce−c
√ nε|z−w|
, whereC=ecs
√ε. This gives that Vj(z)≤Cn
s2 Z
D(zj;s/√ n)
e−c
√ nε|z−w|
dA(w).
By the separation, we then obtain that, whenn≥n0, Fn(z)≤1+Cn
s2 Z
C
e−c
√nε|z−w|
dA(w)=1+C 1 s2ε
Z
C
e−c|ζ|dA(ζ)<∞.
The proof of the lemma is finished.
The following lemma concludes our proof for part (1) of Lemma 2.5.
Lemma 6.4. Let F = {Fn}∞
n=1 be a sequence of Fekete sets. Then the triangular familyF0 given by F0
n=Fn∩Snis(1+2ε)-interpolating for anyε >0.
Proof. Write Fn∩Sn = {zn1, . . . ,znmn} and take a sequence c = (cj)mj=n1. Consider the operator T:Cmn →L1+L∞defined byT(c)=P
jcjLj, whereLjare given by (6.4). In view of Lemma 6.2 kTk
`1mn→L1≤supkLjkL1 ≤C/n, and by Lemma 6.3,
kTk`∞
mn→L∞≤ kFnkL∞≤C.
By the Riesz–Thorin theorem, we conclude that kTk`2
mn→L2≤C/√ n.
We have shown that, if f =T(c), then f ∈H˜n(1+2ε), f(zn j)=cjfor allj≤mnand Z
f
2≤ C n
mn
X
j=1
f(zn j)
2.
I.e.,F0is (1+2ε)-interpolating.
![The proof turns out to be a fairly straightforward extension of the proof of the main result in [4]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)