Table Of ContentORTHOGONAL POLYNOMIALS ON SEVERAL
INTERVALS:
ACCUMULATION POINTS OF RECURRENCE
0 COEFFICIENTS AND OF ZEROS
1
0
2 F. PEHERSTORFER1
n
a
4 J ωA(b∞st)r=act(ω.1L(∞et)E,..=.,ω∪l−lj=1(1∞[a2)j)−,1w,ah2ejr]e,aω1j(<∞a)2i<s t.h..e<haar2ml,oln≥ic2maenadsusreet
of [a2j−1,a2j] at infinity. Let µ be a measure which is on E absolutely
] continuousandsatisfiesSzeg˝o’s-conditionandhasatmostafinitenum-
A ber of point measures outside E, and denote by (Pn) and ( n) the or-
Q
C thonormalpolynomialsandtheirassociatedWeylsolutionswithrespect
h. t√oλd1µ+,nysa−t1i+sfny.inWg ethsehroewcutrhraentctehererlaectiuornre√ncλe2+coneyffi1+cnie=nts(xha−veα1t+onp)oylong−i-
at callythesameconvergencebehaviorasthesequence(nω( ))n∈N mod-
m ulo 1; More precisely, putting (αl1−+1n,λl2−+1n)= (α[l−21]+1+n∞,..., α1+n,...,
[ α−[l−22]+1+n,λ[l−22]+2+n,...,λ2+n,...,λ−[l−21]+2+n)weprovethat(αl1−+1nν,
1 λl2−+1nν)ν∈N converges if and only if (nνω(∞))ν∈N converges modulo 1
v and we give an explicit homeomorphism between the sets of accumu-
8 lation points of (αl−1,λl−1) and (nω( )) modulo 1. As one of the
1+n 2+n ∞
7 consequences there is a homeomorphism from the so-called gaps Xl−1
j=1
04 ([a2j,a2j+1]+∪[a2j,a2j+1]−)ontheRiemannsurfacey2 =Q2j=l1(x−aj)
. into the set of accumulation points of the sequence (αl1−+1n,λl2−+1n) if the
1
harmonic measures ω1( ),...,ωl−1( ), 1 are linearly independent over
0 ∞ ∞
therationalnumbersQ.Furthermoreitisdemonstrated,looselyspeak-
0
ing, that the convergence behavior of the sequence of recurrence coef-
1
ficients (αl−1, λl−1) and of the sequence of zeros of the orthonormal
: 1+n 2+n
v polynomialsandWeylsolutionsoutsidethespectrumistopologicallythe
Xi same. Theaboveresultsareprovedbyderivingfirstcorrespondingstate-
mentsfortheaccumulationpointsofthevectorofmomentsofthediago-
ar nalGreen’sfunctions,thatis,ofthesequence(R xPn2dµ,...,R xl−1Pn2dµ,
√λ2+n R xP1+nPndµ, ..., √λ2+nR xl−1P1+nPndµ)n∈N.
ThisworkwassupportedbytheAustrianScienceFundFWF,projectno. P20413-N18.
1The last modifications and corrections of this manuscript were done by the author in
the two months preceding this passing away in November 2009. The manuscript is not
published elsewhere (submitted by P. Yuditskii and I. Moale).
1
2 F.PEHERSTORFER
1. Introduction
Let l N and a R for k = 1,...,2l,a < a < ... < a . Put
k 1 2 2l
∈ ∈
l 2l
E = E , where E = [a ,a ], and H(x) = (x a )
k k 2k−1 2k j
−
k=1 j=1
[ Y
and henceforth let us choose that branch of √H for which H(x) > 0 for
x (a , ). For convenience we set
2l
∈ ∞ p
(√H)+(x) (√H)−(x)
− = ( 1)l−k H(x) for x E
(1) h(x) := 2i − | | ∈ k
0 elsewhere p
where, as usual, f±(x) denote the limiting values from the upper and lower
half plane, respectively. Note that (√H)−(x) = (√H)+(x) for x E. By
− ∈
φ(z,z ) we denote a so-called complex Green’s function for C E uniquely
0
\
determined up to a multiplication constant of absolute value one (chosen
conveniently below), that is, φ(z,z ) is a multiple valued function which is
0
analytic on C E up to a simple pole at z = z , has no zeros on C E and
0
\ \
satisfies φ(z,z ) 1 for z x E or in other words log φ(z,z ) is the
0 0
| | → → ∈ | |
(potential theoretic) Green’s function with pole at z = z C E, as usual
0
∈ \
denoted by g(z,z ). In the case under consideration, as it is known [17, 34],
0
a complex Green’s function may be represented as
z dx
(2) φ(z) := φ(z, ) = exp r (x)
∞
∞ Za2l H(x)!
where r∞ is the unique monic polynomial of degrepe l 1 such that
−
a2j+1 dx
(3) r (x) = 0 for j = 0,...,l 1;
∞
H(x) −
Za2j
hence p
l−1
(4) r (x) = (x c ), c (a ,a ),j = 1,...,l 1.
∞ j j 2j 2j+1
− ∈ −
j=1
Y
Recall that the so-called capacity of E is given by
z
(5) cap(E) = lim
z→∞ φ(z, )
(cid:12) ∞ (cid:12)
(cid:12) (cid:12)
The density of the equilibrium distributi(cid:12)on of E,(cid:12)denoted by ρ, becomes
(cid:12) (cid:12)
′ ′
1 φ φ r (x)
(6) ρ(x) = ( )+(x) ( )−(x) = ∞ .
2πi φ − φ πh(x)
!
By ω(z,B,C E) we denote the harmonic measure of B E with respect
\ ⊆
to C E at z, which is that harmonic and bounded function on C E which
l
\ \
ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 3
satisfies for ξ E that lim ω(z,B,C E ) = i (ξ), where i denotes the
l z→ξ l B B
∈ \
characteristic function of B. For abbreviation we put
ω(z,E ;C E) = ω (z).
k k
\
Recall that, k = 1,...,l,
a2k
ω ( ) = ρ(x)dx.
k
∞
Za2k−1
In the following we say that the function w is on E from the Szego˝-class,
written w Sz(E), if
∈
(7) log(w(x))ρ(x)dx > .
−∞
ZE
In this paper we consider positive measures of the form
m
(8) dµ(x) = w(x)dx+ µ δ (x)
j dj
j=1
X
where w Sz(E), the mass points d , j = 1,...,m, are from R E and
j
∈ \
mutually disjoint and µ R+. By P we denote the polynomial of degree
j n
∈
n orthonormal with respect to w i.e.,
P (x)P (x)dµ(x) = δ
m n m,n
ZE
and the monic orthogonal polynomial are denoted by p (x). It is well known
n
that the monic orthogonal polynomials satisfy a three term recurrence rela-
tion
(9) p (x) = (x α )p (x) λ p (x)
n n n−1 n n−2
− −
with p (x) := 0 and p (x) = 1. For measures of the form (8) it follows,
−1 0
see [20, Cor. 6.1] (in fact more general sets E and measures are considered
there), that the recurrence coefficients are almost periodic in the limit, more
precisely,thatthereexistrealanalyticfunctionsαandλonthetorus[0,1]l−1
and a constant c Rl−1, depending on the measure µ, such that
∈
(10) α = α(nω( ) c)+o(1) and λ = λ(nω( ) c)+o(1),
n+1 n+2
∞ − ∞ −
where ω( ) = (ω ( ),...,ω ( )); for more details see the remark follow-
1 l−1
∞ ∞ ∞
ing Proposition 2.3 below. By (10) one gets a first, quite good view into the
convergence behavior of the α ’s and λ ’s, but by far not a complete one,
n n
since the functions α and λ and their mapping properties are not known.
The more it is not clear whether the recurrence coefficients and (ω( ))
n∈N
∞
modulo 1 have the same topological convergence behavior; that is, that
subsequences converge simultaneously and that there is a homeomorphism
between the set of accumulation points. The goal of this paper is to show
that, under a proper point of view, there is such a topological equivalence;
also between the recurrence coefficients and the zeros of the polynomials
and Weyl solutions outside the spectrum. As an immediate consequence
4 F.PEHERSTORFER
it follows that the problem of convergence of the recurrence coefficients is
topologically equivalent to the classical problem in number theory to find
diophanticapproximationsoftheharmonicmeasuresω( ),[22,V.Kapitel],
∞
[7, 9].
Instead of continuing to study the accumulation points of the recurrence
coefficients with the help of (10), which looks rather hopeless, we may also
investigate that ones of the sequences of moments of the diagonal Green’s
functions,thatis,( xP2, λ xP P , x2P2, λ x2P P ,...,
n n+2 n+1 n n n+2 n+1 n
xmP2, λ xmP P ,...) since it can be shown (see Proposition
n n+2 R n+1pn nR∈N R p R
5.2) that there is a unique correspondence between such vectors of moments
R p R
andthevectorofrecurrencecoefficients(...,α[m−1]+1+n,λ[m−2]+2+n,...,α1+n,
2 2
λ2+n,..., λ−[m−1]+2+n,α−[m−2]+1+n, ...). In fact it even suffices to consider
2 2
the truncated vector sequence ( xP2, λ xP P , ..., xl−1P2,
n n+2 n+1 n n
λ xl−1P P ) since, by Corollary 2.5, it carries all information
n+2 n+1 n n∈N R p R R
already. We show (Theorem 3.2; concerning the recurrence coefficients see
p R
Theorem 5.3) that there is a homeomorphism between the set of accumu-
lation points of the truncated vector sequence of moments (respectively, of
recurrence coefficients) and of the sequence (nω( )) modulo 1 and that
∞
convergence holds for the same subsequences. In particular this implies that
the set of accumulation points of the truncated vector sequence of moments
is homeomorph to a l 1 dimensional torus if the harmonic measures are
−
linearly independent over Q, where the homeomorphism is given explicitly
even.
Also of special interest is the fact that there is a unique correspondence
between the accumulation points of the discussed truncated vector sequence
of moments and the accumulation points of zeros of the orthogonal polyno-
mials and Weyl solutions outside the spectrum, see Theorem 4.3.
Next let us represent the weight function w in the form
w (x)
0
(11) w(x) =
ρ(x)
and let (z) be that function which is analytic on Ω˜ := C E, has no zeros
o
W \
and poles there, is normalized by ( ) > 0 and satisfies the boundary
W ∞
condition
(12) w (x) = 1/ +(x) −(x).
0 W0 W0
For µ of the form (8) with w Sz(E) it has been shown that P /φn is
n
∈
uniformly asymptotically equivalent to the solution of a certain extremal
ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 5
problem, more precisely, that on compact subsets of Ω := Ω˜ d ,...,d
1 m
\{ }
(13)
1/2
l−1 l−1 l−1
φ(z;c ) φ(z;x )δj,n (z x )
Pn(z) UW0(z)φ′(z) j=1 j j=1 j,n j=1 − j,n
φn(z) ∼ m Q Q r (z) Q
φ(z;d ) ∞
j
j=1
Q
where U is a constant and the points x and the δ 1 are uniquely
j,n j,n
∈ {± }
determined by the conditions that for k = 1,...,l 1
−
(14)
l−1
2 w (ξ) ∂ω (ξ)
0 k
δ ω (x ) = (2n l+1)ω ( ) log dξ
j,n k j,n − − k ∞ − π ρ(ξ) ∂n+
j=1 ZE (cid:18) (cid:19) ξ
X
m
+2 ω (d ) modulo 2,
k j
j=1
X
recall that ω (x) := ω(x;E ,Ω˜) is the harmonic measure of E with respect
k k k
to Ω˜ at the point x and n+ is the normal vector at ξ pointing in the upper
ξ
half plane. It turns out that x [a ,a ] for j = 1,...,l 1 and thus
j,n 2j 2j+1
∈ −
l−1
(15) gˆ (x;w) := gˆ (x) := (x x )
(n) (n) j,n
−
j=1
Y
has exactly one zero in each gap [a ,a ], j = 1,...,l 1. The asymptotic
2j 2j+1
−
representation (13) is due to Widom [34, Theorem 6.2, p. 168 and pp. 175-
176] if there appear no point measures, i.e., if m = 0. The case of point
measures has been first studied in [23] for two intervals. Asymptotics for
more general sets (so-called homogeneous sets) and measures, including the
above ones, have been given by P. Yuditskii and the author in [20, 21].
The so-called Weyl solutions or functions of the second kind
p (x) P (x)
n n
(16) Q (y) = dµ(x), respectively, (y) = dµ(x)
n n
y x Q y x
ZE − ZE −
build another basis system of solutions of the recurrence relation (9). In [20,
Section 3] also asymptotics for Weyl solutions have been derived, more pre-
cisely, usingthe terminologyofthispaper, thefollowinguniformasymptotic
equivalence on compact subsets of Ω has been shown
(17)
m
φ(z;dj) 1/2
gˆ (z)
j=1 (n)
(z)
Qn ∼ UWQ0(z)φn+1(z) r∞(z) jl−=11φ(z;cj) jl−=11φ(z;xj,n)δj,n!
where 0 n is denoted by hn respecQtively h in [2Q0].
W Q
Finally we mention that other questions about finite gap Jacobi matrices
are under investigations by J.S. Christiansen, B. Simon and M. Zinchenko
6 F.PEHERSTORFER
[3] and that a huge part of the forthcoming book by B. Simon [27], see
also [26], is devoted to this topic. The approach is based on automorphic
functions similarly as in [20, 21, 28] and is different from that one used here.
2. Asymptotics for principal and secondary diagonal Green’s
function
The diagonal matrix elements of the resolvent ( z)−1 are the so-called
Green’s functions G(z,n,m) = δ ,( z)−1δ ,Jwh−ere as usual denotes
n m
h J − i J
the Jacobi matrix associated with (α ) and (λ ). More precisely using the
j j
orthogonality of P , respectively, P one obtains
n n+1
P2(x)
(18) P (z) (z) = n dµ(x) =: G(z,n,n)
n n
Q z x
Z −
P (x)P (x)
n+1 n
(19) P (z) (z) = dµ(x) =: G(z,n+1,n)
n n+1
Q z x
Z −
and
P (x)P (x) 1
n+1 n
(20) P (z) (z) = dµ(x)+ =: G(z,n,n+1).
n+1 n
Q z x λ
Z − n+2
Recall that in the case under consideration the λ ’s are bounded away
p n
from zero. Using Schwarz’s inequality it follows that the three sequences
(P ),(P )and(P )areboundedandanalyticoncompactsub-
n n n n+1 n+1 n
Q Q Q
sets ofR supp(µ), hencethey areso-callednormalfamilies. Combining (13)
\
and (17) we obtain with the help of (2) the following asymptotics for the
diagonal Green’s function.
Corollary 2.1. Let µ be given by (8) with w Sz(E). Then uniformly on
∈
compact subsets of Ω there holds
gˆ (z)
(n)
(21) (P )(z) = +o(1).
n n
Q H(z)
For weight functions of the form v(px)ρ(x)dx plus a possible finite number
of mass points outside [a ,a ], where v is positive and analytic on E, the
1 2l
limit relation (21) has been derived recently in [30] independently from the
above asymptotics (13) and (17).
ToderivetheasymptoticrepresentationsofG(z,n+1,n)andG(z,n,n+1)
we will make use of Abel’s Theorem and the solvability and uniqueness of
therealJacobiinversionproblem. ForthisreasonwehavetowriteWidom’s
condition (14) in terms of Abelian integrals, which we will do similarly as
in [2, 30], see also [1].
Let R denote the hyperelliptic Riemann surface of genus l 1 defined by
−
y2 = H with branch cuts [a ,a ],[a ,a ],...,[a ,a ]. Points on R are
1 2 2 3 2l−1 2l
written in the form z and z denotes the projection of z on C also written
pr(z) = z. Furthermore we also use the notation pr(x) = x and pr(y) = y. z
andz∗ denotethepointswhichlieaboveeachotheronR,i.e. pr(z) = pr(z∗).
ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 7
The two sheets of R are denoted by R+ and R−. To indicate that z lies on
the first respectively second sheet we write z+ and z−. On R+ the branch
of √H is chosen for which H(x+) > 0 for x+ > a .
2l
Furthermore (see e.g. [15, 29]) let the cycles α ,β l−1 be the usual
p { j j}j=1
canonical homology basis on R, i.e., the curve α lies on the upper sheet R+
j
ofRandencirclesthereclockwisetheintervalE andthecurveβ originates
j j
at a arrives at a along the upper sheet and turns back to a along the
2j 2l−1 2j
lower sheet. Let ϕ ,...,ϕ , where ϕ = l−1 e zs dz,e C, be
{ 1 l−1} j s=1 j,s√H(z) j,s ∈
a base of the normalized Abelian differential of the first kind, i.e.,
P
(22) ϕ = 2πiδ and ϕ = B for j,k = 1,...,l 1
k jk k jk
−
Zαj Zβj
where δ denotes the Kronecker symbol here. The integrals in (22) are the
jk
so-called periods. Note that the e ’s are real since √H is purely imaginary
j,n
on E and since √H is real on R E the symmetric matrix of periods (B )
j j,k
\
is real also.
Abelian differentials of third kind are differentials with simple poles at
given points x and y on R with residues +1 and 1, respectively, and nor-
−
malized such that integrals along the α cycles vanish for κ = 1,...,l 1.
κ
− −
Representingthedifferentialdlogφ(z,c),c R E,aslinearcombinations
∈ \
ofnormalizedAbeliandifferentialsoffirstandthirdkind(recallthatφ( ,c+),
where c = c+, has a pole at c+ and a zero at c− since the analytic exte·nsion
of the complex Green’s function to the second sheet is given by φ(z−,c+) =
1/φ(z+,c+)) one obtains by integrating along the α and β cycles that
j j
c+ l−1
(23) ϕ = ω (c)B .
j κ jκ
Zc− κ=1
X
Similarly (see [18] for details), since the analytic extension of the harmonic
measure
(24) w = ω +iω∗ on R+ and w = ω +iω∗ on R−
k k k k − k k
is just another basis of Abelian differentials of first kind, ϕ can be repre-
j
sented as linear combination of the w ’s. Integrating along the β cycle,
k κ
κ 1,...,l 1 , and recalling the fact that the integral along a β -cycle is
κ
∈ { − }
the difference of values along the α cycle, which is E , we obtain by (24)
κ κ
that
l−1
1
ϕ = w B .
j κ jκ
−2
Z κ=1
X
Hence, using the fact that ω (z) = 1 on E , κ 1,...,l , and thus by the
k κ
∈ { }
Cauchy-Riemann equations dw (z) = i∂ωκds, we get
κ ∂n
l−1
2 1 ∂ω
(25) ϕ+ = κds B
πi j π ∂n+ jκ
κ=1(cid:18) (cid:19)
X
8 F.PEHERSTORFER
Notation 2.2. Let [a ,a ]± denote the two copies of [a ,a ], j =
2j 2j+1 2j 2j+1
1,...,l 1 in R±. Note that [a ,a ]+ [a ,a ]− is a closed loop
2j 2j+1 2j 2j+1
− ∪
on R and thus Xl−1 ([a ,a ]+ [a ,a ]−) is topologically a l 1
j=1 2j 2j+1 ∪ 2j 2j+1 −
dimensional torus.
By (23) and (25) Widom’s condition (14) becomes,
1 l−1 xj,n l 1 ∞+ i
ϕ = (n − ) ϕ ϕ+logw
2 j=1Zx∗j,n k − − 2 Z∞− k − π ZE k
(26) X
m d+
j
+ ϕ mod periods
k
j=1Zd−j
X
where pr(x ) = x and x Rδj,n := R± for δ = 1; recall that x
j,n j,n j,n j,n j,n
and x∗ lie above each other. F∈urthermore pr(d±) = d .±
j,n j j
Next we note that (26) can be considered as so-called Jacobi inversion
problem, that is, for given (η ,...,η ) Jac R, where Jac R denotes the
1 l−1
∈
Jacobi variety of R, (that is the quotient space Cl−1 (2πin +Bm), B =
→− →−
\
(B ) the matrix of periods, n,m Zl−1) find z ,...,z R such that
jk −→ →− 1 l−1
∈ ∈
l−1 zj
ϕ = η mod periods,
k k
j=1Zej
X
where e ,...,e are given points on R. In this paper the η ’s are real al-
1 l−1 j
ways, that is, we deal with the real Jacobi inversion problem. Denoting by
Jac R/R := Rl−1/Bm the Jacobi variety restricted to reals, the following
−→
important uniqueness property of the real Jacobi inversion problem holds
(see e.g. [10, 17]): The restricted Abel map
(27)
: Xl−1([a ,a ]+ [a ,a ]−) Jac R/R
A j=1 2j 2j+1 ∪ 2j 2j+1 →
1 l−1 zj l−1 zj
(z ,...,z ) ϕ ,..., ϕ
1 l−1 1 l−1
7→ 2 j=1Zz∗j j=1Zz∗j
X X
is a holomorphic bijection.
Now let us show that the solutions x ,...,x of (26) can be described
1,n l−1,n
uniquely with the help of two polynomials which is crucial in what follows.
Proposition 2.3. a) Let g , n N, be given by (15). There exists a monic
(n)
∈
polynomial fˆ of degree l such that
(n+1)
(28) fˆ2 (x) H(x) = L gˆ (x)gˆ (x)
(n+1) − n (n+1) (n)
with
(29)
fˆ (x ) = δ H(x ) and fˆ (x ) = δ H(x )
(n+1) j,n j,n j,n (n+1) j,n+1 j,n+1 j,n+1
− −
q q
ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 9
where x ,x ,δ ,δ are given by (14) and L R is given below.
j,n j,n+1 j,n j,n+1 n
∈
Moreover fˆ can be represented in the form
(n+1)
l−1 l−1 δ H(x )gˆ (x)
(30) fˆ (x) = x+ x c gˆ (x) j,n j,n (n) ,
(n+1) j,n− 1 (n) − gˆ′ (x )(x x )
j=1 j=1 (np) j,n − j,n
X X
where c is given by H(x) = x2l 2c x2l−1+....
1 1
−
Furthermore
±1
(31) (fˆ(n+1)±√H)2(z) = φ2(z; )l−1 φ(z;xj,n+1)δj,n+1
Lngˆ(n)(z)gˆ(n+1)(z) ∞ φ(z;xj,n)δj,n
j=1
Y
and
l−1 φ(x ; )δj,n
(32) L = 4(cap(E))2 j,n ∞ = 4λ (1+o(1))
n φ(xj,n+1; )δj,n+1 n+2
j=1 ∞
Y
b) To each solution (x ,...,x ) of (26) there corresponds a unique
1,n l−1,n
pair of polynomials (gˆ ,fˆ ) given by (15) and by (30) with H(x ) =
(n) (1+n) j,n
δ H(x ).
j,n j,n p
Propof. a) Let us write xδj,n := x± for δ = 1. Considering (26) for n and
j,n n j,n ±
n+1 and substracting both equations we obtain that
l−1 x−δj,n+1 ∞+ l−1 x−δj,n
j,n+1 j,n
ϕ = 2 ϕ + ϕ , k = 1,...,l 1.
k k k
Xj=1Zxδj,jn,n++11 Z∞− Xj=1Zxδj,jn,n −
Thus it follows from Abel’s Theorem that there is a rational function on
n
R
the Riemann surface with the following properties
± is a double pole (zero)
∞
(33) x∓δj,n+1 is a simple zero (pole)
j,n+1
x∓δj,n+1 is a simple pole (zero).
j,n
Thus can be represented in the form
n
R
(f +√H)2
(n+1)
(34) = ,
n
R L gˆ gˆ
n (n) (n+1)
where f is a polynomial of degree l and L is a constant, since the
(n+1) n
function at the RHS in (34) satisfies (33) and has no other zeros or poles
on R as can be checked easily. Since the points x±δj,n+1 and x±δj,n are real
j,n+1 j,n
the rational function (z) has the same properties (33) as (¯z), hence
n n
R R
(¯z) = (z) and therefore f has real coefficients and L R. Taking
n n (n+1) n
R R ∈
10 F.PEHERSTORFER
involution, denoted by z∗, we obtain by (33)
1 (f √H)2
(35) = (z∗) = (n+1)− ,
n
(z) R L gˆ gˆ
n n (n) (n+1)
R
and thus, by multiplying (35) and (34), relation (28)-(29). Concerning (29)
note that H(xδ) = δ H(x).
Next let us note
p p
(√H)−(x) = (√H)+(x) = i( 1)l−k H(x) for x E
k
− − | | ∈
and therefore by (34) and (28) p
±(x) = 1 for x E.
|R | ∈
Thus the function
(z) l−1 φ(z;x )δj,n
n j,n
F(z) := R
φ2(z; ) φ(z;xj,n+1)δj,n+1
∞ j=1
Y
hasneitherzerosnorpolesonΩandsatisfies F± = 1onE.Hencelog f(z)
| | | |
is a harmonic bounded function on Ω which has a continuous extension to
E and thus F = 1, which proves (31).
Considering (31) at z = the first relation in (32) follows. Next denote
∞
by lc(P ) the leading coefficient of P . Obviously λ = p2 / p2 =
n n n+2 n+1 n
(lc(P )/lc(P ))2. Using the fact that lc(P ) = lim P (z)/zn it follows by
n n+1 n z→∞ n R R
(13), in conjunction with (5), that the last equality in (32) holds.
Relation (30) follows by (29), note that the second term at the RHS of
(30) is the unique Lagrange interpolation polynomial which takes on at x
j,n
the value δ H(x ), and by equating the first two leading coefficients
j,n j,n
−
in (28).
b) follows impmediately by a). (cid:3)
We note in passing that (32), which may be derived from (13) also, is the
so-called trace formula for λ and that the other trace formula α =
n+2 n+1
l−1 x c + o(1) follows immediately by (18), (21) and by−equating
j=1 j,n − 1
coefficientsofxl−1 in(30). By(26)and(27)thex ’smaybeexpressedwith
P j,n
the help of the Abel map from which representation (10) can be obtained.
For measures from , defined in Corollary 3.4, which are so-called
G G
reflectionless measures (note that the set of Jacobi matrices associated with
the set of measures is the so called isospectral torus), Proposition 2.3 is
G
essentially known [13, 33, 16, 32] apart from the important relation (31).
But the way of derivation is exactly the opposite one. First one shows that
certain polynomials, which are expressions in Pad´e approximants, satisfy
(28) - (30) - to find the corresponding polynomials in the general case seems
to be hopeless, if they exist at all - and then one derives the correspondence
with the solutions of (26).
