Rakhmanov's theorem establishes a result about the asymptotic behavior of the elements of the Jacobi matrix associated with a measure ¹ which is de¯ned on the interval I = [¡1; 1] with ¹ 0 > 0 almost everywhere on I. In this work we give a weak version of this theorem, for a measure with support on a connected ¯nite union of Jordan arcs on the complex plane, in terms of the Hessenberg matrix, the natural generalization of the tridiagonal Jacobi matrix to the complex plane

Escribano Iglesias, M. del Carmen

Giraldo Carbajo, Antonio

Sastre Rosa, María de la Asunción

Torrano Gimenez, Emilio

Archivo Digital UPM

A Rakhmanov-like theorem for orthogonal polynomials on Jordan arcs in the complex plane

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Proceedings of the 10th International Conference
on Computational and Mathematical Methods
in Science and Engineering, CMMSE 2010
27–30 June 2010.
A Rakhmanov-like theorem for orthogonal polynomials on
Jordan arcs in the complex plane
C. Escribano1, M. A. Sastre1, A. Giraldo1 and E. Torrano1
1 Departamento de Matema´tica Aplicada, Facultad de Informa´tica, Universidad
Polite´cnica de Madrid
emails: cescribano@fi.upm.es, masastre@fi.upm.es, agiraldo@fi.upm.es,
emilio@fi.upm.es
Abstract
Rakhmanov’s theorem establishes a result about the asymptotic behavior of
the elements of the Jacobi matrix associated with a measure µ which is defined on
the interval I = [−1, 1] with µ′ > 0 almost everywhere on I. In this work we give
a weak version of this theorem, for a measure with support on a connected finite
union of Jordan arcs on the complex plane, in terms of the Hessenberg matrix, the
natural generalization of the tridiagonal Jacobi matrix to the complex plane.
Key words: Hessenberg matrix, regular measures, Riemann map.
1 Introduction
In this paper, we consider regular Borel measures µ defined on subsets of the complex
plane which are Jordan arcs, or connected finite union of Jordan arcs, and we show how
the support of µ is determined by the entries of the Hessenberg matrix D associated
with µ. The Hessenberg matrix is the natural generalization of the tridiagonal Jacobi
matrix to the complex plane and, in the particular case of measures with support the
unit circle T, the Hessenberg matrix is a Toeplitz matrix.
Our result represents a broader, although weaker, extension of Rakhmanov’s theo-
rem to C. In the real case, Rakhmanov’s theorem [15, 16] states that, if the support of
a Borel measure is [−1, 1] and µ′ > 0 almost everywhere in [−1, 1], then an → 12 and
bn → 0, where an are the sequences of elements in the subdiagonal and superdiagonal,
and bn are the sequences of elements in the diagonal, in the Jacobi matrix J associated
with µ. Moreover, if the support of µ is the interval [−2a + b, b + 2a], then the above
limits are, respectively, an → a y bn → b. Conversely, if we know that µ′ > 0 and that
the support of µ is a compact connected set of R, knowing the limits of the diagonals
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A Rakhmanov-like theorem for orthogonal polynomials in the complex plane
of J we could obtain the support of µ, i.e., if an → a y bn → b then the support is
[−2a+ b, b+ 2a].
Generalizations of Rakhmanov’s theorem to orthogonal polynomials and to orthog-
onal matrix polynomials on the unit circle has been given in [13] and [22]. The case of
orthogonal polynomials in an arc of circumference has been studied in [2]).
There exist some previous results relating the properties of D and the support of
µ. For example, if the Hessenberg matrix D defines a subnormal operator [12] in `2,
then the closure of the convex hull of its numerical range agrees with the convex hull of
its spectrum. On the other hand, the spectrum of the matrix D contains the spectrum
of its minimal normal extension N = men(D) which is precisely the support of the
measure [6].
In this work we show that, in the case of regular measures µ whose support is a
Jordan arc or a connected union of Jordan arcs in the complex plane C, the limits of the
values at the diagonals of the Hessenberg matrix D of µ, supposing those limits exist,
determine the terms of the coefficients of the series expansion of the Riemann map φ(z)
(see [20]) which applies conformally the exterior of the unit disk in the exterior of the
support of the measure. As a consequence, the support of µ can be determined just
knowing the limits of the values at the diagonals of its Hessenberg matrix D.
For general information on the theory of orthogonal polynomials, we recommend
the books [4, 20] by T. S. Chihara and G. Szego¨, respectively, and the survey [11] by
Golinskii and Totik.
2 Main result
Let µ(z) be a regular positive Borel measure with compact support Ω in the complex
plane. Let P be the space of polynomials. The associated inner product is given by
the expression
〈Q(z), R(z)〉µ =
∫
supp(µ)
Q(z)R(z)dµ(z),
for R,S ∈ P. Then there exists a unique orthonormal polynomials sequence (ONPS)
{Pn(z)}∞n=0 associated to the measure µ (see [4], [8] or [20]).
In the space P2(µ), closure of the polynomials space P in L2µ(Ω), we consider the
multiplication by z operator. Let D = (djk)∞j,k=0 be the infinite upper Hessenberg
matrix of this operator in the basis of ONPS {Pn(z)}∞n=0, hence
zPn(z) =
n+1∑
k=0
dk,nPk(z), n ≥ 0, (1)
with P0(z) = 1 when c00 = 1.
It is a well known fact that the monic polynomials are the characteristic polynomials
of the finite sections of D.
In order to state our main result, we will need that the measure µ is regular with
support a connected finite union of Jordan arcs, and we will also need to consider an
auxiliar Toeplitz matrix. We next recall the definitions of all these notions.
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C. Escribano, A. Giraldo, M. A. Sastre and E. Torrano
A Jordan arc in C is any subset of C homeomorphic to the closed interval [0, 1] on
the real line.
A measure µ is regular if lim
n→∞
1
n
√
γn
= cap(supp(µ)), the capacity of the support
of µ, where the γn are the conductor coefficients of the orthonormal polynomials, i.e.,
Pn(z) = γnzn + . . ..
An infinite matrix T = (ai,j)∞i,j=0 is a Toeplitz matrix if each descending diagonal
from left to right is constant, i.e, there exists (ai)i∈Z such that ai,j = ai−j , for every
i, j ∈ N∪{0}. Given a Toeplitz matrix T , the Laurent series whose coefficients are the
entries ai defines a function known as the symbol of T .
We are now in a position to state and prove the main result of the paper.
Theorem 1. Let D = (dij)∞i,j=1 be a Hessenberg matrix associated with a measure µ
with compact support on the complex plane. Assume that:
1. The measure µ is regular with support supp(µ) a Jordan arc or a connected finite
union of Jordan arcs Γ such that C \ Γ is a simply connected set of the Riemann
sphere C∞.
2. There exists a Hessenberg-Toeplitz matrix T such that D − T defines a compact
operator in `2 with its rows in `1.
Then, the symbol of T is the Riemann function φ : C∞ \ D→ C∞ \ Γ
Proof. Since supp(µ) = Γ is a compact set and C∞ \Γ is connected, we can apply
Merguelyan’s theorem [9, p.97] which asserts that every continuous function in Γ can
be uniformly approximated by polynomials. Since the set of continuous functions with
compact support is dense in L2µ(Γ), then L
2
µ(Γ) = P
2
µ(Γ). Therefore, D defines a normal
operator in `2, hence σ(D) = Γ [5, 21]. Since
σ(D) \ σess(D) = {λ | λ isolated eigenvalue if finite multiplicity},
where σess(D) is the essential spectrum of D (see, for example, [6] for its definition),
and the support is connected, then it has not isolated points, and Γ = σ(D) = σess(D).
Consider now K = D − T which, by hypothesis is a compact operator. Then all
its diagonals converge to 0 [1] and hence the limits
lim
n
dn−k,n = d−k, k = −1, 0, 1, 2, . . .
exist, and the matrix T is
T =

d0 d−1 d−2 . . .
d1 d0 d−1 . . .
0 d1 d0 . . .
0 0 d1 . . .
...
...
...
. . .
 .
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A Rakhmanov-like theorem for orthogonal polynomials in the complex plane
Since the essential spectrum is invariant via compact perturbations [5], we have that
σess(D) = σess(T ). Moreover, T is bounded in `2 and hence the rows and columns of
T are in `2. Therefore, (d1, d0, d−1, d−2, . . .) ∈ `2.
The elements dn,n−1 of the subdiagonal of the matrix D agree with the quotients
γn−1/γn. Since lim
n→∞ dn+1,n = d1, then
d1 = lim
n→∞
γn−1
γn
= lim
n→∞
1
n
√
γn
.
On the other hand, since µ is regular, then [19, p.100]
lim
n→∞
1
n
√
γn
= cap(supp(µ)).
Therefore, d1 = cap(supp(µ)).
Consider now the Laurent series
d(z) = d1z + d0 +
d−1
z
+
d−2
z2
+ · · ·
We see now that the fact that (d1, d0, d−1, . . .) ∈ `2 implies that d(z) is analytic for
every z such that 1 < |z| <∞.
If |z| > 1, then 1|z| < 1 and
∞∑
k=−1
|d−kz−k| ≤
√√√√ ∞∑
k=−1
|d−k|2
√√√√ ∞∑
k=−1
|z−k|2 < +∞.
Therefore d(z) converges absolutely for every 1 < |z| <∞. To see that d(z) is analytic
we have just to show that d′(z) exists for every |z| > 1. But
d ′(z) = d1 −
∞∑
k=1
k
d−k
zk+1
where
∞∑
k=1
k| d−k
zk+1
| ≤
√√√√ ∞∑
k=1
|d−k|2
√√√√ ∞∑
k=1
k2
|z|2k+2 < +∞
if |z| > 1. Hence d′(z) exists for every |z| > 1.
Since (d1, d0, d−1, . . .) ∈ `1, then (d|T)(z) is continuous (where T is the unit circle)
and [3, p.10]
Γ = σess(T ) = d(T) = {d1w + d0 + d−1
w
+
d−2
w2
+ . . . | w ∈ T}.
We can now apply Theorem 1.1 in [14] to conclude that
d : C∞ \ cl(D)→ C∞ \Γ,
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C. Escribano, A. Giraldo, M. A. Sastre and E. Torrano
(where D the unit disk) is an univalent map and, being also analytic, is conformal in
C∞ \ cl(D).
Consider now the Riemann map
φ(z) = c1w + c0 +
c−1
w
+
c2
w2
+ . . .
in C∞ \Γ which is the unique conformal map which applies the exterior of the unit disk
in the exterior of Γ = supp(µ), which preserves the point at infinity and the direction
therein, and which also satisfies cap(Γ) = c1 [20]. The map d satisfies that d1 = cap(Γ).
Moreover, since d′(∞) = φ′(∞) = d1 = c1, then d(z) preserves the point at infinity and
the direction therein. Therefore d = φ.
3 Examples
As an illustration of the previous theorem we consider the following examples.
Example 1. Consider Γ the segment [−1, 1] in C. The Riemann map φ which applies
the exterior of the unit disk in the exterior of Γ is
φ(z) =
1
2
(
z +
1
z
)
.
By Rakhmanov’s theorem, if µ is a Borel measure is [−1, 1] and µ′ > 0 almost every-
where in [−1, 1], then an → 12 and bn → 0, where an are the sequences of elements
in the subdiagonal and superdiagonal, and bn are the sequences of elements in the di-
agonal, in the Jacobi matrix J associated with µ. Note that these are the coefficients
of the Riemann map φ. Although Theorem 1 does not guarantee the existence of the
limits of the diagonals of the Jacobi matrix in any case, in the case that those limits
exist, they must agree with the coefficients of µ, even if µ is not absolutely continuous.
Example 2. Let Γ be a cross-like set, and µ the uniform measure on γ. The Riemann
map is
φ(z) =
√
a2(z2 + 1)2 + b2(z2 − 1)2
2z
,
where a and b are the length of the horizontal and vertical semi-axis, respectively. In
the particular case of a = b,
φ(z) =
a
√
2
2z
√
z4 + 1.
The series expansion of φ is
φ(z) =
√
a2 + b2
2
z +
−2 b2 + 2 a2
4
√
a2 + b2
1
z
+
√
a2 + b2
(
1
2
− (−2 b
2 + 2 a2)2
8 (a2 + b2)2
)
2 z3
+O
(
1
z5
)
,
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A Rakhmanov-like theorem for orthogonal polynomials in the complex plane
where the first coefficient
√
a2 + b2
2
agrees with the capacity of the support. If a = b,
the series expansion is
φ(z) =
a
√
2
2
z +
a
√
2
4
1
z3
+O
(
1
z5
)
.
The image under φ of the unit circle is shown in Figure 1, where we have included
on the right the same result with an interpolation with less steps to give a better insight
of the Riemann map.
Figure 1: φ(T) for a cross-like set
There are many instances, however, when the Hessenberg matrix can not computed
completely, but only finite sections of it, and it is not possible to compute the limits
of the diagonals of D. In this case, it is still possible to compute approximations of
the support of the measure µ obtained computing the image of the unit circle under
suitable approximations of the Riemann map. Specifically, since the coefficients of the
Riemann map are the limits of the elements in each of the diagonals of the Hessenberg
matrix, we may consider, as approximations of the Riemann map φ, the functions
φk(z) = dk,k−1z + dk,k +
k−1∑
i=1
dk−i,k
zi
,
where D = (di,j) is the Hessenberg matrix of µ [7].
The result of approximating supp(µ) using this method, for k = 30, k = 40 and
k = 50, respectively, is shown in Figure 1.
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C. Escribano, A. Giraldo, M. A. Sastre and E. Torrano
Figure 2: φk(T) for k = 30, k = 40 and k = 50, respectively
Example 3. Consider now Γ an arc of circumference. In this case [10] (see also [17, 18]),
there exists a measure for which the diagonals of the Hessenberg matrix stabilize from
the second element on. The monic orthogonal polynomials associated to this measure
satisfy Φ0(0) = 1 and Φn(0) =
1
a
(a > 1), if n ≥ 1, and the corresponding Hessenberg
matrix it the following unitary matrix:
D =

−1
a
−(a
2 − 1)1/2
a2
−(a
2 − 1)2/2
a3
−(a
2 − 1)3/2
a4
−(a
2 − 1)4/2
a5
· · ·
(a2 − 1)1/2
a
− 1
a2
−(a
2 − 1)1/2
a3
−(a
2 − 1)2/2
a4
−(a
2 − 1)3/2
a5
· · ·
0
(a2 − 1)1/2
a
− 1
a2
−(a
2 − 1)1/2
a3
−(a
2 − 1)2/2
a4
· · ·
0 0
(a2 − 1)1/2
a
− 1
a2
(a2 − 1)1/2
a3
· · ·
...
...
...
...
...
. . .

.
Hence we know the limits of the diagonals, and we can obtain the sum of these limits.
It is easy to check that D − T is compact and that the rows of T are in `1, and hence
the expression of the Riemann map is
φ(z) =
z
(
a−√a2 − 1 z
)
√
a2 − 1− az
=
√
a2 − 1
a
z − 1
a2
−
√
a2 − 1
a3z
−O
(
1
z2
)
,
and we can compute the image under φ of the unit circle. The result is shown in Figure
1, where we have included on the right the same result with an interpolation with less
steps to give a better insight of the Riemann map.
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A Rakhmanov-like theorem for orthogonal polynomials in the complex plane
Figure 3: φ(T) for an arc of circumference
In the following figure we compute several approximations of the support of µ, for
the particular case a = 2, using the above method, for k = 10, k = 20 and k = 30,
respectively.
Figure 4: φk(T) for k = 10, k = 20 and k = 30, respectively
Example 4. In the following example we take Γ as the half part of a drop-like set of
parametric equation
z(t) =
(eit)2
1 + 2eit
, t ∈ [0, pi].
and µ the uniform measure on γ. In the following figure we show several approximations
of the support of µ using this method, for k = 5, k = 8 and k = 11, respectively.
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C. Escribano, A. Giraldo, M. A. Sastre and E. Torrano
Figure 5: φk(T) for k = 5, k = 8 and k = 11, respectively
Example 5. For the last example we take Γ as the spiral with parametric equation
z(t) = t
eit
6
, t ∈ [0, 2pi]
and we consider µ the uniform measure on γ. In the following figure we show sev-
eral approximations of the support of µ using this method, for k = 1 and k = 12,
respectively.
Figure 6: φk(T) for k = 1 and k = 12, respectively
Acknowledgements
The authors have been supported by Comunidad Auto´noma de Madrid and Universidad
Polite´cnica de Madrid (UPM-CAM Q061010133).
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A Rakhmanov-like theorem for orthogonal polynomials in the complex plane
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A Rakhmanov-like theorem for orthogonal polynomials on Jordan arcs in the complex plane

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