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V EIPOA 2015
ANALYSIS OF PERTURBATIONS OF MOMENTS
ASSOCIATED WITH ORTHOGONALITY LINEAR
FUNCTIONALS THROUGH THE SZEGO
TRANSFORMATION
EDINSON FUENTES & LUIS E. GARZA
UNIVERSIDAD PEDAGÓGICA Y TECNOLÓGICA DE COLOMBIA
JUNE 2015
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
ORTHONORMAL POLYNOMIAL SEQUENCE (OPS)
Let α be a positive, non trivial Borel measure, supported on asubset E of the real line. The sequence of polynomials{pn(x)}n≥0, with
pn(x) = γnxn + δnx
n−1 + ..., γ > 0,
is said to be an orthonormal polynomial sequence associatedwith α if ∫
Epn(x)pm(x)dα(x) = δm,n, m, n ≥ 0.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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THE STIELTJES FUNCTION
If L is quasi-definite, S(x) admits the following formal seriesexpansion at infinity
S(x) =
∞∑k=0
µkxk+1
,
where µk are the moments associated with α given by
µk =
∫Exkdα(x), k ≥ 0.
The Stieltjes function
S(x) =
∫E
dα(t)
x− t.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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TRANSFORMATION OF THE STIELTJES FUNCTION
A rational spectral transformation of a Stieltjes function S(x) isa transformation of the form
S(x) =A(x)S(x) +B(x)
C(x)S(x) +D(x),
where A(x), B(x), C(x) y D(x) are polynomials in x withAD −BC 6= 0 and such that S(x) has a formal seriesexpansion around infinity. The transformation is said to belinear if C(x) ≡ 0.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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PERTURBATION CANONICAL
Given a measure α supported on the real axis, some canonicalperturbations are
1 The Christoffel transformation dαc = (x− β)dα,β /∈ supp(α).
2 The Uvarov transformation dαu = dα+Mrδ(x− β),β /∈ supp(α), Mr ∈ R.
3 The Geronimus transformation dαg = dαx−β +Mrδ(x− β),
β /∈ supp(α), Mr ∈ R,where δ(x− β) is the Dirac’s delta functional, defined by
〈δ(x− β), q〉 = q(β), q ∈ P, β ∈ R.
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The three canonical perturbations defined above correspond tolinear spectral transformations of the corresponding Stieltjesfunctions
1 The Christoffel canonical transformation
Sc(x) = RC(β)[S(x)] =(x− β)S(x)− 1
µ1 − β.
2 The Uvarov canonical transformation
Su(x) = RU (β,Mr)[S(x)] =S(x) +Mr(x− β)−1
1 +Mr.
3 The Geronimus canonical transformation
SG = RG(β,Mr)[S(x)] =S(β) +Mr − S(x)
(x− β)(Mr + S(β)).
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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The group of spectral linear transformations of the form
S(x) =A(x)S(x) +B(x)
D(x),
It is a non-commutative group generated from Christoffeltransformations and Geronimus described above. furthermore
RC(β) ◦RG(β,Mr)[S(x)] = S(x), Identity transformation
RG(β,Mr) ◦RC(β)[S(x)] = RU (β,Mr)[S(x)]
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
ORTHONORMAL POLYNOMIAL ON THE UNIT CIRCLE
Let σ be a, non trivial Borel measure, supported on the unitcircle T = {z ∈ C : |z| = 1}. Then there exists a sequence{ϕn}n≥0 of orthonormal polynomials
ϕn(z) = κnzn + ..., κn > 0,
which satisfies∫ π
−πϕn(eiθ)ϕm(eiθ)dσ(θ) = δm.n, m, n ≥ 0. (1)
The corresponding monic polynomials are defined by
Φn(z) =ϕn(z)
κn.
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THE CARATHÉODORY FUNCTION
When F (z) is analytic on the unit open disc and ReF (z) > 0 insuch a disk, F (z) is called Carathéodory function and it canrepresented as a Riesz-Herglotz transformatión of σ as follows
F (z) =
∫ π
−π
eiθ + z
eiθ − zdσ(θ).
If L is quasi-definite F (z) admits the following formal in terms ofthe moments {cn}n≥0 as follows
F (z) = c0 + 2
∞∑k=1
c−kzk,
where ck are the moments associated with σ given by
ck =
∫ π
−πeikθdσ(θ).
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PROPERTIES OF THE MEASURE
If we denote by ω(θ) = σ′ the Radon-Nikodyn derivative of σand by σs singular measure, then
dσ(θ) = ω(θ)dθ
2π+ dσs(θ). (2)
Whereω(θ) = ReF (eiθ). (3)
The singular part σs is supported in {θ| lımr↑1Re(reiθ) =∞}.
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TRANSFORMATION OF THE CARATHÉODORY FUNCTION
A rational spectral transformation of a Carathéodory functionF (z) is a transformation of the form
F (z) =A(z)F (z) +B(z)
C(z)F (z) +D(z),
where A(z), B(z), C(z) and D(z) are polynomials in z withAD −BC 6= 0, and such that F (z) is analytic in D and haspositive real part therein. Again, if C(z) ≡ 0, the transformationis said to be linear.
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PERTURBATION CANONICAL
Given a linear functional σ supported on the unit circle , somecanonicals perturbations are
1 The Christoffel transformation dσC = |z − ξ|2dσ,|z| = 1, ξ ∈ C,
2 The Uvarov transformationdσU = dσ+Mcδ(z−ξ)+M cδ(z−ξ
−1), ξ ∈ C−{0},Mc ∈ C,
3 The Geronimus transformationdσG = dσ
|z−ξ|2 +Mcδ(z − ξ) +M cδ(z − ξ−1
), ξ ∈C− {0},Mc ∈ C, |ξ| 6= 1.
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The three transformations defined above correspond to linearspectral transformations,
The Christoffel canonical transformation
FC(z) =A(z)F (z) +B(z)
D(z),
where
A(z) =−ξz2 + (1 + |ξ|2)z − ξ
(1 + |ξ|2)− 2Reξc1,
B(z) =−ξz2 + (ξc1 − ξc1)z + ξ
(1 + |ξ|2)− 2Reξc1,
D(z) = z.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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The Uvarov canonical transformation
FU (z) = F (z) +B(z)
D(z),
where
B(z) = (ξ − ξz2)(Mc +M c)− (1− |ξ|2)(Mc −M c)z,
D(z) = (z − ξ)(ξz − 1).
The Geronimus canonical transformation
FG(z) =A(z)F (z) +B(z)
D(z),
where A(z) = z, D(z) = −ξz2 + (1 + |ξ|2)z − ξ,B(z) = ξz2 − 2iIm(q0)z − ξ, and q0 = c0 − ξc1.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
SZEGO TRANSFORMATION
From a positive, nontrivial Borel measure α supported in[−1, 1], we can define a positive, nontrivial Borel measure σsupported in [−π, π] by
dσ(θ) =1
2|dα(cos θ)|, (4)
in such a way that if dα(x) = ω(x)dx, then
dσ(θ) =1
2ω(cos θ)| sin θ|dθ. (5)
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SZEGO TRANSFORMATION
There exists a relation between the Stieltjes and Carathéodoryfunctions associated with α and σ, respectively, given by
F (z) =1− z2
2z
∫ 1
−1
dα(t)
x− t=
1− z2
2zS(x), (6)
where x = z+z−1
2 and z = x+√x2 − 1, (see [2]). This relation is
known as the Szego transformation.
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[−1, 1]
Christoffel|β| > 1
UvarovIf |β| = 1.If |β| > 1.
Geronimus|β| > 1
TranformationSzego −→
TChristoffelξ = β ±
√β2 − 1
Uvarov|ξ| = 1, ξ = ξaddition of onemass Mr at thepoint ξ = ±1.|ξ| 6= 1,Mc = Mr/2,ξ = β ±
√β2 − 1.
Geronimusξ = β ±
√β2 − 1
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
PERTURBATION j−TH MOMENT
Definition
Let L be a quasi-definite linear functional. The linear functionalLj is defined by
〈Lj , p(x)〉 = 〈L, p(x)〉+ (−1)jmj
j!〈D(j)δ(x− a), p(x)〉
= 〈L, p(x)〉+mj
j!p(j)(a),
(7)
where mj and a are real constants, and p(j)(x) denotes j−thderivative of p(x).
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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PERTURBATION j−TH MOMENT
If both L and Lj are positive definite, then the previoustransformation can be expressed in terms of the orthogonalitymeasures as follows
dαj = dα+ (−1)jmj
j!D(j)δ(x− a). (8)
On the other hand, from (7) it is easily obtained that
νk = 〈Lj , (x− a)k〉 =
νk, if k < j,
νj +mj , if k = j,νk, if k > j.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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PERTURBATION j−TH MOMENT
if Lj is quasi-definite and S(x) denotes its correspondingStieltjes function, S(x) =
∑∞k=0
νk(x−a)k+1 and S(x) are related by
Sj(x) = S(x) +mj
(x− a)j+1. (9)
As a consequence, (9) is a linear spectral transformation ofS(x), where
A(x) = D(x) = (x− a)j+1
andB(x) = mj .
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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DIREC PROBLEM
[−1, 1]
Stieltjes function
Sj(x) = S(x) +mj
(x− 0)j+1.
Measure
dαj = dα+ (−1)jmj
j!D(j)δ(x− 0).
TransormationSzego −→
TCarathéodory
F (z).
Measure
dσj .
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CARATHÉODORY FUNCTION
Applying the Szego transformation (6), and using x = z+z−1
2 ,we obtain the Carathéodory function
F (z) = F (z) + 2jmjzj(1− z2)
∞∑n=0
(−1)n(n+ j
j
)z2n. (10)
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MOMENTS PERTURBED
The moments
c−n =
c−n, if n < j or n = j + 2k + 1,
for k ∈ N,c−n + in−j2j−1mj
(((n+j)/2j
)+((n+j−2)/2
j
)), if n = j + 2k,
for k ∈ N,(11)
with(j−1j
):= 0.
From (10) we conclude that, if the j−th moment associated withα is perturbed and we apply the Szego transformation, then theobtained perturbation in F (z) corresponds to a perturbation ofthe moments associated with σ in the following way
If j is even, all even moments starting from cj areperturbed.If j is odd, all odd moments starting from cj are perturbed.
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MEASURE PERTURBED
The absolutely continuous part of the measure remainsinvariant with respect to the Szego transformation, and we have
dσ = σ′(θ)dθ
2π+ dσs(θ)
= σ′(θ)dθ
2π+ dσs(θ).
(12)
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EXAMPLE
[−1, 1]
If j = 0,Stieltjes function
S0 = S(x) +m0
x.
Meaure
dαj = dα+m0δ(x−0),
Uvarov with|β| = 0 < 1.
Szego−→
TCarathéodory
F0(z) =
F (z)+m0 +2m0
∞∑n=1
(−1)nz2n,
Moments
c−n ={c−n, if n is odd,
c−n + inm0, if n is even.
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EXAMPLE
In matrix form, we have
H =
µ0 +m0 µ1 µ2 µ3 · · ·µ1 µ2 µ3 µ2 · · ·µ2 µ3 µ4 µ1 · · ·µ3 µ4 µ5 µ6 · · ·...
......
.... . .
⇓
Szego transformation⇓
T =
c0 +m0 c1 c2 −m0 c3 · · ·c−1 c0 +m0 c1 c2 −m0 · · ·
c−2 −m0 c−1 c0 +m0 c1 · · ·c−3 c−2 −m0 c−1 c0 +m0 · · ·
......
......
. . .
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EXAMPLE
For instance, for the third kind Chebyshev polynomials, theorthogonality measure is
dα = dα+m0δ(x− 0)
= 2
√1− x1 + x
dx
π+m0δ(x− 0),
and the perturbed measure σ on the unit circle is
dσ = |z − 1|2 dθ2π
+ dσs.
i.e. a Christoffel transformation of the normalized Lebesguemeasure, with parameter 1.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
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PERTURBATION j-TH MOMENT
Definition
Let L be an Hermitian linear functional quasi-definite anddefine a linear functional Lj such that the associated bilinearfunctional satisfies
〈p(z), q(z)〉Lj = 〈p(z), q(z)〉L+Mj〈zjp(z), q(z)〉Lθ +M j〈p(z), zjq(z)〉Lθ ,
(13)
where Mj ∈ C, p, q ∈ P, j ∈ N is fixed, and 〈·, ·〉Lθ is the bilinearfunctional associated with the normalized Lebesgue measure inthe unit circle.
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PERTURBATION j-TH MOMENT
If L is a positive definite linear functional, then the abovetransformation can be expressed in terms of the correspondingmeasures as
dσj = dσ +Mjzj dθ
2π+M jz
−j dθ
2π. (14)
From (13), one easily sees that
ck = 〈Lj , zk〉 = 〈zk, 1〉Lj =
ck, si k /∈ {j,−j},
c−j +Mj , si k = −j,cj +M j , si k = j.
(15)
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PERTURBATION j-TH MOMENT
If Lj is quasi-definite and F (z) denotes its correspondingCarathéodory function, S(x) and F (z) are related by
Fj(z) = F (z) + 2Mjzj , (16)
which is a linear spectral transformation of F (z) with
A(z) = D(z) = 1
andB(z) = 2Mjz
j .
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INVERSE PROBLEM
[−1, 1]
Stieltjes
S(x)
Measure
dαj .
Szego←−
TCarathéodory
Fj(z) = F (z) + 2Mjzj .
Measure
dσj = dσ +Mjzj dθ
2π+M jz
−j dθ
2π.
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MEASURE PERTURBED
MeasureWhere
dαj = dα+ 2MjTj(x)
π
dx√1− x2
, (17)
where Tj(x) := cos(jθ) are the Chebyshev polynomials of thefirst kind. Notice that a measure that changes its sign in theinterval [−1, 1] is added to dα.
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MEASURE PERTURBED
Because
dαj = dα+Mj(x+ i√
1− x2)jdx
π√1− x2
+Mj(x+ i√
1− x2)−jdx
π√1− x2
= dα+Mj(cos(jθ) + i sin(jθ))dx
π√1− x2
+Mj(cos(jθ)− i sin(jθ))dx
π√1− x2
= dα+ 2Mjcos(jθ)dx
π√1− x2
= dα+ 2MjTj(x)
π
dx√1− x2
.
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MOMENTS PERTURBED
The moments perturbed are
µn =
µn +MjB(n, j), if j ≥ n, n+ j is even,
µn, otherwise,(18)
where
B(n, j) = j
[j/2]∑k=0
(−1)k(j − k − 1)!(2)j−2k
k!(j − 2k)!
(j+n−2k)/2∏i=1
j + n− 2k − (2i− 1)
j + n− 2k − 2(i− 1)
.
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MOMENTS PERTURBED
Becauseif j ≥ n and n+ j is even, we have
∫ 1
−1xnTj(x)
dx√1− x2
=j
2
∫ 1
−1xn
[j/2]∑k=0
(−1)k(j − k − 1)!(2x)j−2k
k!(j − 2k)!
dx√1− x2
=j
2
[j/2]∑k=0
(−1)k(j − k − 1)!(2)j−2k
k!(j − 2k)!
∫ 1
−1xj+n−2k dx
√1− x2
,
and j + n− 2k is even, we get
∫ 1
−1xj+n−2k dx
√1− x2
=
(j+n−2k)/2∏i=1
j + n− 2k − (2i− 1)
j + n− 2k − 2(i− 1)
π.
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MOMENTS PERTURBED
From (17) we conclude that, if the j−th moment associatedwith σ is perturbed and we apply the inverse Szegotransformation, then the obtained perturbation in S(x)corresponds to a perturbation of the moments associated withα in the following way
If n+ j is even, all even moments starting from µj areperturbed.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
STIELTJES PERTURBED
The Stieltjes function
Sj(x) =
j−1∑k=0
µkxk+1
+
∞∑k=j+1
µ2k−j−1x2k−j
+
∞∑k=j
µ2k−jx2k−j+1
= S(x) +Mj
∞∑k=j
B(2k − j, j)x2k−j+1
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
EXAMPLE
In matrix form, we have
T =
c0 +M0 c1 c2 c3 · · ·c−1 c0 +M0 c1 c2 · · ·c−2 c−1 c0 +M0 c1 · · ·c−3 c−2 c−1 c0 +M0 · · ·
......
......
. . .
.⇓
Szego transformation inverse⇓
H =
µ0 µ1 µ2 µ3 · · ·µ1 µ2 µ3 µ4 · · ·µ2 µ3 µ4 µ5 · · ·µ3 µ4 µ5 µ6 · · ·...
......
.... . .
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
TABLE OF CONTENTS
1 PERTURBATION OF MEASURES ON THE REAL LINE
2 PERTURBATION OF MEASURES ON THE UNIT CIRCLE
3 THE SZEGO TRANSFORMATION
4 DIRECT PROBLEM
5 INVERSE PROBLEM
6 REFERENCES
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
REFERENCES
Fuentes E., Garza L.E., Analysis of perturbations ofmoments associated with orthogonality linear functionalsthrough the Szego transformation, Rev. Integr. Temas Mat.33 (2015), no. 1, 61-82.
Garza L.E., Hernández J. and Marcellán F., "Spectraltransformations of measures supported on the unit circleand the Szego transformation", Numer. Algorithms 49(2008), no.1, 169-185.
Szego G. Orthogonal Polynomials, Amer. Math. Soc. Coll.Publi. Series, vol 23, Amer. Math. Soc., Providence, RI,1975. Fourth Edition.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
REFERENCES
Simon B., Orthogonal Polynomials on the unit circle, 2 vol.Amer. Math. Soc. Coll. Publi. Series. vol 54, 2005.
Marcellán F., Hernández J., Christoffel transforms andHermitian linear functionals, Mediterr. J. Math. vol. 2, 2005,451-458.
Marcellán F., Quintana Y., Polinomios ortogonales noestándar. Propiedades algebraicas y analíticas, XXIIEscuela Venezolana de Matemáticas, 2009.
Tasis C., Propiedades diferenciales de los polinomiosortogonales relativos a la circunferencia unidad, Tesisdoctoral, Universidad de Cantabria, España, 1989.
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS
PERT. REAL LINE PERT. UNIT CIRCLE SZEGO TRANS. DIR. PROBLEM INV.PROBLEM REFERENCES
Thanks for your attention!
EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS