Analysis of perturbations of moments associated with orthogonality ...

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


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.



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.



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.



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.



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)).



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)]



TABLE OF CONTENTS




4 DIRECT PROBLEM

5 INVERSE PROBLEM

6 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.



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).



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) =∞}.



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.



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.



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.



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.



TABLE OF CONTENTS




4 DIRECT PROBLEM

5 INVERSE PROBLEM

6 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)



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.



[−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



TABLE OF CONTENTS




4 DIRECT PROBLEM

5 INVERSE PROBLEM

6 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).




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.




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 .



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 .



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)



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.



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)



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.



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 · · ·

......

......

. . .

.EDINSON FUENTES & LUIS E. GARZA PERTURBATIONS OF MOMENTS


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.



TABLE OF CONTENTS




4 DIRECT PROBLEM

5 INVERSE PROBLEM

6 REFERENCES



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.




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)




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 .



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.



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.



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

.



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)

.



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.



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.



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



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 · · ·...

......

.... . .



TABLE OF CONTENTS




4 DIRECT PROBLEM

5 INVERSE PROBLEM

6 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.



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.



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