universidad de colima - paginas.matem.unam.mx · 2 ortogonalidad en la circunferencia unidad y...

Polinomios ortogonales: Una introducción a lateoría de transformaciones espectrales

Luis E. GarzaUniversidad de Colima

Encuentro Nacional de Jóvenes Investigadores enMatemáticas, IMATE, UNAM

Diciembre 2, 2015.

LEGG (UdeC) Diciembre 2, 2015. 1 / 42

Contents

1 Polinomios ortogonales en la recta y matrices de Jacobi

2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

3 La representación CMV

4 Algunas generalizaciones

LEGG (UdeC) Diciembre 2, 2015. 2 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Contents

1 Polinomios ortogonales en la recta y matrices de Jacobi

2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

3 La representación CMV

4 Algunas generalizaciones

LEGG (UdeC) Diciembre 2, 2015. 3 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Orthogonal polynomials in R

Given a nontrivial probability measure µ supported on some infinite subset E ofthe real line, a (unique) sequence of orthonormal polynomials {pn}n>0 can bedefined as ∫

Epm(x)pn(x)dµ(x) = δm,n, n,m > 0, (1)

wherepn(x) = γnxn + ζnxn−1 + lower degree terms, (2)

with γn > 0, n > 0.

Classical orthogonal polynomials:

Jacobi dµ(x) = (1 − x)α(1 + x)βdx in [−1, 1]. (Tchebychev, Gegenbauer,Legendre)

Laguerre dµ(x) = xαe−xdx in R+.

Hermite dµ(x) = e−x2dx in R.

LEGG (UdeC) Diciembre 2, 2015. 4 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Some applications

OP appear in a wide range of applications such as:

Approximation theory

Integrable systems

Numerical integration

Signal theory

Image processing

Etc, etc, etc.

LEGG (UdeC) Diciembre 2, 2015. 5 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Three term recurrence relation

Starting from p0(x) = 1 and p−1(x) = 0, {pn}n>0 satisfies

xpn(x) = an+1 pn+1(x) + bn pn(x) + an pn−1(x), n > 0, (3)

where

an =

∫E

xpn−1(x)pn(x)dµ(x) =γn−1

γn> 0, n > 1,

and

bn =

∫E

xp2n(x)dµ(x) =

ζn

γn−ζn+1

γn+1, n > 0.

Favard’s theorem: Given any sequences {an}n>1, {bn}n>0 of real numbers, thepolynomials constructed with (3) are orthogonal with respect to some measuredµ(x).

LEGG (UdeC) Diciembre 2, 2015. 6 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

The monic Jacobi matrix

On the other hand, the monic OP with respect to µ are given by Pn(x) = pn(x)/γn,n > 0. In such a case, (3) becomes

Pn+1(x) = (x − bn)Pn(x) − dnPn−1(x), n > 0, (4)

with dn = a2n, and has the matrix representation

xP(x) = JP(x),

where

J =

b0 1 0 0 · · ·

d1 b1 1 0 · · ·

0 d2 b2 1. . .

0 0 d3 b3. . .

......

. . .. . .

. . .

,

is known as monic Jacobi matrix.

LEGG (UdeC) Diciembre 2, 2015. 7 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

The LU factorization of J

Notice that Pn(0) , 0, n > 1 ⇐⇒ J has a unique LU factorization, where L and Uare bidiagonal matrices

L =

1 0 0 0 · · ·

l1 1 0 0 · · ·

0 l2 1 0. . .

0 0 l3 1. . .

......

. . .. . .

. . .

, U =

u1 1 0 0 · · ·

0 u2 1 0 · · ·

0 0 u3 1. . .

0 0 0 u4. . .

......

. . .. . .

. . .

, (5)

where

l1 =d1

b0, ln =

dn

bn−1 − ln−1, n > 2, (6)

u1 = b0, un = bn−1 − ln−1, n > 2. (7)

LEGG (UdeC) Diciembre 2, 2015. 8 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Darboux transformations

Darboux transformation without parameter

J = LU, Jp := UL

Darboux transformation (not unique)

J = UL, Jd := LU

Notice that Jp and Jd are again tridiagonal matrices with ones as entries on theupper diagonal and, according to Favard’s theorem, they are monic Jacobimatrices associated with some nontrivial measure µ.

LEGG (UdeC) Diciembre 2, 2015. 9 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Canonical spectral transformations on R

Christoffel transformation (RC)

dµ = (x − β)dµ, β < supp(µ).

Uvarov transformation (UU)

dµ = dµ + Mrδ(x − β), Mr ∈ R.

Geronimus transformation (RG)

dµ =dµ

x − β+ Mrδ(x − β), β < supp(µ),Mr ∈ R.

Proposition

RC ◦ RG = I Identity transformation

RG ◦ RC = RU

LEGG (UdeC) Diciembre 2, 2015. 10 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

LST and Stieltjes functions

The Stieltjes function associated with µ is

S (x) =

∫E

dµ(t)x − t

=

∞∑k=0

µk

xk+1 ,

where µk =∫

E xkdµ(x) are the moments of µ. It has been shown that the previoustransformations can be expressed as

S (x) =A(x)S (x) + B(x)

D(x), (8)

where S (x) is the Stieltjes function associated with µ, and A(x), B(x), D(x) arepolynomials in the variable x, which are known. Furthermore,

Proposition (Zhedanov, 97)

All transformations of the form (8) can be obtained as a composition ofChristoffel and Geronimus transformations.

LEGG (UdeC) Diciembre 2, 2015. 11 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Rational spectral transformations

Associated polynomials

From a OPS {Pn}n>0, define the monic associated polynomials or order k,{P(k)

n }n>0, by the shifted recurrence relation

P(k)n+1(x) = (x − bn+k)P(k)

n (x) − dn+kP(k)n−1(x), n > 0,

i.e. removing the first k rows and columns of J.

Anti-associated polynomials

If we "push" the first k rows and columns of J, and introduce newcoefficients b−i (i = k, k − 1, ..., 1) and d−i (i = k − 1, k − 2, ..., 0), then theanti-associated polynomials of order k are defined by

P(−k)n+1 (x) = (x − bn+k)P(−k)

n (x) − dn+kP(−k)n−1 (x), n > 0,

where {bi}i>0 = {b−i}1i=k

⋃{bi}i>0 and {di}i>1 = {d−i}

0i=k−1

⋃{di}i>1.

LEGG (UdeC) Diciembre 2, 2015. 12 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

RST and Stieltjes functions

It has been shown that the previous transformations can be expressed as

S (x) =A(x)S (x) + B(x)C(x)S (x) + D(x)

, (9)

where S (x) is the transformed Stieltjes function, and A(x), B(x), C(x), D(x) arepolynomials in the variable x, which are known. Furthermore,

Proposition (Zhedanov, 97)

All transformations of the form (9) can be obtained as a combination ofChristoffel, Geronimus, associated and anti-associated transformations.

LEGG (UdeC) Diciembre 2, 2015. 13 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

ST and Jacobi matrices

Question

Can we express RC , RU , and RG in terms of thecorresponding monic Jacobi matrices?

Proposition

Let J be the monic Jacobi matrix associated with µ, and β ∈ R such that Pn(β) , 0,n > 1. Then,

J − βI = LU, J := UL + βI,

then J is the monic Jacobi matrix associated with dµ = (x − β)dµ, i.e. theChristoffel transformation.

LEGG (UdeC) Diciembre 2, 2015. 14 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Christoffel transformation

Proposition

Let µ and J be as before. Consider the following transformations

C1 := J − β1I = L1U1, C1 := U1L1 + β1I,C2 := C1 − β2I = L2U2, C2 := U2L2 + β2I,...

Cm := Cm−1 − βmI = LmUm, Cm := UmLm + βmI,

with β1, β2, . . . , βm ∈ R. If {Pn,i} is the MOPS associated with Ci, 1 6 i 6 m − 1, andassuming that Pn(β) , 0, Pn,i(βi+1) , 0, n > 1, 1 6 i 6 m − 1, then Cm is the monicJacobi matrix associated with the measure

dµ = (x − β1)(x − β2) . . . (x − βm)dµ.

LEGG (UdeC) Diciembre 2, 2015. 15 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Uvarov transformation

Proposition

Let J0 be the monic Jacobi matrix associated with µ. Consider

J0 − βI = L1U1, J1 := U1L1,

J1 = U2L2, J2 := L2U2 + βI.

Then J2 is the monic Jacobi matrix associated with the measure

dµ = dµ + Mrδ(x − β),

i.e. the Uvarov transformation of µ, where

Mr =µ0(b0 − β − s)

s,

with µ0 =∫

E dµ(x) and s is the free parameter associated with the UL factorizationof J1.

LEGG (UdeC) Diciembre 2, 2015. 16 / 42

Polinomios ortogonales en la recta y matrices de Jacobi

Geronimus transformation

Proposition

Let J1 be the monic Jacobi matrix associated with µ. Suppose there exists µ s.t.dµ = (x − β)dµ. If

J1 − βI = U1L1, J2 := L1U1 + βI,

then J2 is the monic Jacobi matrix associated with

dµ =dµ

x − β+ Mrδ(x − β),

i.e. the Geronimus transformation of µ, where Mr =

∫E dµ

s and s is the freeparameter associated with the UL factorization of J1.

LEGG (UdeC) Diciembre 2, 2015. 17 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Contents

1 Polinomios ortogonales en la recta y matrices de Jacobi

2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

3 La representación CMV

4 Algunas generalizaciones

LEGG (UdeC) Diciembre 2, 2015. 18 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Measures on T and Toeplitz matrices

If σ is a nontrivial positive Borel measure supported on the unit circle, then we canconsider the inner product

〈p, q〉 =

∫T

p(z)q(z)dσ(z),

The moments are defined by cn := 〈1, zn〉 =∫T

zndσ(z), n ∈ Z.Notice that we have

cn := 〈1, zn〉 =

∫T

zndσ(z) =

∫T

z−ndσ(z) =⟨z−n, 1

⟩= 〈1, z−n〉 = c−n,

and thus the Gram matrix in terms of the standard basis {1, z, z2, . . .} is the Toeplitzmatrix

T =

c0 c1 · · · cn · · ·

c−1 c0 · · · cn−1 · · ·

......

. . ....

c−n c−n+1 · · · c0 · · ·

......

.... . .

(10)

LEGG (UdeC) Diciembre 2, 2015. 19 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Orthogonal polynomials on T

We can apply G-S to get a sequence {ϕn}n>0, where ϕ(z) has the form

ϕ(z) = κnzn + lower order terms.

We have Φn(z) = ϕn(z)/κn, satisfying

Φn+1(z) = zΦn(z) + Φn+1(0)Φ∗n(z), (11)

Φn+1(z) =(1 − |Φn+1(0)|2

)zΦn(z) + Φn+1(0)Φ∗n+1(z), (12)

Φ∗n(z) = znΦn(z−1) (reversed polynomial),

{Φn(0)}n>1 (Verblunsky, Schur, reflection parameters).

|Φn(0)| < 1, n > 1.

Furthermore, if kn = ‖Φn‖2 = κ−2

n , then

kn = (1 − |Φn(0)|2)kn−1

LEGG (UdeC) Diciembre 2, 2015. 20 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Hessenberg matrices

The multiplication operator with respect to {ϕn}n>0 is represented in a matrix formby

zϕ(z) = Hϕϕ(z), (13)

where ϕ(z) =[ϕ0(z), ϕ1(z), . . . , ϕn(z), . . .

]t and Hϕ is a lower Hessenberg matrixwhose entries are

hn, j =

κnκn+1

if j = n + 1,−κ j

κnΦn+1(0)Φ j(0) if j 6 n,

0 if j > n + 1.(14)

Notice that Hϕ is defined in terms of {Φn(0)}n>1.

LEGG (UdeC) Diciembre 2, 2015. 21 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Hessenberg matrices (cont.)

Proposition

Hϕ satisfies

(i) HϕH∗ϕ = I,(ii) H∗ϕHϕ = I − λ∞(0)ϕ(0)ϕ(0)∗,

where I is the semi-infinite identity matrix and λ∞(0) =∏∞

n=0(1 − |Φn+1(0)|2).

Remark

Hϕ is unitary ⇐⇒∑∞

n=0 |Φn(0)|2 = +∞ ⇐⇒ logσ′ < L1(

dθ2π

)(σ < Szego class).

Remark

In the monic case, HΦ has as entries

hn, j =

1 if j = n + 1,−

knk j

Φn+1(0)Φ j(0) if j 6 n,0 if j > n + 1.

(15)

LEGG (UdeC) Diciembre 2, 2015. 22 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Canonical spectral transformations on T

Christoffel transformation (FC)

dσ = |z − α|2dσ, α ∈ C.

Uvarov transformation (FU)

dσ = dσ + Mcδ(z − α) + Mcδ(z − α−1), α ∈ C � {0}, Mc ∈ C.

Geronimus transformation (FG)

dσ =dσ|z − α|2

+ Mcδ(z − α) + Mcδ(z − α−1), α ∈ C � {0}, Mc ∈ C.

Proposition

FC ◦ FG = I Identity transformation

FG ◦ FC = FU

LEGG (UdeC) Diciembre 2, 2015. 23 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

ST and Carathéodory functions

Define

F(z) = c0 + 2∞∑

k=1

c−kzk,

In the positive definite case, F(z) is analytic, Re[F(z)] > 0 in D, and

F(z) =

∫T

w + zw − z

dσ(w).

The previous transformations can be expressed as

F(z) =A(z)F(z) + B(z)

D(z), (16)

where F(z) is associated with σ and A(z), B(z),D(z) are known polynomials in z.

LEGG (UdeC) Diciembre 2, 2015. 24 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Rational spectral transformations

Associated polynomials

Denote by {Φ(N)n }n>0 the associated polynomials of order N, defined by

Φ(N)n+1(z) = zΦ(N)

n (z) + Φn+N+1(0)(Φ(N)n )∗(z), n > 0,

i.e. the first N coefficients are removed.

Anti-associated polynomials

Let ν1, ν2, . . . , νN ∈ C with |ν j| < 1, 1 6 j 6 N. Define{Φn(0)}n>1 = {ν j}

Nj=1

⋃{Φ j(0)}∞j=1. Then, the polynomials

Φ(−N)n+1 (z) = zΦ(−N)

n (z) + Φn+1(0)(Φ(−N)n )∗(z), n > 0,

are called anti-associated polynomials of order N.

LEGG (UdeC) Diciembre 2, 2015. 25 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

RST and Carathéodory functions

Aleksandrov transformation

Define {Φλn(0)}n>1, where Φλ

n(0) = λΦn(0), with λ ∈ C, |λ| = 1. Then,

Φλn+1(z) = zΦλ

n(z) + Φλn+1(0)(Φλ

n)∗(z),

are called Aleksandrov polynomials.

These transformations can be expressed as

F(z) =A(z)F(z) + B(z)C(z)F(z) + D(z)

, (17)

where F(z) is the transformed Carathéodory function and A(z), B(z),C(z),D(z) areknown polynomials in z.

LEGG (UdeC) Diciembre 2, 2015. 26 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

ST and Hessenberg matrices

Question

Can we express FC , FU , and FG in terms of thecorresponding Hessenberg matrices?

LEGG (UdeC) Diciembre 2, 2015. 27 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Christoffel transformation

Let dσC = |z − α|2dσ, and {ψn}n>0 its OPS. The relation between both families ofpolynomials is

(z − α)ψn(z) =

√Kn(α, α)

Kn+1(α, α)ϕn+1(z) −

n∑j=0

ϕn+1(α)ϕ j(α)√

Kn+1(α, α)Kn(α, α)ϕ j(z), (18)

where Kn(z, y) =

n∑k=0

ϕk(z)ϕk(y).

In matrix form

(z − α)ψ(z) = MCϕ(z), (19)

where MC has entries

mi, j =

−

ϕi+1(α)ϕ j(α)√

Ki+1(α,α)Ki(α,α), if j 6 i,√

Ki(α,α)Ki+1(α,α) , if j = i + 1,

0, if j > i + 1.

(20)

LEGG (UdeC) Diciembre 2, 2015. 28 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Christoffel transformation

Proposition

MC satisfies

(i) MCM∗C = I.

(ii) M∗CMC = I − λ∞(α)ϕ(α)ϕ(α)∗,

Proposition

Let MCn be the n × n principal submatrix of MC . Then,

(i) MCnMC∗n = In −

Kn−1(α,α)Kn(α,α) ene∗n, where en = [0, . . . , 0, 1]t ∈ C(n,1).

(ii) MC∗nMCn = In −

1Kn(α,α)ϕ

(n)(α)ϕ(n)∗(α), whereϕ(n)(α) = [ϕ0(α), ϕ1(α), . . . , ϕn−1(α)]t

LEGG (UdeC) Diciembre 2, 2015. 29 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Christoffel transformation (cont.)

Furthermore, if Lϕψ is the lower triangular matrix such that ϕ(z) = Lϕψψ(z), then

Proposition

We have

Hϕ − αI = LϕψMC , (21)

Hψ − αI = MCLϕψ. (22)

An "almost" QR factorization appears, since (MC)n is a quasi-unitary matrix, i.e. itsfirst n − 1 rows constitute an orthonormal set, and the last row is orthogonal withrespect to this set, but is not normalized.

LEGG (UdeC) Diciembre 2, 2015. 30 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Uvarov transformation

Let σU be the Uvarov transformation of σ. If we assume {υn}n>0 is its associatedOPS, and define by Hυ its corresponding Hessenberg matrix, then

Proposition

Hϕ − αI = LϕψMC , (23)

Hυ − αI = LUMU , (24)

where LU = LυϕLϕψ, MU = MCL−1υϕ , and L are the matrices of change of bases for

the orthonormal polynomial families denoted by their subindices.

LEGG (UdeC) Diciembre 2, 2015. 31 / 42

Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

Geronimus transformation

Let σG be the Geronimus transformation of σ. If {Gn}n>0 is its OPS and MG aHessenberg matrix such that

(z − α)Φ(z) = MGG(z).

Then we get

Proposition

Let LG be such that G(z) = LGΦ(z) and denote by HG the Hessenberg matrixassociated with {Gn}n>0. Then,

HΦ − αI = MGLG (25)

andHG − αI = LGMG. (26)

LEGG (UdeC) Diciembre 2, 2015. 32 / 42

La representación CMV

Contents

1 Polinomios ortogonales en la recta y matrices de Jacobi

2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

3 La representación CMV

4 Algunas generalizaciones

LEGG (UdeC) Diciembre 2, 2015. 33 / 42

La representación CMV

Laurent polynomials space

Let Λ(k,l) be span{z j}lj=k, k 6 l, and P(k,l) the orthogonal projection over Λ(k,l) withrespect to a bilinear functional L. Set

Λ(n) =

Λ(−k,k) n = 2k,Λ(−k,k+1) n = 2k + 1,

and let P(n) be the orthogonal projection over Λ(n). Furthermore, define

χ(0)n =

z−k n = 2k,zk+1 n = 2k + 1.

Applying Gram-Schmidt, we obtain the CMV basis from

χn = (1 − P(n−1))χ(0)n .

LEGG (UdeC) Diciembre 2, 2015. 34 / 42

La representación CMV

The CMV basis

{χn}n>0 can be expressed in terms of {Φn(z)}n>0 as follows

χ2n(z) = z−nΦ∗2n(z), n > 0,χ2n−1(z) = z−n+1Φ2n−1(z), n > 1,

and satisfies the following recurrence relations

zχ0 = −Φ1(0)χ0 + ρ0χ1,

z(χ2n−1χ2n

)= ΞT

2n−1

(χ2n−2χ2n−1

)+ Ξ2n

(χ2n

χ2n+1

), n > 1,

with

Ξn :=(−ρn−1Φn+1(0) ρn−1ρn

−Φn(0)Φn+1(0) Φn(0)ρn

), Ξn :=

(−ρn−1Φn+1(0) ρn−1ρn

−Φn(0)Φn+1(0) Φn(0)ρn

),

where ρn = |1 − |Φn+1(0)|2|1/2 and ρn = ςnρn, with ςn = sign(1 − |Φn|2).

LEGG (UdeC) Diciembre 2, 2015. 35 / 42

La representación CMV

A five diagonal matrix

Thus, the five diagonal matrix C of CMV representation is defined as

Ci, j =⟨χi, zχ j

⟩L,

in such a way that

C =

−Φ1(0) −Φ2(0)ρ0 ρ1ρ0 0 0 . . .

ρ0 −Φ2(0)Φ1(0) Φ1(0)ρ1 0 0 . . .

0 −Φ3(0)ρ1 −Φ3(0)Φ2(0) −Φ4(0)ρ2 ρ3ρ2 . . .

0 ρ2ρ1 Φ2(0)ρ2 −Φ4(0)Φ3(0) Φ3(0)ρ3 . . .

0 0 0 −Φ5(0)ρ3 −Φ5(0)Φ4(0) . . .. . . . . . . . . . . . . . . . . .

.

LEGG (UdeC) Diciembre 2, 2015. 36 / 42

La representación CMV

CMV factorization

Furthermore,C =WM,

where

M =

1

Θ1Θ3

. . .

,

W =

Θ0

Θ2Θ4

. . .

,with

Θ j =

(−Φ j+1(0) ρ j

ρ j Φ j+1(0)

).

LEGG (UdeC) Diciembre 2, 2015. 37 / 42

La representación CMV

ST and CMV matrices

Open question

Can we express FC , FU , and FG in terms of thecorresponding CMV matrices?

Partial answer: Yes (Cantero-Marcellán-Velázquez, 2015)

LEGG (UdeC) Diciembre 2, 2015. 38 / 42

Algunas generalizaciones

Contents

1 Polinomios ortogonales en la recta y matrices de Jacobi

2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg

3 La representación CMV

4 Algunas generalizaciones

LEGG (UdeC) Diciembre 2, 2015. 39 / 42

Algunas generalizaciones

Matrix orthogonal polynomials

A matrix polynomial has the form P(x) = Anzn + . . . A0, where Ai are q× q matrices.

A matrix inner product can be defined by∫E

P(x)dµ(x)QT (x),

where dµ(x) is a q × q symmetric matrix of measures with support in E ∈ R.Orthogonality is defined by∫

EPn(x)dµ(x)PT

m(x) = δn.mCn,

where Cn is a nonsingular matrix.

LEGG (UdeC) Diciembre 2, 2015. 40 / 42

Algunas generalizaciones

Spectral transformation for matrix polynomials

Christoffel transformation (Marcellán, Mañas - 2015)

Uvarov transformation (Marcellán, Piñar, 2000s)

Geronimus transformation (Marcellán, LG - 2015)

Other perturbations studied by Choque, Domínguez de la Iglesia, LG.

LEGG (UdeC) Diciembre 2, 2015. 41 / 42

Algunas generalizaciones

¡Gracias por su atención!

LEGG (UdeC) Diciembre 2, 2015. 42 / 42