Post on 04-Jul-2020
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