Una introducción a la teoría de transformaciones espectrales

Transcripción

Polinomios ortogonales: Una introducción a la
teoría de transformaciones espectrales
Luis E. Garza
Universidad de Colima
Encuentro Nacional de Jóvenes Investigadores en
Matemáticas, IMATE, UNAM
Diciembre 2, 2015.
LEGG (UdeC)
Diciembre 2, 2015.
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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)
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Contents
1
2
3
4
LEGG (UdeC)
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Orthogonal polynomials in R
Given a nontrivial probability measure µ supported on some infinite subset E of
the real line, a (unique) sequence of orthonormal polynomials {pn }n>0 can be
defined as
Z
pm (x)pn (x)dµ(x) = δm,n ,
n, m > 0,
(1)
E
where
pn (x) = γn xn + ζn xn−1 + lower degree terms,
(2)
with γn > 0, n > 0.
Classical orthogonal polynomials:
Jacobi dµ(x) = (1 − x)α (1 + x)β dx in [−1, 1]. (Tchebychev, Gegenbauer,
Legendre)
Laguerre dµ(x) = xα e−x dx in R+ .
2
Hermite dµ(x) = e−x dx in R.
LEGG (UdeC)
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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)
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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),
where
an =
Z
xpn−1 (x)pn (x)dµ(x) =
E
and
bn =
Z
xp2n (x)dµ(x) =
E
γn−1
> 0,
γn
ζn ζn+1
−
,
γn γn+1
n > 0,
(3)
n > 1,
n > 0.
Favard’s theorem: Given any sequences {an }n>1 , {bn }n>0 of real numbers, the
polynomials constructed with (3) are orthogonal with respect to some measure
dµ(x).
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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) − dn Pn−1 (x),
n > 0,
(4)
with dn = a2n , and has the matrix representation
xP(x) = JP(x),
where






J = 




b0
d1
1
b1
0
1
0
0
0
d2
b2
1
0
..
.
0
..
.
d3
..
.
b3
..
.
···
···
..
.






,
. . 
. 

. . 
.
is known as monic Jacobi matrix.
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The LU factorization of J
Notice that Pn (0) , 0, n > 1 ⇐⇒ J has a unique LU factorization, where L and U
are bidiagonal matrices






L = 




1
l1
0
1
0
0
0
0
0
l2
1
0
0
..
.
0
..
.
l3
..
.
1
..
.
···
···
..
.












,
U
=


. . 

. 



.. 
.
u1
0
1
u2
0
1
0
0
0
0
u3
1
0
..
.
0
..
.
0
..
.
u4
..
.
···
···
..
.






,
. . 
. 

. . 
.
(5)
where
LEGG (UdeC)
l1
=
u1
=
dn
d1
, ln =
, n > 2,
b0
bn−1 − ln−1
b0 , un = bn−1 − ln−1 , n > 2.
(6)
(7)
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Darboux transformations
Darboux transformation without parameter
J = LU,
J p := UL
Darboux transformation (not unique)
J = UL,
Jd := LU
Notice that J p and Jd are again tridiagonal matrices with ones as entries on the
upper diagonal and, according to Favard’s theorem, they are monic Jacobi
matrices associated with some nontrivial measure µ̃.
LEGG (UdeC)
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Canonical spectral transformations on R
Christoffel transformation (RC )
dµ̃ = (x − β)dµ,
β < supp(µ).
Uvarov transformation (UU )
dµ̃ = dµ + Mr δ(x − β),
Mr ∈ R.
Geronimus transformation (RG )
dµ̃ =
dµ
+ Mr δ(x − β),
x−β
β < supp(µ), Mr ∈ R.
Proposition
LEGG (UdeC)
RC ◦ RG
=
I Identity transformation
RG ◦ RC
=
RU
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LST and Stieltjes functions
The Stieltjes function associated with µ is
S (x) =
∞
Z
E
dµ(t) X µk
=
,
x−t
xk+1
k=0
where µk = E xk dµ(x) are the moments of µ. It has been shown that the previous
transformations can be expressed as
R
A(x)S (x) + B(x)
,
Se(x) =
D(x)
(8)
e(x) is the Stieltjes function associated with µ̃, and A(x), B(x), D(x) are
where S
polynomials in the variable x, which are known. Furthermore,
Proposition (Zhedanov, 97)
All transformations of the form (8) can be obtained as a composition of
Christoffel and Geronimus transformations.
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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
(k)
(k)
P(k)
n+1 (x) = (x − bn+k )Pn (x) − dn+k Pn−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 new
coefficients b−i (i = k, k − 1, ..., 1) and d−i (i = k − 1, k − 2, ..., 0), then the
anti-associated polynomials of order k are defined by
(−k)
(−k)
P(−k)
n+1 (x) = (x − b̃n+k )Pn (x) − d̃n+k Pn−1 (x),
where {b̃i }i>0 = {b−i }1i=k
LEGG (UdeC)
n > 0,
S
S
{bi }i>0 and {d̃i }i>1 = {d−i }0i=k−1 {di }i>1 .
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RST and Stieltjes functions
It has been shown that the previous transformations can be expressed as
A(x)S (x) + B(x)
Se(x) =
,
C(x)S (x) + D(x)
(9)
e(x) is the transformed Stieltjes function, and A(x), B(x), C(x), D(x) are
where S
polynomials in the variable x, which are known. Furthermore,
Proposition (Zhedanov, 97)
All transformations of the form (9) can be obtained as a combination of
Christoffel, Geronimus, associated and anti-associated transformations.
LEGG (UdeC)
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ST and Jacobi matrices
Question
Can we express RC , RU , and RG in terms of the
corresponding 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,
J̃ := UL + βI,
then J̃ is the monic Jacobi matrix associated with dµ̃ = (x − β)dµ, i.e. the
Christoffel transformation.
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Christoffel transformation
Proposition
Let µ and J be as before. Consider the following transformations
C1
:=
J − β1 I = L1 U1 ,
C̃1 := U1 L1 + β1 I,
C2
..
.
:=
C̃1 − β2 I = L2 U2 ,
C̃2 := U2 L2 + β2 I,
Cm
:=
C̃m−1 − βm I = Lm Um ,
C̃m := Um Lm + βm I,
with β1 , β2 , . . . , βm ∈ R. If {Pn,i } is the MOPS associated with C̃i , 1 6 i 6 m − 1, and
assuming that Pn (β) , 0, Pn,i (βi+1 ) , 0, n > 1, 1 6 i 6 m − 1, then C̃m is the monic
Jacobi matrix associated with the measure
dµ̃ = (x − β1 )(x − β2 ) . . . (x − βm )dµ.
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Uvarov transformation
Proposition
Let J0 be the monic Jacobi matrix associated with µ. Consider
J0 − βI =
L1 U1 ,
J1 := U1 L1 ,
=
U2 L2 ,
J2 := L2 U2 + βI.
J1
Then J2 is the monic Jacobi matrix associated with the measure
dµ̃ = dµ + Mr δ(x − β),
i.e. the Uvarov transformation of µ, where
Mr =
with µ0 =
of J1 .
R
E
µ0 (b0 − β − s)
,
s
dµ(x) and s is the free parameter associated with the UL factorization
LEGG (UdeC)
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Geronimus transformation
Proposition
Let J1 be the monic Jacobi matrix associated with µ̂. Suppose there exists µ s.t.
dµ̂ = (x − β)dµ. If
J1 − βI
= U1 L 1 ,
J2 := L1 U1 + βI,
then J2 is the monic Jacobi matrix associated with
dµ̃ =
dµ̂
+ Mr δ(x − β),
x−β
R
dµ̂
i.e. the Geronimus transformation of µ̂, where Mr = E s
parameter associated with the UL factorization of J1 .
LEGG (UdeC)
and s is the free
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Contents
1
2
3
4
LEGG (UdeC)
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Measures on T and Toeplitz matrices
If σ is a nontrivial positive Borel measure supported on the unit circle, then we can
consider the inner product
hp, qi =
Z
p(z)q(z)dσ(z),
R
The moments are defined by cn := h1, z i = T zn dσ(z), n ∈ Z.
Notice that we have
Z
Z
n
n
z dσ(z) =
z−n dσ(z) = z−n , 1 = h1, z−n i = c̄−n ,
cn := h1, z i =
T
n
T
T
and thus the Gram matrix in terms of the standard basis {1, z, z2 , . . .} is the Toeplitz
matrix

 c0
 c
 −1
 .
T =  ..
 c
 −n
 .
..
LEGG (UdeC)
c1
c0
..
.
···
···
..
.
cn
cn−1
..
.
c−n+1
..
.
···
c0
..
.
···
···







· · · 
. . 
.
(10)
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Orthogonal polynomials on T
We can apply G-S to get a sequence {ϕn }n>0 , where ϕ(z) has the form
ϕ(z) = κn zn + lower order terms.
We have Φn (z) = ϕn (z)/κn , satisfying
Φn+1 (z) =
Φn+1 (z) =
zΦn (z) + Φn+1 (0)Φ∗n (z),
1 − |Φn+1 (0)|2 zΦn (z) + Φn+1 (0)Φ∗n+1 (z),
(11)
(12)
Φ∗n (z) = zn Φ̄n (z−1 ) (reversed polynomial),
{Φn (0)}n>1 (Verblunsky, Schur, reflection parameters).
|Φn (0)| < 1,
n > 1.
Furthermore, if kn = kΦn k2 = κn−2 , then
kn = (1 − |Φn (0)|2 )kn−1
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Hessenberg matrices
The multiplication operator with respect to {ϕn }n>0 is represented in a matrix form
by
zϕ(z) = Hϕ ϕ(z),
(13)
where ϕ(z) = ϕ0 (z), ϕ1 (z), . . . , ϕn (z), . . . t and Hϕ is a lower Hessenberg matrix
whose entries are
hn, j
 κn




 κn+1κ j
=
− κn Φn+1 (0)Φ j (0)



 0
if
if
if
j = n + 1,
j 6 n,
j > n + 1.
(14)
Notice that Hϕ is defined in terms of {Φn (0)}n>1 .
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Hessenberg matrices (cont.)
Proposition
Hϕ satisfies
(i) Hϕ H∗ϕ = I,
(ii) H∗ϕ Hϕ = I − λ∞ (0)ϕ(0)ϕ(0)∗ ,
where I is the semi-infinite identity matrix and λ∞ (0) =
Q∞
n=0 (1
− |Φn+1 (0)|2 ).
Remark
Hϕ is unitary ⇐⇒
P∞
n=0
|Φn (0)|2 = +∞ ⇐⇒ log σ0 < L1
dθ
2π
(σ < Szegő class).
Remark
In the monic case, HΦ has as entries
hn, j
LEGG (UdeC)


1



 kn
− k j Φn+1 (0)Φ j (0)
=



 0
if
if
if
j = n + 1,
j 6 n,
j > n + 1.
(15)
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Canonical spectral transformations on T
Christoffel transformation (FC )
dσ̃ = |z − α|2 dσ,
α ∈ C.
Uvarov transformation (FU )
dσ̃ = dσ + Mc δ(z − α) + M c δ(z − ᾱ−1 ),
α ∈ C {0},
Mc ∈ C.
Geronimus transformation (FG )
dσ̃ =
dσ
+ Mc δ(z − α) + M̄c δ(z − ᾱ−1 ),
|z − α|2
α ∈ C {0},
Mc ∈ C.
Proposition
LEGG (UdeC)
FC ◦ FG
=
I Identity transformation
FG ◦ FC
=
FU
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ST and Carathéodory functions
Define
F(z) = c0 + 2
∞
X
c−k zk ,
k=1
In the positive definite case, F(z) is analytic, Re[F(z)] > 0 in D, and
F(z) =
Z
T
w+z
dσ(w).
w−z
The previous transformations can be expressed as
e = A(z)F(z) + B(z) ,
F(z)
D(z)
(16)
e is associated with σ̃ and A(z), B(z), D(z) are known polynomials in z.
where F(z)
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Rational spectral transformations
Associated polynomials
(N)
Denote by {Φn }n>0 the associated polynomials of order N , defined by
(N) ∗
(N)
Φ(N)
n+1 (z) = zΦn (z) + Φn+N+1 (0)(Φn ) (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
S
{Φ̂n (0)}n>1 = {ν j }Nj=1 {Φ j (0)}∞j=1 . Then, the polynomials
(−N)
Φn+1
(z) = zΦ(−N)
(z) + Φ̂n+1 (0)(Φ(−N)
)∗ (z),
n
n
n > 0,
are called anti-associated polynomials of order N .
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RST and Carathéodory functions
Aleksandrov transformation
Define {Φλn (0)}n>1 , where Φλn (0) = λΦn (0), with λ ∈ C, |λ| = 1. Then,
Φλn+1 (z) = zΦλn (z) + Φλn+1 (0)(Φλn )∗ (z),
are called Aleksandrov polynomials.
These transformations can be expressed as
e = A(z)F(z) + B(z) ,
F(z)
C(z)F(z) + D(z)
(17)
e is the transformed Carathéodory function and A(z), B(z), C(z), D(z) are
where F(z)
known polynomials in z.
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ST and Hessenberg matrices
Question
Can we express FC , FU , and FG in terms of the
corresponding Hessenberg matrices?
LEGG (UdeC)
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Let dσC = |z − α|2 dσ, and {ψn }n>0 its OPS. The relation between both families of
polynomials is
s
(z − α)ψn (z) =
where Kn (z, y) =
n
X
n
X
ϕn+1 (α)ϕ j (α)
Kn (α, α)
ϕn+1 (z) −
ϕ j (z),
√
Kn+1 (α, α)
Kn+1 (α, α)Kn (α, α)
j=0
(18)
ϕk (z)ϕk (y).
k=0
In matrix form
(z − α)ψ(z) = MC ϕ(z),
(19)
where MC has entries

ϕ (α)ϕ (α)


− √K i+1(α,α)Kj (α,α) ,



i+1
i


 q
Ki (α,α)
mi, j = 


Ki+1 (α,α) ,




 0,
LEGG (UdeC)
if
j 6 i,
if
j = i + 1,
if
j > i + 1.
(20)
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Proposition
MC satisfies
(i) MC MC∗ = I.
(ii) MC∗ MC = I − λ∞ (α)ϕ(α)ϕ(α)∗ ,
Proposition
Let MC n be the n × n principal submatrix of MC . Then,
(i) MC n MC ∗n = In −
(ii)
Kn−1 (α,α)
∗
t
Kn (α,α) en en , where en = [0, . . . , 0, 1]
1
MC ∗n MC n = In − Kn (α,α)
ϕ(n) (α)ϕ(n)∗ (α), where
(n)
ϕ (α) = [ϕ0 (α), ϕ1 (α), . . . , ϕn−1 (α)]t
LEGG (UdeC)
∈ C(n,1) .
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Christoffel transformation (cont.)
Furthermore, if Lϕψ is the lower triangular matrix such that ϕ(z) = Lϕψ ψ(z), then
Proposition
We have
Hϕ − αI =
Lϕψ MC ,
(21)
Hψ − αI =
MC Lϕψ .
(22)
An "almost" QR factorization appears, since (MC )n is a quasi-unitary matrix, i.e. its
first n − 1 rows constitute an orthonormal set, and the last row is orthogonal with
respect to this set, but is not normalized.
LEGG (UdeC)
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Uvarov transformation
Let σU be the Uvarov transformation of σ. If we assume {υn }n>0 is its associated
OPS, and define by Hυ its corresponding Hessenberg matrix, then
Proposition
Hϕ − αI =
Lϕψ MC ,
(23)
Hυ − αI =
LU MU ,
(24)
where LU = Lυϕ Lϕψ , MU = MC L−1
υϕ , and L are the matrices of change of bases for
the orthonormal polynomial families denoted by their subindices.
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Geronimus transformation
Let σG be the Geronimus transformation of σ. If {Gn }n>0 is its OPS and MG a
Hessenberg matrix such that
(z − α)Φ(z) = MG G(z).
Then we get
Proposition
Let LG be such that G(z) = LG Φ(z) and denote by HG the Hessenberg matrix
associated with {Gn }n>0 . Then,
HΦ − αI = MG LG
(25)
HG − αI = LG MG .
(26)
and
LEGG (UdeC)
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Contents
1
2
3
4
LEGG (UdeC)
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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) with
respect to a bilinear functional L. Set
Λ
(n)



n = 2k,
Λ(−k,k)
=

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
=

zk+1
n = 2k,
n = 2k + 1.
Applying Gram-Schmidt, we obtain the CMV basis from
χn = (1 − P(n−1) )χ(0)
n .
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The CMV basis
{χn }n>0 can be expressed in terms of {Φn (z)}n>0 as follows
χ2n (z)
χ2n−1 (z)
= z−n Φ∗2n (z),
= z
−n+1
n > 0,
Φ2n−1 (z),
n > 1,
and satisfies the following recurrence relations
zχ0
!
χ2n−1
z
χ2n
=
−Φ1 (0)χ0 + ρ0 χ1 ,
!
!
χ2n−2
χ2n
T
b
= Ξ2n−1
+ Ξ2n
,
χ2n−1
χ2n+1
n > 1,
with
−ρn−1 Φn+1 (0)
Ξn :=
−Φn (0)Φn+1 (0)
!
ρn−1 ρn
,
Φn (0)ρn
−ρ̂n−1 Φn+1 (0)
b
Ξn :=
−Φn (0)Φn+1 (0)
!
ρ̂n−1 ρ̂n
,
Φn (0)ρ̂n
where ρn = |1 − |Φn+1 (0)|2 |1/2 and ρ̂n = ςn ρn , with ςn = sign(1 − |Φn |2 ).
LEGG (UdeC)
Diciembre 2, 2015.
35 / 42
A five diagonal matrix
Thus, the five diagonal matrix C of CMV representation is defined as
D
E
Ci, j = χi , zχ j ,
L
in such a way that

 −Φ1 (0)

ρ0


0
C = 
0


0

...
LEGG (UdeC)
−Φ2 (0)ρ̂0
−Φ2 (0)Φ1 (0)
−Φ3 (0)ρ1
ρ2 ρ1
0
...
ρ̂1 ρ̂0
Φ1 (0)ρ̂1
−Φ3 (0)Φ2 (0)
Φ2 (0)ρ2
0
...
0
0
−Φ4 (0)ρ̂2
−Φ4 (0)Φ3 (0)
−Φ5 (0)ρ3
...
0
...
0
...
ρ̂3 ρ̂2
...
Φ3 (0)ρ̂3
...
−Φ5 (0)Φ4 (0) . . .
...
...
Diciembre 2, 2015.





 .




36 / 42
CMV factorization
Furthermore,
C = WM,
where

 1


M = 



 Θ0


W = 


Θ1
Θ2
with
Θj =
LEGG (UdeC)
−Φ j+1 (0)
ρ̂ j
Θ3
Θ4




 ,

. . 
.




 ,
. . 
.
ρj
Φ j+1 (0)
!
.
Diciembre 2, 2015.
37 / 42
ST and CMV matrices
Open question
Can we express FC , FU , and FG in terms of the
corresponding CMV matrices?
Partial answer: Yes (Cantero-Marcellán-Velázquez, 2015)
LEGG (UdeC)
Diciembre 2, 2015.
38 / 42
Contents
1
2
3
4
LEGG (UdeC)
Diciembre 2, 2015.
39 / 42
Matrix orthogonal polynomials
A matrix polynomial has the form P(x) = An zn + . . . A0 , where Ai are q × q matrices.
A matrix inner product can be defined by
Z
P(x)dµ(x)QT (x),
E
where dµ(x) is a q × q symmetric matrix of measures with support in E ∈ R.
Orthogonality is defined by
Z
Pn (x)dµ(x)PTm (x) = δn.mCn ,
E
where Cn is a nonsingular matrix.
LEGG (UdeC)
Diciembre 2, 2015.
40 / 42
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
¡Gracias por su atención!
LEGG (UdeC)
Diciembre 2, 2015.
42 / 42

Una introducción a la teoría de transformaciones espectrales

Transcripción

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