Comportamiento Asint ´ otico de Secuencias de Polinomios Ortogonales e Interpretaci ´ on Electrost ´ atica sus Ceros Edmundo J. Huertas Universidad Polit´ ecnica de Madrid - Grupo SERPA-HGA March 12, 2015- Seminario del Departamento de Matem ´ aticas e Inform ´ atica Aplicadas a las II. Civil y Naval (UPM 2015) Polinomios Ortogonales UPM 2015 1 / 50
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Comportamiento Asintotico de Secuencias dePolinomios Ortogonales e
Interpretacion Electrostatica sus Ceros
Edmundo J. Huertas
Universidad Politecnica de Madrid - Grupo SERPA-HGA
March 12, 2015- Seminario del Departamento de Matematicas eInformatica Aplicadas a las II. Civil y Naval
(UPM 2015) Polinomios Ortogonales UPM 2015 1 / 50
Outline
1 Electrostatic Interpretation of Zeros of Orthogonal PolynomialsIntroductionBasic background for MOPS and its zerosCanonical perturbations of a measure (Christoffel, Uvarov and Geronimus)The interacting particle model (M. Ismail)Example for the Laguerre-Geronimus measure with c =−1Another example for the Uvarov modification of the Laguerre measure
2 Asymptotic Behavior of Ratios of Laguerre Orthogonal PolynomialsAsymptotics for Classical Laguerre PolynomialsMotivation of the problemAn alternative (algorithmic) approachAn expansion of 1F1(a;c;z) by BuchholzA first strong asymptotic expansion valid in the whole C
A second strong asymptotic expansion valid in C\R+
Asymptotics of ratios of Laguerre polynomials
(UPM 2015) Polinomios Ortogonales UPM 2015 2 / 50
Outline
1 Electrostatic Interpretation of Zeros of Orthogonal PolynomialsIntroductionBasic background for MOPS and its zerosCanonical perturbations of a measure (Christoffel, Uvarov and Geronimus)The interacting particle model (M. Ismail)Example for the Laguerre-Geronimus measure with c =−1Another example for the Uvarov modification of the Laguerre measure
2 Asymptotic Behavior of Ratios of Laguerre Orthogonal PolynomialsAsymptotics for Classical Laguerre PolynomialsMotivation of the problemAn alternative (algorithmic) approachAn expansion of 1F1(a;c;z) by BuchholzA first strong asymptotic expansion valid in the whole C
A second strong asymptotic expansion valid in C\R+
Asymptotics of ratios of Laguerre polynomials
(UPM 2015) Polinomios Ortogonales UPM 2015 3 / 50
FIRST PART OF THE TALK:
ELECTROSTATIC INTERPRETATION OF ZEROS OF ORTHOGONALPOLYNOMIALS
References:
A. Branquinho, E.J. Huertas, and F.R. Rafaeli, Zeros of orthogonal polynomialsgenerated by the Geronimus perturbation of measures, Lecture Notes inComputer Science (LNCS), 8579 (2014), 44–59.
E.J. Huertas, F. Marcellan and H. Pijeira, An electrostatic model for zeros ofperturbed Laguerre polynomials, Proceedings of the American MathematicalSociety, 142 (5) (2014), 1733–1747.
E.J. Huertas, F. Marcellan and F.R. Rafaeli, Zeros of orthogonal polynomialsgenerated by canonical perturbations of measures, Applied Mathematics andComputation, 218 (13) (2012), 7109–7127.
(UPM 2015) Polinomios Ortogonales UPM 2015 4 / 50
The work of Stieltjes.
Theorem (Stieltjes 1885-1889): Suppose n unit charges at points x1,x2, . . . ,xn aredistributed in the interval [−1,1]. The energy of the system
E(x) = E(x1,x2, . . . ,xn) =n
∑k=1
V (xn,k)− ∑1≤ j≤k≤n
ln∣
∣xn, j − xn,k∣
∣ .
The above expression becomes a minimum when x1,x2, . . . ,xn are the zeros of theJacobi polynomials P(2p−1,2q−1)
n (x)
Similar results hold for the zeros of Laguerre and Hermite polynomials.
(UPM 2015) Polinomios Ortogonales UPM 2015 5 / 50
Motivation
1. Zeros of orthogonal polynomials are the nodes of the Gaussian quadraturerules and its extensions (Gauss–Radau, Gauss–Lobatto, Gauss–Kronrodrules,...etc)
∫
f (x)dµ(x)∼n
∑k=1
λk,n f (xk,n)
2. Zeros of classical orthogonal polynomials are the electrostatic equilibriumpoints of positive unit charges interacting according to a logarithmic potentialunder the action of an external field.
3. Zeros of orthogonal polynomials are used in collocation methods for boundaryvalue problems of 2nd order linear differential operators .
4. Global properties of zeros of orthogonal polynomials can be analyzed when theysatisfy 2nd order differential equations with polynomial coefficients, using theWKB method.
5. Zeros of orthogonal polynomials are eigenvalues of Jacobi matrices and its rolein Numerical Linear Algebra is very well known.
(UPM 2015) Polinomios Ortogonales UPM 2015 6 / 50
Basic background - MOPS
Let us consider the inner product 〈·, ·〉µ : P×P→ R
〈 f ,g〉µ =∫ b
af (x)g(x)dµ(x), n ≥ 0, f ,g ∈ P,
and supp(dµ) = (a,b)⊆ R.
Let {Pn(x)}n≥0 be a Monic Orthogonal Polynomial Sequence (MOPS) with respectto the above inner product.
Three-term recurrence relation (TTRR)
xPn(x) = Pn+1(x)+βnPn(x)+ γnPn−1(x), n ≥ 0,
with P−1(x) = 0, P0(x) = 1, and recurrence coefficients
βn =〈xPn,Pn〉µ
‖Pn‖2µ
, n ≥ 0 and γn =‖Pn‖2
µ
‖Pn−1‖2µ> 0, n ≥ 1.
(UPM 2015) Polinomios Ortogonales UPM 2015 7 / 50
Properties of the zeros of the MOPS
1 For each n ≥ 1, the polynomial Pn(x) has n real and simple zeros in the interior ofC0(supp(dµ)).
2 Interlacing property: The zeros of Pn+1(x) interlace with the zeros of Pn(x).
3 Between any two zeros of Pn(x) there is at least one zero of Pm(x), for m > n ≥ 2.
4 Each point of supp(dµ) attracts zeros of the MOPS. In other words, the zeros aredense in supp(dµ).
(UPM 2015) Polinomios Ortogonales UPM 2015 8 / 50
Basic background - Reproducing Kernel
nth-Kernel
Kn(x,y) =n
∑j=0
Pj(y)Pj(x)
‖Pj‖2µ
, ∀n ∈ N
Christoffel-Darboux formula
Kn(x,y) =1
‖Pn‖2µ
Pn+1(x)Pn(y)−Pn(x)Pn+1(y)x− y
, ∀n ∈ N
Confluent form of Kn
Kn(x,x) =P′
n+1(x)Pn(x)−P′n(x)Pn+1(x)
‖Pn‖2µ
, ∀n ∈ N
(UPM 2015) Polinomios Ortogonales UPM 2015 9 / 50
Christoffel perturbation of a measure dµ
Let {Pc,[1]n (x)}n≥0 be the MOPS associated with the measure
dµ [1] = (x− c)dµ ,
with (any complex or real number) c 6∈C0(supp(dµ)).
In the former modification by a linear divisor, we add a mass point exactly at thepoint c. Then we obtain a Geronimus transformation of the measure dµ .
Let {Qc,Nn (x)}n≥0 be the MOPS associated with the measure
dνN =1
(x− c)dµ +Nδ (x− c),
with c 6∈C0(supp (dµ)), and let yc,Nn,k := yc,N
n,k (c) be the zeros of Qc,Nn (x).
Geronimus (1940), conclude that the sequences associated to dνN must be of theform
Pn(x)+anPn−1(x), an 6= 0,
for certain numbers an ∈ R.
Maroni (1990), stated that the sequence {Pn+1(x)}n≥0, orthogonal with respect tou = δc +λ (x− c)−1L, can be represented as
The mass point c attracts exactly one zero of Qc,Nn (x), when N → ∞.
When either c < a or c > b, at most one of the zeros of Qc,Nn (x) is located outside of
C0(supp (dµ)) = (a,b). In the next result, we will give explicitly the value N0 of themass such that for N > N0 one of the zeros is located outside (a,b).If C0(supp (dµ)) = (a,b) and c < a, then the largest zero yc,N
Electrostatic model for zeros of Laguerre and Jacobi Geronimusperturbed MOPS
Let introduce a system of n movable unit charges in (a,b) in the presence of aexternal potential V (x)To find V (x) is enough to consider the polynomial coefficients of [Qc,N(x)]′′ and[Qc,N(x)]′, evaluated in the zeros of Qc,N(x), such that
[Qc,N(yc,Nn,k )]
′′
[Qc,N(yc,Nn,k )]
′=−R(yc,N
n,k ;n),
and after some computations we obtain
[Qc,N(yc,Nn,k )]
′′
[Qc,N(yc,Nn,k )]
′= D [lnu(x)] |x=yc,N
n,k−
ψ(yc,Nn,k )
φ(yc,Nn,k )
.
The total external potential V (x) is given by two external fields
Electrostatic model for zeros of Laguerre and Jacobi Geronimusperturbed MOPS
The equilibrium position for the zeros of {Qc,Nn (x)}n≥0 occurs under the presence
of a total external potential V (x) = υlong(x)+υshort(x).
υshort(x) = (1/2) lnu(x;n) represents a short range potential (or varying externalpotential) corresponding to unit charges located at the zeros of u(x).
The polynomial u(x) plays a remarkable role in the behavior of the zeros ofQc,N
n (x). As an example, we show below total external potentials VJ(x) and VL(x)when the measure dµ(x) is the classical Jacobi and Laguerre measuresrespectively. In this examples we have deg(u(x)) = 1.
1 Electrostatic Interpretation of Zeros of Orthogonal PolynomialsIntroductionBasic background for MOPS and its zerosCanonical perturbations of a measure (Christoffel, Uvarov and Geronimus)The interacting particle model (M. Ismail)Example for the Laguerre-Geronimus measure with c =−1Another example for the Uvarov modification of the Laguerre measure
2 Asymptotic Behavior of Ratios of Laguerre Orthogonal PolynomialsAsymptotics for Classical Laguerre PolynomialsMotivation of the problemAn alternative (algorithmic) approachAn expansion of 1F1(a;c;z) by BuchholzA first strong asymptotic expansion valid in the whole C
A second strong asymptotic expansion valid in C\R+
ASYMPTOTIC BEHAVIOR OF RATIOS OF LAGUERRE ORTHOGONALPOLYNOMIALS
Reference:
A. Deano, E.J. Huertas, and F. Marcellan, Strong and ratio asymptotics forLaguerre polynomials revisited, Journal of Mathematical Analysis andApplications, 403 (2) (2013), 477–486.
Cited by:
R.J. Furnstahl, S.N. More, T. Papenbrock. Systematic expansion for infraredoscillator basis extrapolations. Physical Review C 89, 044301 (2014)
K.I. Ishikawa, D. Kimura, K. Shigaki, A. Tsuji. A numerical evaluation of vacuumpolarization tensor in constant external magnetic fields. International Journal ofModern Physics A, 28, 1350100 (2013)
S. Konig, S. K. Bogner, R. J. Furnstahl, S. N. More, and T. Papenbrock. Ultravioletextrapolations in finite oscillator bases. Physical Review C, 2014 - APS90, 064007(2014)
n=0 (sometimes called Soninpolynomials) are orthogonal with respect to the weight function w(x) = xα e−x,α >−1, on the interval (0,+∞), so they satisfy
〈L(α)m ,L(α)
n 〉=∫ +∞
0L(α)
m L(α)n xα e−xdx = ‖L(α)
n ‖2 ·δm,n, α >−1.
We consider the normalization (not monic)
L(α)n (x) =
(−1)n
n!xn + lower degree terms.
They are the polynomial solutions of the second order differential equation
x[L(α)n (x)]′′+(α +1− x)[L(α)
n (x)]′+nL(α)n (x) = 0.
This polynomials can be given in terms of an 1F1 confluent hypergeometricfunction
Outer strong asymptotics: Perron’s asymptotic formula in C\R+.For α >−1 we get
L(α)n (x) =
12√
πex/2 (−x)−α/2−1/4 nα/2−1/4e2(−nx)1/2
·{
d−1
∑m=0
Cm(α ;x) n−m/2+O(n−d/2)
}
.
Here Cm(α ;x) is independent of n. This relation holds for x in the complex planewith a cut along the positive real semiaxis. The bound for the remainder holdsuniformly in every closed domain of the complex plane with empty intersectionwith R+.
C0(α ;x) = 1, but in the original paper by Perron do not appear higher ordercoefficients Cm(α ;x), m > 1.
Mehler-Heine type formula . Fixed j, with j ∈ N∪{0} and Jα the Bessel functionof the first kind, then
where Ak(α ;x) and Bk(α ;x) are certain functions of x independent of n and regularfor x > 0. The bound for the remainder holds uniformly in [ε,ω]. For k = 0 we haveA0(α ;x) = 1 and B0(α ;x) = 0.
Main reference:
G. Szego, Orthogonal Polynomials, Coll. Publ. Amer. Math. Soc. Vol. 23, (4thed.), Amer. Math. Soc. Providence, RI (1975).
Higher order coefficients in the asymptotic expansions are important whenone deals with Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials.
They play a key role in the analysis of the asymptotic behavior of these newfamilies of “perturbed” orthogonal polynomials.
One needs to estimate ratios of Laguerre orthogonal polynomials like
L(α)n+ j(x)
L(β )n (x)
,
where n = 0,1,2, . . ., j ∈ Z. Additionally, we require α ,β >−1.
More precisely, we need to know exactly the coefficient of n−d/2 to estimate theabove expressions correctly.
For example, if d = 1 we need to know the coefficient of n−1/2, if d = 2 thecoefficient of n−1, and so on.
There are some expressions in the literature, but not accurate enough.
Remark: For more precise asymptotic expressions of
L(α)n+ j(x)
L(β )n (x)
we need more coefficients Cm(α ;x) in the Perron’s asymptotic formulas.
The main advantage of Perron’s expansions for Laguerre polynomials is thesimplicity of the asymptotic sequence (inverse powers of n), but it has the problemthat the coefficients Cm(α ;x) soon become very cumbersome to compute .
One possibility is to use the generating function for Laguerre polynomials:
(1− z)−α−1exp(
xzz−1
)
=∞
∑m=0
L(α)m (x)zm, |z|< 1,
write the coefficients as contour integrals and apply the standard method ofsteepest descent.
However, the computations soon become complicated, since parametrizing thepath of steepest descent is not easy in explicit form.
To the best of our knowledge, the only sources of information for higher ordercoefficients in the Perron expansion are
W. Van Assche, Erratum to Weighted zero distribution for polynomialsorthogonal on an infinite interval. SIAM J. Math. Anal., 32 (2001), 1169–1170.
D. Borwein, J. M. Borwein, and R. E. Crandall, Effective Laguerreasymptotics. SIAM J. Numer. Anal., 46 (6) (2008), 3285–3312.
The PAMO coefficient:
C1(α ;z) =1
4√−z
(
14−α2−2(α +1)z+
z2
3
)
,
named after O. Perron, W. van Assche, T. Muller and F. Olver.
Borwein et al. use complex integral representations with strict error bounds. Thisprovides a powerful (and very technical) method to generate the coefficientsCm(α ;z).
In this paper we propose an alternative to this approach, based solely on using anexpansion of the Laguerre polynomials that involves Bessel functions of the firstkind.
This type of expansions go back to the works of Tricomi and Buchholz.
In this way, the different behaviors of L(α)n (x) in the complex plane are better
captured, and thus the coefficients are simpler to compute.
Moreover, apart from the large n asymptotic property, the resulting approximationis convergent in the complex plane.
Using this approach it is possible to recover easily the results in the work ofBorwein et al. (and more).
In order to rewrite the above expression in negative powers of n, we also use thefollowing asymptotic approximation for the Bessel function of large argument,
Jα (z)∼(
2πz
)12
(
cosω∞
∑k=0
(−1)k a2k(α)
z2k −sinω∞
∑k=0
(−1)k a2k+1(α)
z2k+1
)
,
where |argz|< π, ω = ω(α) = z− απ2
− π4
and the coefficients are, starting with
a0(α) = 1,
ak(α) =(4α2−1)(4α2−9) . . .(4α2− (2k−1)2)
8kk!, k ≥ 1.
We need Jm+α (2√
κz) for integers m ≥ 0. Consequently, for |arg(κz)|< 2π
Implementing this grouping carefully, what we have at the end will be twosummations, depending on d = 2M (even) or d = 2M+1 (odd) , for M = 0,1, . . ..
If d = 2M, we have an alternating sum of the form
S2M(α ,z) = (−1)M2M
∑m=0
(−1)m( z
4κ
)m/2Pm(α ,z)
a2M−m(α +m)
(2√
κz)2M−m
= (−4κz)−M2M
∑m=0
(−1)mzmPm(α ,z)a2M−m(α +m).
If d = 2M+1, we have a similar situation, and finally we obtain
Main advantages (wrt the Perron and Fejer formulas):
The coefficients are still complicated but they can be computed systematically, up to theaccuracy desired.
The expansion given in Theorem 1 is convergent on the whole complex plane.
Retaining the Bessel functions instead of expanding them in negative powers of n, itprovides a useful representation of the Laguerre polynomials for large degree.
Disadvantages:
One difficulty of the previous expansion is that it contains cosω and sinω terms.
These cosω and sinω terms can be grouped together away from [0,∞).
Theorem 2: Alternative strong asymptotics for L(α)n (z): Let α >−1, the Laguerre
polynomial L(α)n (z) admits the following asymptotic expansion as n → ∞:
L(α)n (z) =
12√
πΓ(n+α +1)
n!ez/2(−κz)−α/2−1/4e2
√−κz
∞
∑m=0
Bm(α ,z)n−m/2,
where the error term is uniform for z in bounded sets of C\ [0,∞), and thecoefficients Bm(α ,z) are related to the original ones Bm(α ,z) in the following way:
Next, we use the former results to obtain the asymptotic behavior as n → ∞ ofarbitrary ratios of Laguerre polynomials with greater accuracy than formulasavailable in the literature.
We begin rewriting Theorem 2 as
L(α)n (z) = f (α)
n (z)
(
d−1
∑m=0
Bm(α ,z)n−m/2+O(n−d/2)
)
,
with the prefactor
f (α)n (z) =
12√
πΓ(n+α +1)
n!ez/2(−κ(n,α)z)−α/2−1/4e2
√−nz.
Note that we emphasize that κ depends both on n and on α , since we want toconsider different degree and different parameter of the Laguerre polynomials.