New examples of matrix orthogonal polynomials satisfying second - - PowerPoint PPT Presentation

Introduction A new family Rodrigues formula Recurrence relations

New examples of matrix orthogonal polynomials satisfying second order differential equations. Structural formulas

• J. Borrego Morell(∗) , joint work with A. Dur´

an(∗∗) and M. Castro(∗∗)

(∗) Carlos III University of Madrid (∗∗) University of Seville

IMUS Doc-course, University of Seville, March-May 2010

Introduction A new family Rodrigues formula Recurrence relations

Outline

Introduction A new family Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

Outline

Introduction A new family Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

Outline

Introduction A new family Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

Outline

Introduction A new family Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

Matrix orthogonal polynomials

Let W be an N × N positive definite matrix of measures. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P(t) and Q(t) by the numerical matrix P, Q = P, QW =

• R

P(t)W(t)Q∗(t)dt, where Q∗(t) denotes the conjugate transpose of Q(t). There exists a sequence (Pn)n of matrix polynomials, orthonormal with respect to W and with Pn of degree n. The sequence (Pn)n is unique up to a product with a unitary matrix.

Introduction A new family Rodrigues formula Recurrence relations

Matrix orthogonal polynomials

Let W be an N × N positive definite matrix of measures. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P(t) and Q(t) by the numerical matrix P, Q = P, QW =

• R

P(t)W(t)Q∗(t)dt, where Q∗(t) denotes the conjugate transpose of Q(t). There exists a sequence (Pn)n of matrix polynomials, orthonormal with respect to W and with Pn of degree n. The sequence (Pn)n is unique up to a product with a unitary matrix.

Introduction A new family Rodrigues formula Recurrence relations

Matrix orthogonal polynomials

Let W be an N × N positive definite matrix of measures. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P(t) and Q(t) by the numerical matrix P, Q = P, QW =

• R

P(t)W(t)Q∗(t)dt, where Q∗(t) denotes the conjugate transpose of Q(t). There exists a sequence (Pn)n of matrix polynomials, orthonormal with respect to W and with Pn of degree n. The sequence (Pn)n is unique up to a product with a unitary matrix.

Introduction A new family Rodrigues formula Recurrence relations

Matrix orthogonal polynomials

Property Any sequence of orthonormal matrix valued polynomials (Pn)n satisfies a three term recurrence relation A∗

nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular. An are nonsingular matrices and Bn hermitian.

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations. Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying Pnℓ2,R = P

′′

n F2(t) + P

′

nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

Right hand side differential operator ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Pn eigenfunctions, Λn eigenvalues: Pnℓ2,R = ΛnPn

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations. Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying Pnℓ2,R = P

′′

n F2(t) + P

′

nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

Right hand side differential operator ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Pn eigenfunctions, Λn eigenvalues: Pnℓ2,R = ΛnPn

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations. Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying Pnℓ2,R = P

′′

n F2(t) + P

′

nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

Right hand side differential operator ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Pn eigenfunctions, Λn eigenvalues: Pnℓ2,R = ΛnPn

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

We will say that a weight W does not reduce to scalar if there exists a non singular matrix T independent of t such that W(t) = TD(t)T ∗ with D(t) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

We will say that a weight W does not reduce to scalar if there exists a non singular matrix T independent of t such that W(t) = TD(t)T ∗ with D(t) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.

Introduction A new family Rodrigues formula Recurrence relations

The matrix Bochner’s problem

We will say that a weight W does not reduce to scalar if there exists a non singular matrix T independent of t such that W(t) = TD(t)T ∗ with D(t) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.

Introduction A new family Rodrigues formula Recurrence relations

• Dr. Jekyll Dur´

an’s course

A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0} F0 = A2−      2(N − 1) 2(N − 2) ...      , W(t) = e−t2eAteA∗t The weight matrix has a symmetric second order differential

• perator of the form

d dt 2 + d dt

• (2A − 2It) + F0

Introduction A new family Rodrigues formula Recurrence relations

• Dr. Jekyll Dur´

an’s course

A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0} F0 = A2−      2(N − 1) 2(N − 2) ...      , W(t) = e−t2eAteA∗t The weight matrix has a symmetric second order differential

• perator of the form

d dt 2 + d dt

• (2A − 2It) + F0

Introduction A new family Rodrigues formula Recurrence relations

The goal

We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as

Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

The goal

We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as

Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

The goal

We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as

Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

The goal

We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as

Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

The goal

We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as

Rodrigues formula Recurrence relations

Introduction A new family Rodrigues formula Recurrence relations

The method to find MOP satisfying 2nd order differential eq.s

(Dur´ an-Gr¨ unbaum), 2004 Simmetry Eqs: differential equations for the weight function W and the coefficients of ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Symmetry Equations F2W = WF ∗

2

2(F2W)′ = F1W + WF ∗

1

(F2W)

′′ − (F1W)′

= (WF ∗

0 − F0W)

Introduction A new family Rodrigues formula Recurrence relations

The method to find MOP satisfying 2nd order differential eq.s

(Dur´ an-Gr¨ unbaum), 2004 Simmetry Eqs: differential equations for the weight function W and the coefficients of ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Symmetry Equations F2W = WF ∗

2

2(F2W)′ = F1W + WF ∗

1

(F2W)

′′ − (F1W)′

= (WF ∗

0 − F0W)

Introduction A new family Rodrigues formula Recurrence relations

MOP related to second order differential op.

Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,

• rthonormal with respect to a weight matrix W

A∗

nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that lim

n→∞

An ϕ(n) = aI, lim

n→∞

Bn ϕ(n) = bI, for a convenient continuous function ϕ(t). The example given here satisfies the property that the previous limits give no longer a scalar matrix

Introduction A new family Rodrigues formula Recurrence relations

MOP related to second order differential op.

Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,

• rthonormal with respect to a weight matrix W

A∗

nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that lim

n→∞

An ϕ(n) = aI, lim

n→∞

Bn ϕ(n) = bI, for a convenient continuous function ϕ(t). The example given here satisfies the property that the previous limits give no longer a scalar matrix

Introduction A new family Rodrigues formula Recurrence relations

MOP related to second order differential op.

Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,

• rthonormal with respect to a weight matrix W

A∗

nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that lim

n→∞

An ϕ(n) = aI, lim

n→∞

Bn ϕ(n) = bI, for a convenient continuous function ϕ(t). The example given here satisfies the property that the previous limits give no longer a scalar matrix

Introduction A new family Rodrigues formula Recurrence relations

MOP related to second order differential op.

Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,

• rthonormal with respect to a weight matrix W

A∗

nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that lim

n→∞

An ϕ(n) = aI, lim

n→∞

Bn ϕ(n) = bI, for a convenient continuous function ϕ(t). The example given here satisfies the property that the previous limits give no longer a scalar matrix

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 where D is a diagonal matrix independent of t D =            −(v+1)

2

−(N−1)(v+1)

2(v+N−1)

... −(N−1)(v+1)

2(jv+N−1)

...

−1 2

           where v ∈ (−1, ∞) \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 where D is a diagonal matrix independent of t D =            −(v+1)

2

−(N−1)(v+1)

2(v+N−1)

... −(N−1)(v+1)

2(jv+N−1)

...

−1 2

           where v ∈ (−1, ∞) \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 A = [ N

2 ]−1

• j=0

(−1)j

• 1

4(N − 1) j (2j + 1)j−1 j!

• v

v + 1 j A2j+1 A is the N × N nilpotent matrix A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 A = [ N

2 ]−1

• j=0

(−1)j

• 1

4(N − 1) j (2j + 1)j−1 j!

• v

v + 1 j A2j+1 A is the N × N nilpotent matrix A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 A = [ N

2 ]−1

• j=0

(−1)j

• 1

4(N − 1) j (2j + 1)j−1 j!

• v

v + 1 j A2j+1 A is the N × N nilpotent matrix A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

W(t) = TT ∗, T(t) = eAteDt2 A = [ N

2 ]−1

• j=0

(−1)j

• 1

4(N − 1) j (2j + 1)j−1 j!

• v

v + 1 j A2j+1 A is the N × N nilpotent matrix A =       a1 · · · a2 · · · . . . . . . . . . . . . . . . . . . . · · · aN       a1, · · · aN ∈ C \ {0}

Introduction A new family Rodrigues formula Recurrence relations

General case

ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). F2(t) = Ψ2 + [ N

2 ]−1

• j=0

(−1)jv

• v

v + 1 j 2j + 1 N − 1 (2j + 1)j−1 j! A2j+1t Ψ2 =           1

v+N−1 N−1

...

jv+N−1 N−1

... v + 1          

Introduction A new family Rodrigues formula Recurrence relations

General case

Using the Theorem 2.3 A.Duran, Constructive Approx, 2009 we

• btain the expression for

F1(t) = −2(k + 1)tI + [ N

2 ]−1

• j=0

αj

• v

v + 1 j Γ2j+1A2j+1+ [ N

2 ]−1

• j=0

βj

• v

v + 1 j A2jvt

Introduction A new family Rodrigues formula Recurrence relations

General case

where Γj =               

jv+N−1 N−1

...

kv+N−1 N−1

...

(N−2)v+N−1 N−1

...               

Introduction A new family Rodrigues formula Recurrence relations

General case

where Γj =               

jv+N−1 N−1

...

kv+N−1 N−1

...

(N−2)v+N−1 N−1

...               

Introduction A new family Rodrigues formula Recurrence relations

General case

For the matrix F0(t) we have F0(t) = (v+1)Ψ0+ [ N

2 ]−1

• j=1

(−1)j+1 jj−1 (N − 1)j2j−1j!

• v

v + 1 j Γ2jA2j where Ψ0 is the diagonal matrix Ψ0 =           2 ... 2j ... 2(N − 1)          

Introduction A new family Rodrigues formula Recurrence relations

Particular cases

W(t) = TT ∗, T(t) = eAteDt2 For N = 2 we have A = a1

• , D =

−v+1

2

−1

2

• W =
• |a1|2e−t2t2 + e−(v+1)t2

a1e−t2t a1e−t2t e−t2

Introduction A new family Rodrigues formula Recurrence relations

Particular cases

The second order differential equation that satisfy the MOP is P

′′

n (t)

1 a1vt v + 1

• + P

′

n(t)

−2(v + 1)t 2a1(v + 1) −2(v + 1)t

• +

Pn(t) 2(v + 1)

• =

−2(v + 1)N −2(v + 1)(N − 1)

• Pn(t)

Introduction A new family Rodrigues formula Recurrence relations

Rodrigues formula

Property (A. Dur´ an, Int. Math. Research Notices, 2009) Fix n and consider the second order differential equation (XF ∗

2 )

′′ − (X[F ∗

1 + n(F ∗ 2 )

′]) ′ + X

• F ∗

0 + n(F ∗ 1 )

′ +

n 2

• (F ∗

2 )

′′

= ΛnX Write Rn = X for a solution of this equation. Then the function Pn(t) = R(n)

n (t)W −1(t) satisfies

P

′′

n (t)F2(t) + P

′

n(t)F1(t) + Pn(t)F0(t) = ΛnPn(t)

Introduction A new family Rodrigues formula Recurrence relations

Rodrigues formula

Consider A = a

• , D =
• − b

2

−1

2

• W =
• |a|2e−t2t2 + e−bt2

ae−t2t ae−t2t e−t2

• Rn(t) = e−t2
• b−ne(1−b)t2 + a2

2 (n + 2t2)

at a(2t + et2√π(1 + Erf( √ bt) − Erf(t)) 2

• where Erf(t) =

2 √π t e−x2dx. Pn(t) = R(n)

n (t)W −1(t),

||Pn||2 = 2n−1√πn!

• αn+1

bn+ 1

2

4αn

• αn = 2 + a2bn− 1

2 n,

b = v + 1

Introduction A new family Rodrigues formula Recurrence relations

Rodrigues formula

Consider A = a

• , D =
• − b

2

−1

2

• W =
• |a|2e−t2t2 + e−bt2

ae−t2t ae−t2t e−t2

• Rn(t) = e−t2
• b−ne(1−b)t2 + a2

2 (n + 2t2)

at a(2t + et2√π(1 + Erf( √ bt) − Erf(t)) 2

• where Erf(t) =

2 √π t e−x2dx. Pn(t) = R(n)

n (t)W −1(t),

||Pn||2 = 2n−1√πn!

• αn+1

bn+ 1

2

4αn

• αn = 2 + a2bn− 1

2 n,

b = v + 1

Introduction A new family Rodrigues formula Recurrence relations

Rodrigues formula

Consider A = a

• , D =
• − b

2

−1

2

• W =
• |a|2e−t2t2 + e−bt2

ae−t2t ae−t2t e−t2

• Rn(t) = e−t2
• b−ne(1−b)t2 + a2

2 (n + 2t2)

at a(2t + et2√π(1 + Erf( √ bt) − Erf(t)) 2

• where Erf(t) =

2 √π t e−x2dx. Pn(t) = R(n)

n (t)W −1(t),

||Pn||2 = 2n−1√πn!

• αn+1

bn+ 1

2

4αn

• αn = 2 + a2bn− 1

2 n,

b = v + 1

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

The sequence of polynomials (Pn)n≥0 orthogonal with respect to W generated by the above Rodrigues formula satisfy the three term recurrence relation tPn(t) = An+1Pn+1(t) + BnPn(t) + CnPn−1(t), n ≥ 1, An = −1 2 1

αn−1 αn

• Bn = a(−n + (n + 1)b)
• 1

2bαn 2bn− 1

2

αn+1

• Cn = −n

αn+1

bαn

1

• ,

αn = 2 + a2bn− 1

2 n

with the initial conditions P−1 = 0, P0 = 1 2

• .

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

The sequence of polynomials (Pn)n≥0 orthogonal with respect to W generated by the above Rodrigues formula satisfy the three term recurrence relation tPn(t) = An+1Pn+1(t) + BnPn(t) + CnPn−1(t), n ≥ 1, An = −1 2 1

αn−1 αn

• Bn = a(−n + (n + 1)b)
• 1

2bαn 2bn− 1

2

αn+1

• Cn = −n

αn+1

bαn

1

• ,

αn = 2 + a2bn− 1

2 n

with the initial conditions P−1 = 0, P0 = 1 2

• .

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

t ˜ Pn(t) = ˜ An+1 ˜ Pn+1(t) + ˜ Bn ˜ Pn(t) + ˜ A∗

n ˜

Pn−1(t), , n ≥ 1, ˜ Pn(t) orthonormal w.r.t. W(t) ˜ An = √n  

√αn+1 √ 2b√αn √αn−1 √ 2√αn

  ˜ Bn = ab

2n−3 4 (b + (1 − b)n)

√αnαn+1 1 1

• with the initial conditions P−1 = 0, P0 = (π)− 1

4

√

2b

1 4

α1

1

• .

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

t ˜ Pn(t) = ˜ An+1 ˜ Pn+1(t) + ˜ Bn ˜ Pn(t) + ˜ A∗

n ˜

Pn−1(t), , n ≥ 1, ˜ Pn(t) orthonormal w.r.t. W(t) ˜ An = √n  

√αn+1 √ 2b√αn √αn−1 √ 2√αn

  ˜ Bn = ab

2n−3 4 (b + (1 − b)n)

√αnαn+1 1 1

• with the initial conditions P−1 = 0, P0 = (π)− 1

4

√

2b

1 4

α1

1

• .

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

Observe that lim

n→∞

˜ An √n =           

• 1

√ 2 1 √ 2b

• if b > 1
• 1

√ 2b 1 √ 2

• for 0 < b < 1

. That is, we have obtained an example where the limits of ˜ An ϕ(n), for certain convenient continuous function ϕ(x), give a diagonal matrix and not a scalar matrix of the form cI, c ∈ R. However lim

n→∞

˜ Bn = 0.

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

Observe that lim

n→∞

˜ An √n =           

• 1

√ 2 1 √ 2b

• if b > 1
• 1

√ 2b 1 √ 2

• for 0 < b < 1

. That is, we have obtained an example where the limits of ˜ An ϕ(n), for certain convenient continuous function ϕ(x), give a diagonal matrix and not a scalar matrix of the form cI, c ∈ R. However lim

n→∞

˜ Bn = 0.

Introduction A new family Rodrigues formula Recurrence relations

Recurrence relations

Observe that lim

n→∞

˜ An √n =           

• 1

√ 2 1 √ 2b

• if b > 1
• 1

√ 2b 1 √ 2

• for 0 < b < 1

. That is, we have obtained an example where the limits of ˜ An ϕ(n), for certain convenient continuous function ϕ(x), give a diagonal matrix and not a scalar matrix of the form cI, c ∈ R. However lim

n→∞

˜ Bn = 0.