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Orthogonal Rational Functions, Associated Rational Functions and - - PowerPoint PPT Presentation

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Introduction ORFs and ARFs Relations Favard theorem Orthogonal Rational Functions, Associated Rational Functions and Functions of the Second Kind Karl Deckers Department of Computer Science, Katholieke Universiteit Leuven, Heverlee,

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal Rational Functions, Associated Rational Functions and Functions of the Second Kind

Karl Deckers

Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium. PhD student. Supervisor: Adhemar Bultheel.

July 2008

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Inner product

Inner product consider an inner product defined by a linear functional M: f , g = M{fg∗}, g∗(x) = g(x). assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀f , g : M{fg∗} = M{f∗g} (Hermitian) ∀f = 0 : M{ff∗} > 0 (positive-definite) M{1} = 1 (normalized).

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Inner product

Inner product consider an inner product defined by a linear functional M: f , g = M{fg∗}, g∗(x) = g(x). assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀f , g : M{fg∗} = M{f∗g} (Hermitian) ∀f = 0 : M{ff∗} > 0 (positive-definite) M{1} = 1 (normalized).

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Inner product

Inner product consider an inner product defined by a linear functional M: f , g = M{fg∗}, g∗(x) = g(x). assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀f , g : M{fg∗} = M{f∗g} (Hermitian) ∀f = 0 : M{ff∗} > 0 (positive-definite) M{1} = 1 (normalized).

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Inner product

Inner product consider an inner product defined by a linear functional M: f , g = M{fg∗}, g∗(x) = g(x). assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀f , g : M{fg∗} = M{f∗g} (Hermitian) ∀f = 0 : M{ff∗} > 0 (positive-definite) M{1} = 1 (normalized).

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Inner product

Inner product consider an inner product defined by a linear functional M: f , g = M{fg∗}, g∗(x) = g(x). assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀f , g : M{fg∗} = M{f∗g} (Hermitian) ∀f = 0 : M{ff∗} > 0 (positive-definite) M{1} = 1 (normalized).

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal polynomials

Orthonormal polynomials (OPs) Pn = space of polynomials of degree ≤ n. canonical basis for Pn: Pn = span{1, x, x2, . . . , xn}. suppose a sequence of polynomials exists so that ∀n ≥ 0 : pn ∈ Pn \ Pn−1 and M{pnpj∗} = 0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal polynomials

Orthonormal polynomials (OPs) Pn = space of polynomials of degree ≤ n. canonical basis for Pn: Pn = span{1, x, x2, . . . , xn}. suppose a sequence of polynomials exists so that ∀n ≥ 0 : pn ∈ Pn \ Pn−1 and M{pnpj∗} = 0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal polynomials

Orthonormal polynomials (OPs) Pn = space of polynomials of degree ≤ n. canonical basis for Pn: Pn = span{1, x, x2, . . . , xn}. suppose a sequence of polynomials exists so that ∀n ≥ 0 : pn ∈ Pn \ Pn−1 and M{pnpj∗} = 0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal polynomials

Orthonormal polynomials (OPs) These OPs satisfy the following 3-term recurrence relation: p−1(x) ≡ 0, p0(x) ≡ 1, pn(x) = En(x + Fn)pn−1(x) + Cnpn−2(x), n > 0, En = 0, Cn = − En

E n−1 = 0.

Associated polynomials (APs) APs p(k)

n−k ∈ Pn−k of order k are defined by:

p(k)

−1(x) ≡ 0,

p(k)

0 (x) ≡ 1,

p(k)

n−k(x) = En(x + Fn)p(k) n−1−k(x) + Cnp(k) n−2−k(x),

n > k.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Orthogonal polynomials

Orthonormal polynomials (OPs) These OPs satisfy the following 3-term recurrence relation: p−1(x) ≡ 0, p0(x) ≡ 1, pn(x) = En(x + Fn)pn−1(x) + Cnpn−2(x), n > 0, En = 0, Cn = − En

E n−1 = 0.

Associated polynomials (APs) APs p(k)

n−k ∈ Pn−k of order k are defined by:

p(k)

−1(x) ≡ 0,

p(k)

0 (x) ≡ 1,

p(k)

n−k(x) = En(x + Fn)p(k) n−1−k(x) + Cnp(k) n−2−k(x),

n > k.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Associated polynomials (APs) Note that APs of order 0 = OPs, ∀n ≥ k ≥ 0 : p(k)

n−k ∈ Pn−k \ Pn−k−1

⇒ Favard theorem: there exists a HPDN linear functional M(k) so that M(k){p(k)

n−kp(k) j−k∗} =

0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Associated polynomials (APs) Note that APs of order 0 = OPs, ∀n ≥ k ≥ 0 : p(k)

n−k ∈ Pn−k \ Pn−k−1

⇒ Favard theorem: there exists a HPDN linear functional M(k) so that M(k){p(k)

n−kp(k) j−k∗} =

0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Associated polynomials (APs) Note that APs of order 0 = OPs, ∀n ≥ k ≥ 0 : p(k)

n−k ∈ Pn−k \ Pn−k−1

⇒ Favard theorem: there exists a HPDN linear functional M(k) so that M(k){p(k)

n−kp(k) j−k∗} =

0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Relation APs of different order (R1) p(k)

n−k(x) = Ek+1(x + Fk+1)p(k+1) n−(k+1)(x) + Ck+2p(k+2) n−(k+2)(x),

n > k (R2) p(k)

n−k(x) = p(j) n−j(x)p(k) j−k(x) + Cj+1p(j+1) n−(j+1)(x)p(k) (j−1)−k(x),

k + 1 ≤ j ≤ n − 1 APs and functions of the second kind (R3) p(k)

n−k(x) = 1 Ek · M(k−1) t

  • p(k−1)

n−(k−1)(t)−p(k−1) n−(k−1)(x)

t−x

  • Karl Deckers

ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Relation APs of different order (R1) p(k)

n−k(x) = Ek+1(x + Fk+1)p(k+1) n−(k+1)(x) + Ck+2p(k+2) n−(k+2)(x),

n > k (R2) p(k)

n−k(x) = p(j) n−j(x)p(k) j−k(x) + Cj+1p(j+1) n−(j+1)(x)p(k) (j−1)−k(x),

k + 1 ≤ j ≤ n − 1 APs and functions of the second kind (R3) p(k)

n−k(x) = 1 Ek · M(k−1) t

  • p(k−1)

n−(k−1)(t)−p(k−1) n−(k−1)(x)

t−x

  • Karl Deckers

ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Associated polynomials

Relation APs of different order (R1) p(k)

n−k(x) = Ek+1(x + Fk+1)p(k+1) n−(k+1)(x) + Ck+2p(k+2) n−(k+2)(x),

n > k (R2) p(k)

n−k(x) = p(j) n−j(x)p(k) j−k(x) + Cj+1p(j+1) n−(j+1)(x)p(k) (j−1)−k(x),

k + 1 ≤ j ≤ n − 1 APs and functions of the second kind (R3) p(k)

n−k(x) = 1 Ek · M(k−1) t

  • p(k−1)

n−(k−1)(t)−p(k−1) n−(k−1)(x)

t−x

  • Karl Deckers

ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Ln = space of rational functions with poles in An = {α1, . . . , αn} ⊂ (C ∪ {∞}) \ {0} f ∈ Ln ⇔ f is of the form f (x) = pn(x) (1 − x/α1) · . . . · (1 − x/αn), pn ∈ Pn.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Ln = space of rational functions with poles in An = {α1, . . . , αn} ⊂ (C ∪ {∞}) \ {0} f ∈ Ln ⇔ f is of the form f (x) = pn(x) (1 − x/α1) · . . . · (1 − x/αn), pn ∈ Pn.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Define factors Zk(x) = x 1 − x/αk , k = 1, 2, . . . and products b0(x) ≡ 1, bk(x) = Zk(x)bk−1(x), k = 1, 2, . . . . Then, basis for Ln: Ln = span{b0(x), b1(x), b2(x), . . . , bn(x)}. Note that, whenever ∀k : αk = ∞, then Zk(x) = x and bk(x) = xk.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Define factors Zk(x) = x 1 − x/αk , k = 1, 2, . . . and products b0(x) ≡ 1, bk(x) = Zk(x)bk−1(x), k = 1, 2, . . . . Then, basis for Ln: Ln = span{b0(x), b1(x), b2(x), . . . , bn(x)}. Note that, whenever ∀k : αk = ∞, then Zk(x) = x and bk(x) = xk.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Define factors Zk(x) = x 1 − x/αk , k = 1, 2, . . . and products b0(x) ≡ 1, bk(x) = Zk(x)bk−1(x), k = 1, 2, . . . . Then, basis for Ln: Ln = span{b0(x), b1(x), b2(x), . . . , bn(x)}. Note that, whenever ∀k : αk = ∞, then Zk(x) = x and bk(x) = xk.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Define factors Zk(x) = x 1 − x/αk , k = 1, 2, . . . and products b0(x) ≡ 1, bk(x) = Zk(x)bk−1(x), k = 1, 2, . . . . Then, basis for Ln: Ln = span{b0(x), b1(x), b2(x), . . . , bn(x)}. Note that, whenever ∀k : αk = ∞, then Zk(x) = x and bk(x) = xk.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Suppose a sequence of rational functions exists so that ϕn ∈ Ln \ Ln−1 ⇔ pn(αn) = 0, αn = ∞ pn / ∈ Pn−1, αn = ∞ M{ϕnϕj∗} = 0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) Suppose a sequence of rational functions exists so that ϕn ∈ Ln \ Ln−1 ⇔ pn(αn) = 0, αn = ∞ pn / ∈ Pn−1, αn = ∞ M{ϕnϕj∗} = 0, n = j 1, n = j.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) These ORFs satisfy the following 3-term recurrence relation: ϕ−1(x) ≡ 0, ϕ0(x) ≡ 1, ϕn(x) = En

  • Zn(x) + Fn

Zn(x) Zn−1(x)

  • ϕn−1(x) + Cn

Zn(x) Zn−2∗(x)ϕn−2(x),

n > 0, En = 0, Cn = − En(1+Fn/Zn−1(αn−1))

E n−1

= 0 iff the ORFs are regular, i.e. ∀n > 0 : pn(αn−1) = 0 and pn(αn−1) = 0.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ORFs

Orthonormal rational functions (ORFs) These ORFs satisfy the following 3-term recurrence relation: ϕ−1(x) ≡ 0, ϕ0(x) ≡ 1, ϕn(x) = En

  • Zn(x) + Fn

Zn(x) Zn−1(x)

  • ϕn−1(x) + Cn

Zn(x) Zn−2∗(x)ϕn−2(x),

n > 0, En = 0, Cn = − En(1+Fn/Zn−1(αn−1))

E n−1

= 0 iff the ORFs are regular, i.e. ∀n > 0 : pn(αn−1) = 0 and pn(αn−1) = 0.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Associated rational functions (ARFs) Ln\k = space of rational functions with poles in An\k = {αk+1, . . . , αn} ⊂ (C ∪ {∞}) \ {0}. ARFs ϕ(k)

n\k ∈ Ln\k of order k are defined by:

ϕ(k)

(k−1)\k(x) ≡ 0,

ϕ(k)

k\k(x) ≡ 1,

ϕ(k)

n\k(x) = En

  • Zn(x) + Fn

Zn(x) Zn−1(x)

  • ϕ(k)

(n−1)\k(x)

+Cn

Zn(x) Zn−2∗(x)ϕ(k) (n−2)\k(x),

n > k.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Associated rational functions (ARFs) Ln\k = space of rational functions with poles in An\k = {αk+1, . . . , αn} ⊂ (C ∪ {∞}) \ {0}. ARFs ϕ(k)

n\k ∈ Ln\k of order k are defined by:

ϕ(k)

(k−1)\k(x) ≡ 0,

ϕ(k)

k\k(x) ≡ 1,

ϕ(k)

n\k(x) = En

  • Zn(x) + Fn

Zn(x) Zn−1(x)

  • ϕ(k)

(n−1)\k(x)

+Cn

Zn(x) Zn−2∗(x)ϕ(k) (n−2)\k(x),

n > k.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Favard theorem Let the ARFs of order k (ϕ(k)

n\k) be as defined before, and assume

αk−1 ∈ (R ∪ {∞}) \ {0}, ϕ(k)

n\k ∈ Ln\k \ L(n−1)\k, n = k + 1, k + 2, . . ..

Then there exists a HPDN linear functional M(k) so that M(k){ϕ(k)

n\kϕ(k) (j\k)∗} =

0, n = j 1, n = j.

  • K. Deckers and A. Bultheel, “Orthogonal rational functions

with complex poles: The Favard theorem,” Technical Report TW518, Depart. Comp. Sc., K.U.Leuven, February 2008.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Favard theorem Let the ARFs of order k (ϕ(k)

n\k) be as defined before, and assume

αk−1 ∈ (R ∪ {∞}) \ {0}, ϕ(k)

n\k ∈ Ln\k \ L(n−1)\k, n = k + 1, k + 2, . . ..

Then there exists a HPDN linear functional M(k) so that M(k){ϕ(k)

n\kϕ(k) (j\k)∗} =

0, n = j 1, n = j.

  • K. Deckers and A. Bultheel, “Orthogonal rational functions

with complex poles: The Favard theorem,” Technical Report TW518, Depart. Comp. Sc., K.U.Leuven, February 2008.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Favard theorem Let the ARFs of order k (ϕ(k)

n\k) be as defined before, and assume

αk−1 ∈ (R ∪ {∞}) \ {0}, ϕ(k)

n\k ∈ Ln\k \ L(n−1)\k, n = k + 1, k + 2, . . ..

Then there exists a HPDN linear functional M(k) so that M(k){ϕ(k)

n\kϕ(k) (j\k)∗} =

0, n = j 1, n = j.

  • K. Deckers and A. Bultheel, “Orthogonal rational functions

with complex poles: The Favard theorem,” Technical Report TW518, Depart. Comp. Sc., K.U.Leuven, February 2008.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Favard theorem Let the ARFs of order k (ϕ(k)

n\k) be as defined before, and assume

αk−1 ∈ (R ∪ {∞}) \ {0}, ϕ(k)

n\k ∈ Ln\k \ L(n−1)\k, n = k + 1, k + 2, . . ..

Then there exists a HPDN linear functional M(k) so that M(k){ϕ(k)

n\kϕ(k) (j\k)∗} =

0, n = j 1, n = j.

  • K. Deckers and A. Bultheel, “Orthogonal rational functions

with complex poles: The Favard theorem,” Technical Report TW518, Depart. Comp. Sc., K.U.Leuven, February 2008.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

ARFs

Favard theorem Let the ARFs of order k (ϕ(k)

n\k) be as defined before, and assume

αk−1 ∈ (R ∪ {∞}) \ {0}, ϕ(k)

n\k ∈ Ln\k \ L(n−1)\k, n = k + 1, k + 2, . . ..

Then there exists a HPDN linear functional M(k) so that M(k){ϕ(k)

n\kϕ(k) (j\k)∗} =

0, n = j 1, n = j.

  • K. Deckers and A. Bultheel, “Orthogonal rational functions

with complex poles: The Favard theorem,” Technical Report TW518, Depart. Comp. Sc., K.U.Leuven, February 2008.

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Relation 1 APs p(k)

n−k(x) = Ek+1(x + Fk+1)p(k+1) n−(k+1)(x)

+ Ck+2p(k+2)

n−(k+2)(x),

n > k ARFs ϕ(k)

n\k(x) = Ek+1

  • Zk+1(x) + Fk+1

Zk+1(x) Zk(x)

  • ϕ(k+1)

n\(k+1)(x)

+ Ck+2 Zk+2(x) Zk∗(x) ϕ(k+2)

n\(k+2)(x),

n > k

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Relation 1 APs p(k)

n−k(x) = Ek+1(x + Fk+1)p(k+1) n−(k+1)(x)

+ Ck+2p(k+2)

n−(k+2)(x),

n > k ARFs ϕ(k)

n\k(x) = Ek+1

  • Zk+1(x) + Fk+1

Zk+1(x) Zk(x)

  • ϕ(k+1)

n\(k+1)(x)

+ Ck+2 Zk+2(x) Zk∗(x) ϕ(k+2)

n\(k+2)(x),

n > k

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Relation 2 APs p(k)

n−k(x) = p(j) n−j(x)p(k) j−k(x) + Cj+1p(j+1) n−(j+1)(x)p(k) (j−1)−k(x),

k + 1 ≤ j ≤ n − 1 ARFs ϕ(k)

n\k(x) = ϕ(j) n\j(x)ϕ(k) j\k(x)+Cj+1

Zj+1(x) Zj−1∗(x)ϕ(j+1)

n\(j+1)(x)ϕ(k) (j−1)\k(x),

k + 1 ≤ j ≤ n − 1

Karl Deckers ORFs and ARFs

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Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Relation 2 APs p(k)

n−k(x) = p(j) n−j(x)p(k) j−k(x) + Cj+1p(j+1) n−(j+1)(x)p(k) (j−1)−k(x),

k + 1 ≤ j ≤ n − 1 ARFs ϕ(k)

n\k(x) = ϕ(j) n\j(x)ϕ(k) j\k(x)+Cj+1

Zj+1(x) Zj−1∗(x)ϕ(j+1)

n\(j+1)(x)ϕ(k) (j−1)\k(x),

k + 1 ≤ j ≤ n − 1

Karl Deckers ORFs and ARFs

slide-47
SLIDE 47

Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Outline proof relation 1 and 2 similar to the proof for the polynomial case, i.e. by induction, with the aid of the 3-term recurrence relation.

Karl Deckers ORFs and ARFs

slide-48
SLIDE 48

Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Outline proof relation 1 and 2 similar to the proof for the polynomial case, i.e. by induction, with the aid of the 3-term recurrence relation.

Karl Deckers ORFs and ARFs

slide-49
SLIDE 49

Introduction ORFs and ARFs Relations Favard theorem

Relation ARFs of different order

Outline proof relation 1 and 2 similar to the proof for the polynomial case, i.e. by induction, with the aid of the 3-term recurrence relation.

Karl Deckers ORFs and ARFs

slide-50
SLIDE 50

Introduction ORFs and ARFs Relations Favard theorem

Outline

1

Introduction Preliminaries

2

ORFs and ARFs Orthogonal rational functions Associated rational functions

3

Relations ARFs of different order ARFs and functions of the second kind

4

Favard theorem

Karl Deckers ORFs and ARFs

slide-51
SLIDE 51

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 APs p(k)

n−k(x) = 1

Ek · M(k−1)

t

   p(k−1)

n−(k−1)(t) − p(k−1) n−(k−1)(x)

t − x    ARFs suppose the ϕ(k−1)

n\(k−1) are orthonormal w.r.t. M(k−1), and let

ψ(k)

n (x) = 1

ck · M(k−1)

t

   ϕ(k−1)

n\(k−1)(t) − ϕ(k−1) n\(k−1)(x)

t − x   

Karl Deckers ORFs and ARFs

slide-52
SLIDE 52

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 APs p(k)

n−k(x) = 1

Ek · M(k−1)

t

   p(k−1)

n−(k−1)(t) − p(k−1) n−(k−1)(x)

t − x    ARFs suppose the ϕ(k−1)

n\(k−1) are orthonormal w.r.t. M(k−1), and let

ψ(k)

n (x) = 1

ck · M(k−1)

t

   ϕ(k−1)

n\(k−1)(t) − ϕ(k−1) n\(k−1)(x)

t − x   

Karl Deckers ORFs and ARFs

slide-53
SLIDE 53

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ψ(k)

n

= ϕ(k)

n\k?

∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ψ(k)

k−1(x) ≡ 0; but generally,

∀n > k − 1 : ψ(k)

n

∈ Ln\(k−1) \ Ln\k, ψ(k)

j

, j = k + 1, k, k − 1, do not satisfy the 3-term recurrence relation. ⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-54
SLIDE 54

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ψ(k)

n

= ϕ(k)

n\k?

∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ψ(k)

k−1(x) ≡ 0; but generally,

∀n > k − 1 : ψ(k)

n

∈ Ln\(k−1) \ Ln\k, ψ(k)

j

, j = k + 1, k, k − 1, do not satisfy the 3-term recurrence relation. ⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-55
SLIDE 55

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ψ(k)

n

= ϕ(k)

n\k?

∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ψ(k)

k−1(x) ≡ 0; but generally,

∀n > k − 1 : ψ(k)

n

∈ Ln\(k−1) \ Ln\k, ψ(k)

j

, j = k + 1, k, k − 1, do not satisfy the 3-term recurrence relation. ⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-56
SLIDE 56

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ψ(k)

n

= ϕ(k)

n\k?

∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ψ(k)

k−1(x) ≡ 0; but generally,

∀n > k − 1 : ψ(k)

n

∈ Ln\(k−1) \ Ln\k, ψ(k)

j

, j = k + 1, k, k − 1, do not satisfy the 3-term recurrence relation. ⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-57
SLIDE 57

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ψ(k)

n

= ϕ(k)

n\k?

∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ψ(k)

k−1(x) ≡ 0; but generally,

∀n > k − 1 : ψ(k)

n

∈ Ln\(k−1) \ Ln\k, ψ(k)

j

, j = k + 1, k, k − 1, do not satisfy the 3-term recurrence relation. ⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-58
SLIDE 58

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 Redefine ψ(k)

n :

ψ(k)

n (x) = (1 − x/αk)

ck ×  M(k−1)

t

   ϕ(k−1)

n\(k−1)(t) − ϕ(k−1) n\(k−1)(x)

t − x    − δn,k−1dk(x)   Problem : ψ(k)

k−1(x) = − (1−x/αk)dk(x) ck

= 0

Karl Deckers ORFs and ARFs

slide-59
SLIDE 59

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 Redefine ψ(k)

n :

ψ(k)

n (x) = (1 − x/αk)

ck ×  M(k−1)

t

   ϕ(k−1)

n\(k−1)(t) − ϕ(k−1) n\(k−1)(x)

t − x    − δn,k−1dk(x)   Problem : ψ(k)

k−1(x) = − (1−x/αk)dk(x) ck

= 0

Karl Deckers ORFs and ARFs

slide-60
SLIDE 60

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 Let Φn\(k−1)(x, t) = (1 − t/αk−1)ϕ(k−1)

n\(k−1)(x)

and redefine ψ(k)

n :

ψ(k)

n (x) = (1 − x/αk)

ck ×

  • M(k−1)

t

Φn\(k−1)(t, x) − Φn\(k−1)(x, t) t − x

  • − δn,k−1dk(x)
  • Karl Deckers

ORFs and ARFs

slide-61
SLIDE 61

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ∀n > k − 1 : ψ(k)

n

∈ Ln\k, setting dk(x) ≡ 1/αk−1 → ψ(k)

k−1(x) ≡ 0 and ψ(k) j

, j = k + 1, k, k − 1, satisfy the 3-term recurrence relation, setting ck = Ek(1 + Fk/Zk−1(αk−1)) → ψ(k)

k (x) ≡ 1.

⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-62
SLIDE 62

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ∀n > k − 1 : ψ(k)

n

∈ Ln\k, setting dk(x) ≡ 1/αk−1 → ψ(k)

k−1(x) ≡ 0 and ψ(k) j

, j = k + 1, k, k − 1, satisfy the 3-term recurrence relation, setting ck = Ek(1 + Fk/Zk−1(αk−1)) → ψ(k)

k (x) ≡ 1.

⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-63
SLIDE 63

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ∀n > k − 1 : ψ(k)

n

∈ Ln\k, setting dk(x) ≡ 1/αk−1 → ψ(k)

k−1(x) ≡ 0 and ψ(k) j

, j = k + 1, k, k − 1, satisfy the 3-term recurrence relation, setting ck = Ek(1 + Fk/Zk−1(αk−1)) → ψ(k)

k (x) ≡ 1.

⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-64
SLIDE 64

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ∀n > k − 1 : ψ(k)

n

∈ Ln\k, setting dk(x) ≡ 1/αk−1 → ψ(k)

k−1(x) ≡ 0 and ψ(k) j

, j = k + 1, k, k − 1, satisfy the 3-term recurrence relation, setting ck = Ek(1 + Fk/Zk−1(αk−1)) → ψ(k)

k (x) ≡ 1.

⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-65
SLIDE 65

Introduction ORFs and ARFs Relations Favard theorem

ARFs and functions of the second kind

Relation 3 ∀n > k + 1, ψ(k)

j

, j = n, n − 1, n − 2, satisfy the 3-term recurrence relation, ∀n > k − 1 : ψ(k)

n

∈ Ln\k, setting dk(x) ≡ 1/αk−1 → ψ(k)

k−1(x) ≡ 0 and ψ(k) j

, j = k + 1, k, k − 1, satisfy the 3-term recurrence relation, setting ck = Ek(1 + Fk/Zk−1(αk−1)) → ψ(k)

k (x) ≡ 1.

⇒ ψ(k)

n

= ϕ(k)

n\k

Karl Deckers ORFs and ARFs

slide-66
SLIDE 66

Introduction ORFs and ARFs Relations Favard theorem

Favard theorem for ARFs

Favard theorem Suppose the ϕ(k−1)

n\(k−1) are orthonormal w.r.t. M(k−1). Assume

αk−1 ∈ (R ∪ {∞}) \ {0}, and let ϕ(k)

n\k(x) =

(1 − x/αk) Ek(1 + Fk/Zk−1(αk−1))×

  • M(k−1)

t

Φn\(k−1)(t, x) − Φn\(k−1)(x, t) t − x

  • − δn,k−1/αk−1
  • .

Then the ϕ(k)

n\k are orthonormal w.r.t. M(k) iff M(k−1) Zn Zk−1

  • = 0

for every n > k.

Karl Deckers ORFs and ARFs