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Introduction Preliminaries Measures on T Examples Measures on I

Orthogonal rational functions and rational modifications of a measure

Karl Deckers

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

September 30, 2008

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Outline

1

Introduction

2

Preliminaries

3

Measures on T

4

Examples

5

Measures on I

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Motivation

Explicit examples of orthogonal rational functions (ORFs) Chebyshev ORFs on I = [−1, 1] w.r.t. w(x) = (1 − x)a(1 + x)b, a, b ∈

• ±1

2

• .

Takenaka-Malmquist basis on T = {z : |z|2 = 1} w.r.t. Lebesgue measure

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Motivation

Relating I and T ˚ w(θ) = w(cos θ)| sin θ|, θ ∈ [−π, π], z = eiθ (a, b) w(x) ˚ w(θ) ˚ µ′(z)

• − 1

2, − 1 2

• 1 − x2−1/2

1

1 iz

1

2, 1 2

• 1 − x21/2

sin2 θ

• (z2−1)2

4iz3

1

2, − 1 2

• 1−x

1+x

1/2 1 − cos θ

• (z−1)2

2iz2

• − 1

2, 1 2

• 1+x

1−x

1/2 1 + cos θ

(z+1)2 2iz2

φn(z; ˚ µ) → φn(x; w)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Orthogonal polynomials

Polynomial modifications d ˜ µ = (1 ± cos θ)d˚ µ = |z ± 1|2d˚ µ d ˆ µ = sin2 θd˚ µ = |z2 − 1|2d˚ µ , z = eiθ d ˜ µ = |pm(z)|2d˚ µ = |(z − γ1) · . . . · (z − γm)|2d˚ µ, |γj| ≤ 1, j = 1, . . . , m φn(z; ˜ µ) ⊥˜

µ Pn−1

⇒ pm(z)φn(z; ˜ µ) ⊥˚

µ pm(z)Pn−1

⇒ relation φn(z; ˜ µ) and φn+m(z; ˚ µ)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Orthogonal rational functions

Rational modifications d ˜ µ = |rm(z)|2d˚ µ =

• (z−γ1)·...·(z−γm)

(1−α1z)·...·(1−αmz)

• 2

d˚ µ, |γj| ≤ 1, and |αj| < 1, j = 1, . . . , m φn(z; ˜ µ) ⊥˜

µ ˚

Ln−1 ⇒ rm(z)φn(z; ˜ µ) ⊥˚

µ rm(z) ˚

Ln−1 ⇒ relation φn(z; ˜ µ) and φn+m(z; ˚ µ)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Canonical basis for ˚ L

Blaschke factors and Blaschke products canonical basis for ˚ Ln: ζβk(z) = ηβk

z−βk 1−βkz ,

ηβk =

• βk

|βk|,

βk = 0 1, βk = 0 Bk(z) = ζβk(z)Bk−1(z), B0(z) ≡ 1 canonical basis for ˜ Lm: ˜ Bk(z) = ζαk(z)˜ Bk−1(z) canonical basis for ˆ Ln+m: ˆ Bk(z) = ˜ Bk(z), k ≤ m ˜ Bm(z)Bk−m(z), k > m

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Definitions

Monic ORFs φn(z) is called monic iff φn(z) = 1 · Bn(z) + fn−1(z), fn−1 ∈ ˚ Ln−1

• r equivalently φ∗

n(βn) = 1 where

φ∗

n(z) = Bn(z)φn∗(z)

and φn∗(z) = φn(1/z). Suppose φn(z) = a · Bn(z) + fn−1(z), then φ∗

n(z)

= Bn(z) (a · Bn∗(z) + fn−1∗(z)) = a + ζβn(z)f ∗

n−1(z),

• Bk∗(z) = B−1

k (z)

• .

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Definitions

From now on we assume φk’s are monic ϕk’s are orthonormal, with ϕk = ˚ κkφk Reproducing kernel ˚ kn(z, u; ˚ µ) = n

k=0 ϕk(z; ˚

µ)ϕk(u; ˚ µ) = |˚ κn+1|2 φ∗

n+1(z;˚

µ)φ∗

n+1(u;˚

µ)−φn+1(z;˚ µ)φn+1(u;˚ µ) 1−ζβn+1(z)ζβn+1(u)

, n > 0

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modification

Relating monic ORFs φn(z; ˜ µ) ⊥˜

µ ˚

Ln−1 and φn+m(z; ˚ µ) ⊥˚

µ ˆ

Ln+m−1 φn+m(z; ˚ µ) − rm(z) r∗

m(βn)φn(z; ˜

µ) ∈ ˆ Ln+m−1 because φ∗

n+m(βn; ˚

µ) = 1 and [rm(z)φn(z; ˜ µ)]∗ = ˆ Bn+m(z)rm∗(z)φn∗(z; ˜ µ) = ˜ Bm(z)rm∗(z)Bn(z)φn∗(z; ˜ µ) = r∗

m(z)φ∗ n(z; ˜

µ).

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modification

Basis for ˆ Ln+m−1 Consider orthogonal decomposition ˆ Ln+m−1 = rm ˚ Ln−1 ⊕

• rm ˚

Ln−1 ⊥˚

µ

{rmφk(z; ˜ µ)}n−1

k=0 is orthogonal basis for rm ˚

Ln−1 w.r.t. ˚ µ

• {gi,k(z)}mi−1

k=0

j

i=1 is basis for

• rm ˚

Ln−1 ⊥˚

µ where

gi,k(z) = ∂kˆ kn+m−1(z, u; ˚ µ) ∂uk

• u=γi

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modification

φn+m(z; ˚ µ) − rm(z) r∗

m(βn)φn(z; ˜

µ) =

n−1

• k=0

Λkrm(z)φk(z; ˜ µ) +

j

• i=1

mi−1

• k=0

λi,kgi,k(z) φn+m(z; ˚ µ) ⊥˚

µ rm(z)φk(z; ˜

µ) and rm(z)φn(z; ˜ µ) ⊥˚

µ rm(z)φk(z; ˜

µ), so that Λk = 0. ⇒ φn+m(z; ˚ µ) − rm(z) r∗

m(βn)φn(z; ˜

µ) =

j

• i=1

mi−1

• k=0

λi,kgi,k(z)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modification

Suppose the zeros γi are simple: φn+m(z; ˚ µ) − rm(z) r∗

m(βn)φn(z; ˜

µ) =

m

• j=1

λjˆ kn+m−1(z, γj; ˚ µ). φn+m(γi; ˚ µ) =

m

• j=1

λj ˆ kn+m−1(γi, γj; ˚ µ), i = 1, . . . , m. λ = K−1φn+m(˚ µ).

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modification

Theorem Let d ˜ µ = |rm(z)|2d˚ µ where rm ∈ ˜ Lm \ ˜ Lm−1 with simple zeros in {γj}m

j=1. Let φn(z; ˜

µ) denote the monic ORF in ˚ Ln w.r.t. ˜ µ. Similarly, let φn+m(z; ˚ µ) denote the monic ORF in ˆ Ln+m w.r.t. ˚ µ. Then rm(z) r∗

m(βn)φn(z; ˜

µ) = 1 det K

• φn+m(z; ˚

µ) ˆ k

T m+n−1(z; ˚

µ) φn+m(˚ µ) K

• Karl Deckers

ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Computational aspects

Computing the monic ORFs computing φn(z; ˚ µm) by means of intermediate results; i.e. rational modifications of degree 1: d˚ µ1 =

• z − γ

1 − αz

• d˚

µ z − γ 1 − αz

• φn(z; ˚

µ1) = ηα 1 − γβn 1 − αβn

• φn+1(z; ˚

µ) − φn+1(γ; ˚ µ) ˆ kn(γ, γ; ˚ µ) ˆ kn(z, γ; ˚ µ)

• .

distinction between |γ| = 1 and |γ| < 1

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Computational aspects

|γ| = 1 (z − γ)2 1 − αz φn(z; ˚ µ1) = ηα

1−γβn 1−αβn

• φn+1(γ; ˚

µ) φ∗

n+1(γ; ˚

µ) φ

′

n+1(γ; ˚

µ) φ∗′

n+1(γ; ˚

µ)

• ×
• (z − γ)φn+1(z; ˚

µ) φn+1(z; ˚ µ) φ∗

n+1(z; ˚

µ) φn+1(γ; ˚ µ) φ∗

n+1(γ; ˚

µ)

• 1−βnz

1−βnγ

• φn+1(γ; ˚

µ) φ

′

n+1(γ; ˚

µ) φ∗′

n+1(γ; ˚

µ)

• ,

where φ

′ represents the derivative of φ. Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Computational aspects

|γ| < 1 (1 − γz)(z − γ) 1 − αz φn(z; ˚ µ1) = ηα

1−γβn 1−αβn

• φn+1(γ; ˚

µ) φ∗

n+1(γ; ˚

µ) φ∗

n+1 (γ; ˚

µ) φn+1 (γ; ˚ µ)

• ×
• (1 − γz)φn+1(z; ˚

µ) cn(z)φn+1(z; ˚ µ) cn(z)φ∗

n+1(z; ˚

µ)

• 1−|γ|2

1−βnγ

• φn+1(γ; ˚

µ) φn+1(γ; ˚ µ) φ∗

n+1(γ; ˚

µ) φ∗

n+1 (γ; ˚

µ) φn+1 (γ; ˚ µ)

• ,

where cn(z) = (1 − βnz).

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Examples

Chebyshev ORFs on T ˚ w(θ) = 1 ± cos θ φn(z; ˚ w) = cn an + z(z − bn)zBn−1(z)

1−βnz

(z ± 1)2 ˚ w(θ) = sin2 θ φn(z; ˚ w) = hn dn + enz + z2(fn + gnz + z2)zBn−1(z)

1−βnz

(z2 − 1)2

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Application

OPs and rational modifications of a measure Alternative way to compute OPs w.r.t. rational modification of a measure d ˜ µ = d˚ µ |hm(z)|2 , hm(z) = (1 − α1z) · . . . · (1 − αmz), |αj| < 1 (cfr. E. Godoy and F. Marcell´ an: based on functions of the second kind) φn(z; ˜ µ) ∈ Pn is related with φn+m(z; ˚ µ) ∈ ˜ Lm · Pn if n > m and φn(z; ˚ µ) = Pn(z)

hm(z) ∈ ˜

Lm · Pn−m, then φn(z; ˜ µ) = Pn(z) if φn(z; ˚ µ) is monic, then

Pn(z) ηα1·...·ηαm is monic

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modifications of a measure on I

d ˜ µ = rm(x)dµ, rm(x) = pm(x) (1 − x/α1) · . . . · (1 − x/αm), rm(x) ≥ 0 on I, αi / ∈ I and pm(αi) · pm(βj) = 0 rm(x) = c p

k=1(x − γk) r k=1 |x − ˜

γk|2 q

j=1(1 − x/αj) s j=1 |1 − x/˜

αj|2 , p + 2r ≤ q + 2s = m = rS(x) · |rT (x)|2, S = max{p, q} and T = max{r, s} with rS(x) = c1 p

k=1(x − γk)

(1 − x/ǫ)t1 q

j=1(1 − x/αj), ǫ = ∞ and t1 = max{0, p − q}

rT(x) = c2 r

k=1(x − ˜

γk) (1 − x/ǫ)t2 s

j=1(1 − x/˜

αj), t2 = max{0, r − s}

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modifications of a measure on I

relation I and T: φn(x; µ) → φn(z; ˚ µ) where ˚ µ(E) = µ ({cos θ, θ ∈ E ∩ [0, π)})+µ ({cos θ, θ ∈ E ∩ [−π, 0)}) Joukowski Transform x = J(z) = (z + z−1)/2: rm(x) → rm(z) rm(z) = c

m

• k=1

z − εk 1 − δkz , c = 0, γk = J(εk), αk = J(δk) rational modification d˚ µ∗ = |rm(z)|2d˚ µ relation I and T: φn(z; ˚ µ∗) → φn(x; ˜ µ)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Rational modifications of a measure on I

Alternative: ˆ Ln+T+S → {˜ α1, . . . , ˜ αT , β1, . . . , βn−1, α1, . . . , αS, βn} φS+T+n(x; µ) − c−1

n rS(x)rT (x)φn(x; ˜

µ) =

n−1

• l=0

0 · rT(x)φl(x; ˜ µ)

S−1

• k=0

Λkφn+T+k(x; µ) +

T

• j=1

λj ˆ kT+S+n−1(x, γj; µ) where cn = rS(x)rT (x) xS+T

• x=βn

. Evaluate in zeros of rS(x) and rT(c) to eliminate the Λk’s and λj’s.

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Measures on I

Rational modifications of a measure on I For computational purposes → rational modifications of degree 1: R1

• x−γ

1−x/α

• , γ ∈ R \ (−1, 1) and α ∈ (R ∪ {∞}) \ I
• γ

1−x/α

• , γ ∈ C0 and α ∈ (R ∪ {∞}) \ I

R2

• x−γ

1−x/α

• 2

, γ ∈ C and α ∈ (R ∪ {∞}) \ I

• γ

1−x/α

• 2

, γ ∈ C0 and α ∈ (R ∪ {∞}) \ I R3 R2 with α ∈ (C \ R)

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Measures on I

R2: S = 0 and T = 1 c−1

n r1(x)φn(x; ˜

µ) = φn+1(x; µ) − φn+1(γ; µ) ˆ kn(γ, γ; µ) ˆ kn(x, γ; µ). and use Christoffel-Darboux relation ˆ kn(x, y; µ) =

• φn+1(y;µ)

Zβn(y) φn(x;µ) Zξn(x)

• −
• φn+1(x;µ)

Zβn(x) φn(y;µ) Zξn(y)

• En+1H(x, y)

where H(x, y) = y − x xy , Zβ(x) = x 1 − x/β , with β ∈ (R ∪ {∞}) \ I En+1 =

• φn+1(x; µ)

Zβn(x)φn(x; µ)

• x=ξn

, ξ1 = α, ξn = βn−1, n > 1

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

Measures on I

R1: S = 1 and T = 0 c−1

n r1(x)φn(x; ˜

µ) = φn+1(x; µ) − φn+1(γ; µ) φn(γ; µ) φn(x; µ). R3: S = 0 and T = 1 c−1

n |r1(x)|2φn(x; ˜

µ) = φn+2(x; µ) + anφn+1(x; µ) + bnφn(x; µ) and evaluate in γ and γ to eliminate an and bn. ? Christoffel-Darboux relation for arbitrary complex poles ?

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

References

• K. Deckers, J. Van Deun, and A. Bultheel, “An extended

relation between orthogonal rational functions on the unit circle and the interval [−1, 1]”, Journal of Mathematical Analysis and Applications, 334(2):1260–1275, 2007.

• E. Godoy and F. Marcell´

an, “An analog of the Christoffel formula for polynomial modification of a measure on the unit circle”, Bollettino della Unione Mathematica Italiana, 7(5–A),1–12, 1991.

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

and rational modifications of a measure on the unit circle”, Journal of Approximation Theory, 2008. To appear.

Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I

References

• E. Godoy and F. Marcell´

an, “Orthogonal polynomials and rational modifications of measures”, Canadian Journal of Mathematics, 45(5):930–943, 1993.

• A. Bultheel, R. Cruz-Barroso, K. Deckers, and P.

Gonz´ alez-Vera, “Rational Szeg˝

• quadratures associated with

Chebyshev weight functions”, Mathematics of Computation,

• 2008. To appear.

Karl Deckers ORFs and RM