Takanaka-Malmquist basis and general Toeplitz matrices Adhemar - - PowerPoint PPT Presentation

Takanaka-Malmquist basis and general Toeplitz matrices

Adhemar Bultheel1 and Pierre Carrette2 December 2003

1Department Computer Science

K.U.Leuven, Belgium adhemar.bultheel@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/∼ade/

2Shell Oil Company

Houston, Texas USA pierre.carrette@shell.com

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Takenaka-Malmquist basis

Orthogonal rational functions w.r.t. Lebesgue measure on T ϕn(z) =

• 1 − |αn|2

1 − αnz

n−1

• k=0

z − αk 1 − αkz

• Bn(z)

, n ≥ 0, αk ∈ D, α0 = 0, ϕ−n = ϕn(1/z) {ϕn}k∈Z complete in L2(T) iff (1 − |αk|) = ∞ ⇒ Bn(z) → 0 Assumption |αk| ≤ c < 1, α0 = 0 .

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Convergence in Cq

2π, q > 2

f(ω) := F(eiω) ∈ Cq

2π, q > 2

Classical theory = all αk = 0: denote with a hat. ˆ ϕk(eiω) = eikω, ˆ ck = f, ˆ ϕk =

1 2π

2π f(ω) ˆ ϕk(eiω)dω Thm: f(ω) =

k ˆ

ck ˆ ϕk(eiω) absolutely convergent Thm: f ∈ Cq

2π, q > 2 then |k|<p ckϕk(eiω) → f(ω),

ck = f, ϕk cvg uniform in ω; rate of cvg is at least 1/pq−1 Proof is technical, based on |c0| ≤ f1 and |c±k| ≤ Cf (q)1/(ǫk)q, k ≥ 1 C and ǫ depend on q, c, k

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Example

The function is f(ω) = (cos 2ω + 2)/(4 cos 2ω + 5). f(ω) = ReF(eiω), F(z) = (z2 + 2)−1. Poles at ±i √ 2.

1 2 3 4 5 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 2 4 6 8 10 12 14 x 10

−3

Figure: left: f and f2, random zeros; right: f − f4 and f − f6 for poles αk = (−1)k0.6.

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Example convergence

The function is f(ω) = (cos 2ω + 2)/(4 cos 2ω + 5). f(ω) = ReF(eiω), F(z) = (z2 + 2)−1. Poles at ±i √ 2.

2 4 6 8 10 12 14 16 18 20 10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 2 4 6 8 10 12 14 16 18 20 10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

Figure: left: f −fp∞ for (1) αk random, (2) αk = 0, (3) αk = (−1)k0.6, right: Fourier coefs for case (3)

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Convergence in C2π

Here convergence need not be uniform ⇒ Ces` aro means. Example: The function is f(ω) = |ω − π|.

1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure: left: |f − fn|, n = 2, 4, 6 for poles αk = 0. right: Ces` aro sum for n = 6.

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Ces` aro sums and Toeplitz matrix

ˆ Mp(f) = 1 2π 2π eikωf(ω)eilωdω p−1

k,l=0

= 1 2π 2π ˆ Γp(ω)f(ω)ˆ Γ∗

p(ω)dω

ˆ Γp(ω) = [1, e−iω, . . . , e−i(p−1)ω]∗ = [ ˆ ϕ0(eiω), . . . , ˆ ϕp−1(eiω)]∗ Note ˆ Γ∗

p(µ)ˆ

Γp(σ) =

p−1

• k=0

ˆ ϕk(eiσ) ˆ ϕk(eiµ) = ˆ Kp(µ, σ) ˆ γp = ˆ Γ∗

p(ω)ˆ

Γp(ω) = ˆ Kp(ω, ω) = p Recall ˆ Mp(f) = 1 2π 2π ˆ Γp(ω)f(ω)ˆ Γ∗

p(ω)dω

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Set ˆ Gp(ω) = ˆ Γp(ω)/

• ˆ

γp. Then ˆ G∗

p(ω) ˆ

Mp(f) ˆ Gp(ω) =

• |k|<p
• 1 − |k|

p

• ˆ

ck ˆ ϕk(eiω) This is Ces` aro sum and lim

p→∞

ˆ G∗

p(ω) ˆ

Mp(f) ˆ Gp(ω) = f(ω) lim

p→∞

ˆ G∗

p(ω) ˆ

Mp(f) ˆ Mp(g) ˆ Gp(ω) = f(ω)g(ω)

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Generalized Ces` aro sums

Assume f ∈ Cq

2π with q > 4.

Generalized Toeplitz: Mp(fi) =

1 2π

2π Γp(ω)fi(ω)Γ∗

p(ω)dω

Γp(ω) = [ϕ0(eiω), . . . , ϕp−1(eiω)]∗ Gp(ω) = Γp(ω)

• γp(ω)

, γp(ω) = Γ∗

p(ω)Γp(ω)

G∗

p(σ)T(Mp(f1), . . . , Mp(fn))Gp(µ) =

• T(f1, . . . , fn),

σ = µ 0, σ = µ T(x1, . . . , xn) analytic function in n variables. Rate of convergence as fast as 1/p if σ = µ or ln p

p if σ = µ.

Depends on analysis of the generalized Toeplitz matrix Mp(f).

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Spectral properties

Classical case lim

p→∞

1 p

p−1

• i=0

λi( ˆ Mp(f)) = 1 2π 2π f(ω)dω Generalized to f ∈ Cq

2π, q > 4

lim

p→∞

1 p

p−1

• i=0

T (λi(Mp(f))) = 1 2π 2π T

• f(χ−1(ω))
• dω

χ(ω) = limp→∞

χp(ω) p

, χp(ω) = phase of Bp(ω).

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Example

2 4 6 8 10 12 14 16 18 20 0.57 0.575 0.58 0.585 0.59 0.595 0.6 0.605

f(ω) = cos 2ω + 2 4 cos 2ω + 5 lim

p→∞

1 p

p−1

• k=0

λk(Mp(f)) = 1 2π 2π f(˜ χ−1(ω))dω = 2π f(µ)˜ χ(µ)dµ left-hand side for p = 20 is 0.6045, right-hand side integral = 0.6098

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Quadrature formulas

The basic idea in the proofs is to approximate integrals by quadratures. Mp(f) = 1 2π 2π Γp(ω)f(ω)Γ∗

p(ω)dω

Use 1 2π 2π g(ω)dω ≈

p−1

• k=0

Hkg(ωk). Then Mp(f) ≈ Wp(f) = ΥpFpΥ∗

p

Fp = diag(f(ω0), . . . , f(ωp−1)), Υp = [˜ Γp(ω0), . . . , ˜ Γp(ωp−1)], ˜ Γp(ωk) = Γp(ωk)

• Hk

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The quadrature formula is exact in Lp · L−p = span{ϕk(eiω)ϕl(eiω) : k, l = 0, . . . , p − 1} if ωk are zeros of para-orthogonals ϕp(eiω) − ηBp(eiω)ϕp(eiω), η ∈ T and Hk = 1/γp(ωk). The classical case = Szeg˝

• quadrature: ωk equidistant, Hk = 1/p. Then

Υp is the (unitary) FFT matrix. Mp(f) ≈ Wp(f) = ΥpFpΥ∗

p

EVD of Wp(f) ≈ Mp(f), so f(ωk) ≈ fp(ωk) approximate the eigenvalues

• f Mp(f).

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The general case = rational Szeg˝

• : ωk ∈ Ωp(θ), Hk = 1/γp(ωk).

Ωp(θ) = {ω ∈ [0, 2π) : χp(ω) = θ mod 2π; η = eiθ; Bp(eiω) = eiχp(ω)}.

1 2 3 4 5 6 −3 −2 −1 1 2 3 1 2 3 4 5 6 −3 −2 −1 1 2 3

Figures: χ10(ω), left αk = ±0.6i, right αk = 0.9eikπ/10

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Spectral approximation

2 4 6 8 10 12 14 16 18 20 0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 6 0.4 0.5 0.6 0.7 0.8 0.9 1

f(ω) = (cos 2ω+2)/(4 cos 2ω+5). Left: 2-norm of the vector of differences λi(Mp(f)) − f(ωi) as a function of p. Right diag(Υ∗

pMp(f)Υp) (circles)

and f(ω).

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Spectral approximation

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7

cumulative distribution of f(ωk) for ωk ∈ Ω20(1) (dashed) and cumulative distribution of λk(M20(f)) (solid).

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Reference

[1] A. Bultheel, P. Carrette. Algebraic and spectral properties of general Toeplitz matrices. SIAM J. Control Optim., 41 (2003), pp. 1413–1439.

CDC, Maui, Hawaii, December, 2003

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