Orthogonal functions with a skew-Hermitian differentiation matrix - - PowerPoint PPT Presentation

Orthogonal functions with a skew-Hermitian differentiation matrix

Adhemar Bultheel

• Dept. Computer Science, KU Leuven

ETNA25, Recent Advances in Scientific Computation

Santa Margherita di Pula, Sardinia, May 27-30, 2019

http://nalag.cs.kuleuven.be/papers/ade/ETNA25

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 1 / 28

Motivation: stability of methods for time-dependent PDE

Example (diffusion eq.) ∂u ∂t = ∂ ∂x

• a(x)∂u

∂x

• ,

x ∈ [−1, 1], t > 0, a(x) > 0 u(0) = u0. Discretisation (D = differentiation matrix): u′ = DADu, t > 0 A > 0, u(0) = u0. Stability: if D⊤ = −D then 1 2 du2 dt = u⊤u′ = u⊤DADu = (D⊤u)⊤A(Du) ≤ 0 Challenge: find (orthogonal) basis Φ = [φ0, φ1, . . .]⊤, u(x) = ukφk such that Φ′ = DΦ with D⊤ = −D.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 2 / 28

Example: Divided differences

D = 1 2∆x       1 · · · −1 1 ... −1 ... ... . . . ... ... ...       Note: only first order!

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 3 / 28

Example: Hermite functions

φn(x) = (−1)n √ 2nn!n1/4 e−x2/2Hn(x), n = 0, 1, . . . , x ∈ R

• R

φm(x)φn(x)dx = δm,n, m, n ∈ Z+ φ′

n(x) = −

n 2φn−1(x) +

• n + 1

2 φn+1(x), n ∈ Z+ Φ′ = DΦ, D =       b0 · · · −b0 b1 ... −b1 ... ... . . . ... ... ...       , bn =

• n + 1

2 Note: φn eigenfunction of Fourier operator.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 4 / 28

How to arrive at such a system?

OPS {pn} dense in L2(R, dµ), dµ(x) = w(x)dx ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), δn ∈ R, βn > 0 skew Hermitian ×in+1 iξ{inpn(ξ)} = −βn−1{in−1pn−1(ξ)} + iδn{inpn(ξ)} + βn{in+1pn+1(ξ)} get derivative with Fourier transform φn(x) := in √ 2π ∞

−∞

eixξpn(ξ)|w(ξ)|1/2dξ φ′

n(x) = −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x)

The {φn} is orthogonal in L2(R).

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system?

OPS {pn} dense in L2(R, dµ), dµ(x) = w(x)dx ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), δn ∈ R, βn > 0 skew Hermitian ×in+1 iξ{inpn(ξ)} = −βn−1{in−1pn−1(ξ)} + iδn{inpn(ξ)} + βn{in+1pn+1(ξ)} get derivative with Fourier transform φn(x) := in √ 2π ∞

−∞

eixξpn(ξ)|w(ξ)|1/2dξ φ′

n(x) = −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x)

The {φn} is orthogonal in L2(R).

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system?

OPS {pn} dense in L2(R, dµ), dµ(x) = w(x)dx ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), δn ∈ R, βn > 0 skew Hermitian ×in+1 iξ{inpn(ξ)} = −βn−1{in−1pn−1(ξ)} + iδn{inpn(ξ)} + βn{in+1pn+1(ξ)} get derivative with Fourier transform φn(x) := in √ 2π ∞

−∞

eixξpn(ξ)|w(ξ)|1/2dξ φ′

n(x) = −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x)

The {φn} is orthogonal in L2(R).

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system?

OPS {pn} dense in L2(R, dµ), dµ(x) = w(x)dx ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), δn ∈ R, βn > 0 skew Hermitian ×in+1 iξ{inpn(ξ)} = −βn−1{in−1pn−1(ξ)} + iδn{inpn(ξ)} + βn{in+1pn+1(ξ)} get derivative with Fourier transform φn(x) := in √ 2π ∞

−∞

eixξpn(ξ)|w(ξ)|1/2dξ φ′

n(x) = −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x)

The {φn} is orthogonal in L2(R).

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

General form

The previous construct is a canonical choice. More generally we have:

Theorem (Iserles & Webb, 20191)

With {φn} as before, also ˜ φn(x) = Aei(ωx+κn)φn(Bx + C) ω, A, B, C, κn ∈ R, AB = 0 satisfies the same properties.

• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 6 / 28

Paley-Wiener

Theorem (Iserles & Webb, 20192)

With {φn} as before, assume the differentiation matrix is irreducible, then {φn} is dense in PWΩ(R) where Ω = supp (dµ).

• 2A. Iserles & M. Webb, Orthogonal systems with a skew-symmetric differentiation

matrix, Foundations of Computational Mathematics, 2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 7 / 28

Some notes and computation

If φ′

n(x) = −bn−1φn−1(x) + icnφn(x) + bnφn+1(x) then

cn = 0 iff dµ is symmetric: w(−ξ) = w(ξ). If

1 √ 2π

• w(ξ)dξ = 1 then p0 = 1 and φ0 =

1 √ 2π

∞

−∞ eixξ|w(ξ)|1/2dξ

φ1 = 1

b0 [φ′ 0 − ic0φ0]

φ2 = 1

b1 [φ′ 1 + b0φ0 − ic1φ1] = 1 b0b1 [∗φ0 + ∗φ′ 0 + ∗φ′′ 0]

· · · φn =

1 b0b1···bn−1 [βn,0φ0 + βn,1φ′ 0 + · · · + βn,nφ(n) 0 ]

β0,0 = β1,1 = 1, β1,0 = −ic0 + recurrence for βn,ℓ if n > 11 If pn(ξ) = n

ℓ=0 pn,ℓξℓ, then1

φn = in p0,0

n

• ℓ=0

(−i)ℓpn,ℓφ(ℓ) = in p0,0 pn

• −i d

dx

• φ0
• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

Some notes and computation

If φ′

n(x) = −bn−1φn−1(x) + icnφn(x) + bnφn+1(x) then

cn = 0 iff dµ is symmetric: w(−ξ) = w(ξ). If

1 √ 2π

• w(ξ)dξ = 1 then p0 = 1 and φ0 =

1 √ 2π

∞

−∞ eixξ|w(ξ)|1/2dξ

φ1 = 1

b0 [φ′ 0 − ic0φ0]

φ2 = 1

b1 [φ′ 1 + b0φ0 − ic1φ1] = 1 b0b1 [∗φ0 + ∗φ′ 0 + ∗φ′′ 0]

· · · φn =

1 b0b1···bn−1 [βn,0φ0 + βn,1φ′ 0 + · · · + βn,nφ(n) 0 ]

β0,0 = β1,1 = 1, β1,0 = −ic0 + recurrence for βn,ℓ if n > 11 If pn(ξ) = n

ℓ=0 pn,ℓξℓ, then1

φn = in p0,0

n

• ℓ=0

(−i)ℓpn,ℓφ(ℓ) = in p0,0 pn

• −i d

dx

• φ0
• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

Some notes and computation

If φ′

n(x) = −bn−1φn−1(x) + icnφn(x) + bnφn+1(x) then

cn = 0 iff dµ is symmetric: w(−ξ) = w(ξ). If

1 √ 2π

• w(ξ)dξ = 1 then p0 = 1 and φ0 =

1 √ 2π

∞

−∞ eixξ|w(ξ)|1/2dξ

φ1 = 1

b0 [φ′ 0 − ic0φ0]

φ2 = 1

b1 [φ′ 1 + b0φ0 − ic1φ1] = 1 b0b1 [∗φ0 + ∗φ′ 0 + ∗φ′′ 0]

· · · φn =

1 b0b1···bn−1 [βn,0φ0 + βn,1φ′ 0 + · · · + βn,nφ(n) 0 ]

β0,0 = β1,1 = 1, β1,0 = −ic0 + recurrence for βn,ℓ if n > 11 If pn(ξ) = n

ℓ=0 pn,ℓξℓ, then1

φn = in p0,0

n

• ℓ=0

(−i)ℓpn,ℓφ(ℓ) = in p0,0 pn

• −i d

dx

• φ0
• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

Some notes and computation

If φ′

n(x) = −bn−1φn−1(x) + icnφn(x) + bnφn+1(x) then

cn = 0 iff dµ is symmetric: w(−ξ) = w(ξ). If

1 √ 2π

• w(ξ)dξ = 1 then p0 = 1 and φ0 =

1 √ 2π

∞

−∞ eixξ|w(ξ)|1/2dξ

φ1 = 1

b0 [φ′ 0 − ic0φ0]

φ2 = 1

b1 [φ′ 1 + b0φ0 − ic1φ1] = 1 b0b1 [∗φ0 + ∗φ′ 0 + ∗φ′′ 0]

· · · φn =

1 b0b1···bn−1 [βn,0φ0 + βn,1φ′ 0 + · · · + βn,nφ(n) 0 ]

β0,0 = β1,1 = 1, β1,0 = −ic0 + recurrence for βn,ℓ if n > 11 If pn(ξ) = n

ℓ=0 pn,ℓξℓ, then1

φn = in p0,0

n

• ℓ=0

(−i)ℓpn,ℓφ(ℓ) = in p0,0 pn

• −i d

dx

• φ0
• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

An Example: Fourier-Laguerre

measure: dµ(ξ) = χ[0,∞)(ξ)ξαe−ξdξ, α > −1 OPS: Laguerre L(α)

n (ξ) after renormalization:

pn(ξ) = (−1)n

• n!

Γ(n + 1 + α)L(α)

n (ξ)

Symmetric recurrence ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α Computing machinery ⇒ φ0(x) = 1 √ 2π Γ(1 + 1

2α)

• Γ(1 + α)
• 1

1 − 2ix 1+α/2 φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• Adhemar Bultheel (KU leuven)

OF with skew differentiation matrix 9 / 28

An Example: Fourier-Laguerre

measure: dµ(ξ) = χ[0,∞)(ξ)ξαe−ξdξ, α > −1 OPS: Laguerre L(α)

n (ξ) after renormalization:

pn(ξ) = (−1)n

• n!

Γ(n + 1 + α)L(α)

n (ξ)

Symmetric recurrence ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α Computing machinery ⇒ φ0(x) = 1 √ 2π Γ(1 + 1

2α)

• Γ(1 + α)
• 1

1 − 2ix 1+α/2 φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• Adhemar Bultheel (KU leuven)

OF with skew differentiation matrix 9 / 28

An Example: Fourier-Laguerre

measure: dµ(ξ) = χ[0,∞)(ξ)ξαe−ξdξ, α > −1 OPS: Laguerre L(α)

n (ξ) after renormalization:

pn(ξ) = (−1)n

• n!

Γ(n + 1 + α)L(α)

n (ξ)

Symmetric recurrence ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α Computing machinery ⇒ φ0(x) = 1 √ 2π Γ(1 + 1

2α)

• Γ(1 + α)
• 1

1 − 2ix 1+α/2 φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• Adhemar Bultheel (KU leuven)

OF with skew differentiation matrix 9 / 28

An Example: Fourier-Laguerre

measure: dµ(ξ) = χ[0,∞)(ξ)ξαe−ξdξ, α > −1 OPS: Laguerre L(α)

n (ξ) after renormalization:

pn(ξ) = (−1)n

• n!

Γ(n + 1 + α)L(α)

n (ξ)

Symmetric recurrence ξpn(ξ) = βn−1pn−1(ξ) + δnpn(ξ) + βnpn+1(ξ), βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α Computing machinery ⇒ φ0(x) = 1 √ 2π Γ(1 + 1

2α)

• Γ(1 + α)
• 1

1 − 2ix 1+α/2 φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• Adhemar Bultheel (KU leuven)

OF with skew differentiation matrix 9 / 28

An Example: Fourier-Laguerre1

φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• φ′

n(x)

= −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x) βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α with x = 1

2 tan θ 2 ⇔ eiθ = 1+2ix 1−2ix : polynomials Π(α) n

satisfy 1 2π π

−π

Π(α)

n (eiθ)Π(α) m (eiθ) cosα θ 2dθ = δn,m

• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 10 / 28

An Example: Fourier-Laguerre1

φn(x) = in

• 2

π

• 1

1 − 2ix 1+α/2 Π(α)

n

1 + 2ix 1 − 2ix

• φ′

n(x)

= −βn−1φn−1(x) + iδnφn(x) + βnφn+1(x) βn =

• (n + 1)(n + 1 + α),

δn = 2n + 1 + α with x = 1

2 tan θ 2 ⇔ eiθ = 1+2ix 1−2ix : polynomials Π(α) n

satisfy 1 2π π

−π

Π(α)

n (eiθ)Π(α) m (eiθ) cosα θ 2dθ = δn,m

• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 10 / 28

Special case α = 0 → Takenaka-Malmquist basis

1 2π

π

−π Πn(eiθ)Πm(eiθ)dθ = δn,m ⇒ Πn(z) = zn and thus

φn(x) = in

• 2

π

• 1

1 − 2ix 1 + 2ix 1 − 2ix n , n ∈ Z+ dense in PW[0,∞)(R). Similarly φ−n−1(x) = −(−i)n+1

• 2

π

• 1

1 + 2ix 1 − 2ix 1 + 2ix n , n ∈ Z+ dense in PW(−∞,0](R).

φ0 = −

• 2

π 1 1−2ix ,

φ1 = i

• 2

π 1+2ix (1−2ix)2 ,

... φ−1 = −

• 2

π 1 1+2ix ,

φ−2 = i

• 2

π 1−2ix (1+2ix)2 ,

...

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 11 / 28

Special case α = 0 → Takenaka-Malmquist basis

1 2π

π

−π Πn(eiθ)Πm(eiθ)dθ = δn,m ⇒ Πn(z) = zn and thus

φn(x) = in

• 2

π

• 1

1 − 2ix 1 + 2ix 1 − 2ix n , n ∈ Z+ dense in PW[0,∞)(R). Similarly φ−n−1(x) = −(−i)n+1

• 2

π

• 1

1 + 2ix 1 − 2ix 1 + 2ix n , n ∈ Z+ dense in PW(−∞,0](R).

φ0 = −

• 2

π 1 1−2ix ,

φ1 = i

• 2

π 1+2ix (1−2ix)2 ,

... φ−1 = −

• 2

π 1 1+2ix ,

φ−2 = i

• 2

π 1−2ix (1+2ix)2 ,

...

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 11 / 28

Special case α = 0 → Takenaka-Malmquist basis

1 2π

π

−π Πn(eiθ)Πm(eiθ)dθ = δn,m ⇒ Πn(z) = zn and thus

φn(x) = in

• 2

π

• 1

1 − 2ix 1 + 2ix 1 − 2ix n , n ∈ Z+ dense in PW[0,∞)(R). Similarly φ−n−1(x) = −(−i)n+1

• 2

π

• 1

1 + 2ix 1 − 2ix 1 + 2ix n , n ∈ Z+ dense in PW(−∞,0](R).

φ0 = −

• 2

π 1 1−2ix ,

φ1 = i

• 2

π 1+2ix (1−2ix)2 ,

... φ−1 = −

• 2

π 1 1+2ix ,

φ−2 = i

• 2

π 1−2ix (1+2ix)2 ,

...

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 11 / 28

Special case α = 0 → Takenaka-Malmquist basis

Differentiation matrix Φ′ = DΦ, Φ = [. . . , φ−1, φ0, φ1, . . .]⊤ D =                 ... ... ... −5i −2 2 −3i −1 1 −i i 1 −1 3i 2 −2 5i ... ... ...                 and with eiθ = 1+2ix

1−2ix computations basically FFT.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 12 / 28

Special case α = 0 → Takenaka-Malmquist basis

More generally TM basis for T is φn(z) =

• 1 − |αn|2

1 − αnz

n−1

• k=0

z − αk 1 − αkz , |αk| < 1, n ∈ Z+, φ−n = φn(1/z) and for R φn(z) =

• Im αn

π

x − αn

n−1

• k=0

x − αk x − αk , Im αk > 0, n ∈ Z+, φ−n(z) = φn(z) But then we loose the skew Hermitian structure of the differentiation matrix ⇒ Choose all αk = α (not necessarily α = i/2)

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 13 / 28

Most general form

Theorem

Most general form of the TM basis family is φn(z) = γn

• |Im α|

π eiωx (x − α)n+δ (x − α)n+δ+1 , n ∈ Z, with α ∈ C \ R, ω, δ ∈ R, |γn| = 1 D is skew-Hermitian and

• R φn(x)φm(x)dx = δn,m is orthogonal system
• 1A. Iserles & M. Webb, A family of orthogonal rational functions and other
• rthogonal systems with a skew-Hermitian differentiation matrix, J. Fourier Anal. Appl.,

2019, to appear.

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 14 / 28

Influence of α, n, ω

α = 1 + i, n = 2, ω = 0 α = 1 + i, n = 2, ω = 5 α = 1 + 5i, n = 2, ω = 0 α = 1 + i, n = 5, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 15 / 28

Influence of α, n, ω

α = 1 + 0.2i, n = 2, ω = 0 α = 1 + i, n = 2, ω = 0 α = 1 + 5i, n = 2, ω = 0 α = 1 + 25i, n = 5, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 16 / 28

skew-Hermitian

Set α = A + iB then : φn(x) =

• |B|

π eiωx (x − α)n (x − α)n+1 , n ∈ Z, B = 0 φ′

n = 1

2B [nφn−1 + i(1 + 2Bω)φn − (n + 1)φn+1] Set ω = 0 and −eiθ = x − α x − α ⇔ x(θ) = A − B tan θ

2 ⇒ x − α = iB(1 + i tan θ 2)

φn(x) =

• |B|

π (−1)neinθ B(1 + i tan θ

2),

n ∈ Z, θ ∈ [−π, π]

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 17 / 28

skew-Hermitian

Set α = A + iB then : φn(x) =

• |B|

π eiωx (x − α)n (x − α)n+1 , n ∈ Z, B = 0 φ′

n = 1

2B [nφn−1 + i(1 + 2Bω)φn − (n + 1)φn+1] Set ω = 0 and −eiθ = x − α x − α ⇔ x(θ) = A − B tan θ

2 ⇒ x − α = iB(1 + i tan θ 2)

φn(x) =

• |B|

π (−1)neinθ B(1 + i tan θ

2),

n ∈ Z, θ ∈ [−π, π]

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 17 / 28

Discrete orthogonality

Theorem

If f , g =

• R

f (x)g(x)dx and f , gN =

π 2NB

2N

k=1 f (xk)g(xk)|xk − α|2

x(θ) = A − B tan θ

2, xk = x(θk), θk = k π N , k = 0, . . . , 2N,

then f , g = f , gN for all f , g ∈ span{φk : k = −N, . . . , N − 1}

Corollary

Discrete orthogonality of the φn: φn, φmN = δn,m

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 18 / 28

Fourier computation

If f (x) =

k∈Z fkφk(z),

with fk = φk, f then DFT (and possibly FFT) could be used to compute fk approximately as fk = −i √ Bπ 2N

2N

• j=1

f (xj)(1 − i tan θj

2 )e−ikθjeiωxj

for −N ≤ k ≤ N − 1 and where θj = jπ

N ,

xj = A − B tan(θj)

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 19 / 28

Examples

f (x) = 1 |x −

√ 10 2 (1 + i)|2

α =

√ 10 2 (1 + i), N = 1,

ω = 0 α = i/2, N = 50, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 20 / 28

Examples

f (x) = cos 10x x2 + 0.01 α = 0.1i, N = 1, ω = 10 α = −i/2, N = 50, ω = 10

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 21 / 28

Examples

f (x) = sin 10x x2 + 0.01 α = 0.1i, N = 500, ω = 10 α = 3i, N = 500, ω = 0

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Examples

f (x) = sech(x) α = 5i, N = 50, ω = 0 α = i/2, N = 50, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 23 / 28

Examples

f (x) =

1 (x2+4)(x2+9)

α = 2.5i, N = 10, ω = 0 α = i/2, N = 10, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 24 / 28

Examples

f (x) = cos(10x) e−x2 α = 10i, N = 200, ω = 10 α = 0.5i, N = 200, ω = 0

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Convergence

|φn| ≤ |φ0| =

• |Im α|

√π|x − α| ⇒ how fast |fk| ց 0? The fk are (up to constant factor) the classical Fourier coefficients for F(θ) = f (x(θ))(1 − i tan θ

2)e−iωx(θ),

θ ∈ [−π, π] with x(θ) = A − B tan θ

2, α = A + Bi

If singularities concentrated near α then convergence is faster.

• 1J. Weideman. Computing the Hilbert transform on the real line, Math. Comp.

64(210) 745-762, 1995

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 26 / 28

Convergence

|φn| ≤ |φ0| =

• |Im α|

√π|x − α| ⇒ how fast |fk| ց 0? The fk are (up to constant factor) the classical Fourier coefficients for F(θ) = f (x(θ))(1 − i tan θ

2)e−iωx(θ),

θ ∈ [−π, π] with x(θ) = A − B tan θ

2, α = A + Bi

If singularities concentrated near α then convergence is faster.

• 1J. Weideman. Computing the Hilbert transform on the real line, Math. Comp.

64(210) 745-762, 1995

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 26 / 28

Convergence

|φn| ≤ |φ0| =

• |Im α|

√π|x − α| ⇒ how fast |fk| ց 0? The fk are (up to constant factor) the classical Fourier coefficients for F(θ) = f (x(θ))(1 − i tan θ

2)e−iωx(θ),

θ ∈ [−π, π] with x(θ) = A − B tan θ

2, α = A + Bi

If singularities concentrated near α then convergence is faster.

• 1J. Weideman. Computing the Hilbert transform on the real line, Math. Comp.

64(210) 745-762, 1995

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 26 / 28

Examples

f (x) = cos(10x) e−x2 f (x) = N−1

k=−N fkφk(x)

α = i α = 5i α = 10i fk, N = 100, ω = 0

Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 27 / 28

The end

Thank you for your attention

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