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Rational approximation of semigroups

(and beyond)

Patricio JaraLouisiana State University

November 28, 2007

Oliver Heaviside (1850-1925)

Oliver Heaviside(1850-1925)

What is aSemigroup?

Cauchy Problem

Idea

PadéHille-PhillipsFunctional CalculusHersh - Kato,Brenner - Thomée.Translationsemigroup

Laplace TransformInversion

Examples

Thanks

Where: LSU For: Student Seminar on Control and Optimization – slide 2

“Orthodox Mathematicians, when theycannot find the solution of a problem in aplain agebraical form, are apt to take refugein a definite integral, and call that thesolution.”

What is a Semigroup?


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Def: A strongly continuous semigroups is a functionT : [0,∞) → L (X) that satisfies

■ T (t)T (s) = T (t + s), T (0) = I



What is aSemigroup?

Cauchy Problem

Idea



Examples

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■ T (t)T (s) = T (t + s), T (0) = I

■ The maps t → T (t)x are continuous for each x ∈ X.



What is aSemigroup?

Cauchy Problem

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■ T (t)T (s) = T (t + s), T (0) = I


■ The semigroup is of type (M, ω), i.e., There is M ≥ 1and ω ∈ R such that ‖T (t)‖L (X) ≤ Me

ωt.



What is aSemigroup?

Cauchy Problem

Idea



Examples

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■ T (t)T (s) = T (t + s), T (0) = I


■ The semigroup is of type (M, ω), i.e., There is M ≥ 1and ω ∈ R such that ‖T (t)‖L (X) ≤ Me

ωt.

If T is strongly continuous, then the generator of T isdefined by

Ax := limt→0

T (t)x − x

t

Cauchy Problem


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


■ Heat Equation

Cauchy Problem


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


■ Heat Equation

■ Wave Equation

Idea


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r be a rational function such that

■ |r(z)| ≤ 1 when Re(z) ≤ 0.

■ r(z) = ez + O(zm+1) when z → 0.

Then

Idea


What is aSemigroup?

Cauchy Problem

Idea



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■ |r(z)| ≤ 1 when Re(z) ≤ 0.

■ r(z) = ez + O(zm+1) when z → 0.

Then∣∣∣∣r

n

(tz

n

)− etz

∣∣∣∣ =∣∣∣∣r

n

(tz

n

)− (e

tzn )n

∣∣∣∣

Idea


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



■ |r(z)| ≤ 1 when Re(z) ≤ 0.

■ r(z) = ez + O(zm+1) when z → 0.

Then∣∣∣∣r

n

(tz

n

)− etz

∣∣∣∣ =∣∣∣∣r

n

(tz

n

)− (e

tzn )n

∣∣∣∣

=

∣∣∣∣∣

n−1∑

j=0

rn−1−j(

tz

n

)e

tzj

n

∣∣∣∣∣ ·∣∣∣∣r

(tz

n

)− e

tzn

∣∣∣∣

Idea


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



■ |r(z)| ≤ 1 when Re(z) ≤ 0.

■ r(z) = ez + O(zm+1) when z → 0.

Then∣∣∣∣r

n

(tz

n

)− etz

∣∣∣∣ =∣∣∣∣r

n

(tz

n

)− (e

tzn )n

∣∣∣∣

=

∣∣∣∣∣

n−1∑

j=0

rn−1−j(

tz

n

)e

tzj

n

∣∣∣∣∣ ·∣∣∣∣r

(tz

n

)− e

tzn

∣∣∣∣

≤ nC

(tz

n

)m+1

Idea


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



■ |r(z)| ≤ 1 when Re(z) ≤ 0.

■ r(z) = ez + O(zm+1) when z → 0.

Then∣∣∣∣r

n

(tz

n

)− etz

∣∣∣∣ =∣∣∣∣r

n

(tz

n

)− (e

tzn )n

∣∣∣∣

=

∣∣∣∣∣

n−1∑

j=0

rn−1−j(

tz

n

)e

tzj

n

∣∣∣∣∣ ·∣∣∣∣r

(tz

n

)− e

tzn

∣∣∣∣

≤ nC

(tz

n

)m+1= Ctm+1

1

nmzm+1

Padé


What is aSemigroup?

Cauchy Problem

Idea



Examples

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Padé’s approximants rj,l of the exponential function are of

the form rj,l =Pl(z)Qj(z)

where

Pj,l(z) =l∑

k=0

(l + j − k)!l!

j!k!(l − k)!zk

and

Qj,l(z) =

j∑

k=0

(l + j − k)!

k!(j − k)!(−z)k.

Hille-Phillips Functional Calculus


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


F ω := {f(z) =´∞

0eztdα(t), where α ∈ NBVω and

‖α‖ω =´∞

0eωtd|α|(t)} is an algebra .



What is aSemigroup?

Cauchy Problem

Idea



Examples

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F ω := {f(z) =´∞


‖α‖ω =´∞


Examples: The constant functions are in F ω for every ω andrational function that are bounded on Re(z) ≤ ω are also inF ω.



What is aSemigroup?

Cauchy Problem

Idea



Examples

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F ω := {f(z) =´∞


‖α‖ω =´∞


Examples: The constant functions are in F ω for every ω andrational function that are bounded on Re(z) ≤ ω are also inF ω.

Theorem: If A is the generator of a strongly continuoussemigroup T of type (M, ω) in X, then Ψ : F ω → L (X)defined by

Ψ(f)x :=

∞

0

T (t)xdα(t)

is an homomorphism between the algebras F ω and L (X).Moreover, if Φ(f) := f(A), then ‖f(A)‖ ≤ ‖α‖ω.

Hersh - Kato, Brenner - Thomée.


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


SIAM J. Num. Anal. ’79



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



Theorem: If A a strongly continuous semigroup T of type(M, 0), then

∥∥∥∥rn

(t

nA

)f − T (t)f

∥∥∥∥ ≤ Ctm+1 1

nm‖Am+1f‖



What is aSemigroup?

Cauchy Problem

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Examples

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Theorem: If A a strongly continuous semigroup T of type(M, 0), then

∥∥∥∥rn

(t

nA

)f − T (t)f

∥∥∥∥ ≤ Ctm+1 1

nm‖Am+1f‖

So What?

Translation semigroup


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let T (t)f := f(t + ·) on C0(R+0 , X). Then A =

dds

:= D.



What is aSemigroup?

Cauchy Problem

Idea



Examples

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dds

:= D.Moreover, in general

R(λ, D)f =

ˆ ∞

0

e−λtT (t)fdt.



What is aSemigroup?

Cauchy Problem

Idea



Examples

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dds


R(λ, D)f =

ˆ ∞

0

e−λtT (t)fdt.

R(λ, D)f(0) =

ˆ ∞

0

e−λtf(t)dt = x̂(λ).



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



dds


R(λ, D)f =

ˆ ∞

0

e−λtT (t)fdt.

R(λ, D)f(0) =

ˆ ∞

0


R(λ, D)n+1f(0) =(−1)n

n!R(λ, D)(n)f(0)



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



dds


R(λ, D)f =

ˆ ∞

0

e−λtT (t)fdt.

R(λ, D)f(0) =

ˆ ∞

0


R(λ, D)n+1f(0) =(−1)n

n!R(λ, D)(n)f(0)

=(−1)n

n!

ˆ ∞

0

e−λt(−t)nf(t)dt



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks



dds


R(λ, D)f =

ˆ ∞

0

e−λtT (t)fdt.

R(λ, D)f(0) =

ˆ ∞

0


R(λ, D)n+1f(0) =(−1)n

n!R(λ, D)(n)f(0)

=(−1)n

n!

ˆ ∞

0

e−λt(−t)nf(t)dt

=(−1)n

n!f̂ (n)(λ).

Laplace Transform Inversion


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j

.



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j

.There there are Cn,i,j ∈ C

such that rn(z) = Cn,0,0 +∑

1≤i≤s1≤j≤nr

Cn,i,j(bi−z)j

.



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j



1≤i≤s1≤j≤nr

Cn,i,j(bi−z)j

.

In this way,

rn(

tnD

)f(0) = Cn,0,0f(0) +

∑1≤i≤s

1≤j≤nrCnn,i,jR

j(bi,

tnD

)f(0)



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j



1≤i≤s1≤j≤nr

Cn,i,j(bi−z)j

.

In this way,

rn(

tnD

)f(0) = Cn,0,0f(0) +

∑1≤i≤s

1≤j≤nrCnn,i,jR

j(bi,

tnD

)f(0)

= Cn,0,0f(0) +∑

1≤i≤s1≤j≤nr

Cn,i,j(

nt

)jRj

(bi

nt, D

)f(0)



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j



1≤i≤s1≤j≤nr

Cn,i,j(bi−z)j

.

In this way,

rn(

tnD

)f(0) = Cn,0,0f(0) +

∑1≤i≤s

1≤j≤nrCnn,i,jR

j(bi,

tnD

)f(0)

= Cn,0,0f(0) +∑

1≤i≤s1≤j≤nr

Cn,i,j(

nt

)jRj

(bi

nt, D

)f(0)

= Cn,0,0f(0) +∑

1≤i≤s1≤j≤nr

Cn,i,j(

nt

)j (−1)j−1(j−1)!

f̂ (j−1)(

ntbi

)



What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Let r(z) = B0,0 +∑

1≤i≤s1≤j≤r

Bi,j(bi−z)j



1≤i≤s1≤j≤nr

Cn,i,j(bi−z)j

.

In this way,

rn(

tnD

)f(0) = Cn,0,0f(0) +

∑1≤i≤s

1≤j≤nrCnn,i,jR

j(bi,

tnD

)f(0)

= Cn,0,0f(0) +∑

1≤i≤s1≤j≤nr

Cn,i,j(

nt

)jRj

(bi

nt, D

)f(0)

= Cn,0,0f(0) +∑

1≤i≤s1≤j≤nr

Cn,i,j(

nt

)j (−1)j−1(j−1)!

f̂ (j−1)(

ntbi

)

Moreover,∣∣∣∣r

n

(t

nD

)f(0) − f(t)

∣∣∣∣ ≤ Ctm+1 1

nm‖f (m+1)‖.

Examples


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


Internet

Thanks


What is aSemigroup?

Cauchy Problem

Idea



Examples

Thanks


References

■ Rational approximation of bi-continuous semigroups.

■ Rational inversion of the Laplace transform (with F.Neubrander and K. Ozer). Preprint.

■ Rational approximation of C-semigroups and thesecond order abstract Cauchy problem. Preprint

Oliver Heaviside (1850-1925)What is a Semigroup?Cauchy ProblemIdeaPadéHille-Phillips Functional CalculusHersh - Kato, Brenner - Thomée.Translation semigroupLaplace Transform InversionExamplesThanks

GoBack - Louisiana State University · 2007. 11. 28. · Hille-Phillips Functional Calculus Hersh -...

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