From a Canonical Factorization to a J-spectral...
Transcript of From a Canonical Factorization to a J-spectral...
From a Canonical Factorizationto a J-spectral Factorization
for a Class of Infinite-Dimensional Systems
Orest V. Iftime∗
∗University of Groningen
Outline
1 Preliminaries
2 Algorithmic construction
3 Proof: Algorithmic construction
4 Algorithmic construction: remarks
5 Approximation of theJ-spectral factorization
6 Conclusions
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
Preliminaries
Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.
Three kinds of factorization are discussed:
(right-)standard factorization (also called noncanonicalfactorization)
canonical factorization
J-spectral factorization
In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.
In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.
TheH∞-control problem
The standardH∞-suboptimal control problem has a solution if andonly if there exist bistable matricesW andV such that
N1(jω)N∼1 (jω)−γ2D1(jω)D
∼1 (jω) = W(jω)Jny,nzW
∼(jω), for ω ∈ R,
with bistable lower-right block ofM := W−1(
−N1 D1)
,and
R∼(jω)Jnw,nzR(jω) = V∼(jω)Jny ,nuV(jω) for ω ∈ R,
whereR =
[
0 Inw
Inz 0
]
W−1[−N2 D2]
[
0 Iny
Inu 0
]
,
and the lower-right block of the matrixRV−1 is bistable.Moreover, the set of all stabilizing controllers is given by
[
Kn
Kd
]
= V−1[
UIny
]
,
with U ∈ A such that‖U‖H∞< 1 and detKd 6= 0.
TheJ-spectral factorization
For the definition of theJ-spectral factorization we shall introduce the
signature matrixJp,q =
(
Ip 00 −Iq
)
, wherep, q ∈ N.
J-spectral factorization
Let Z = Z∼ ∈ Wn×n be a matrix-valued function.Z has aJ-spectralfactorization if there exists a matrix-functionV ∈ GAn×n such that
Z(s) = V∼(s)JV(s) for all s ∈ iR.
Such a matrixV is called aJ-spectral factor.
Necessary and sufficient conditions for the existence of aJ-spectralfactorization are known in the literature.
TheJ-spectral factorization
For the definition of theJ-spectral factorization we shall introduce the
signature matrixJp,q =
(
Ip 00 −Iq
)
, wherep, q ∈ N.
J-spectral factorization
Let Z = Z∼ ∈ Wn×n be a matrix-valued function.Z has aJ-spectralfactorization if there exists a matrix-functionV ∈ GAn×n such that
Z(s) = V∼(s)JV(s) for all s ∈ iR.
Such a matrixV is called aJ-spectral factor.
Necessary and sufficient conditions for the existence of aJ-spectralfactorization are known in the literature.
The Wiener classConsider the setA of functionsf with the representation
f (t) =
{
fa(t) + f0δ(t), t ≥ 00, t < 0
wherefa(t) andf0 are inC,∫
∞
0 |fa(t)|dt < ∞, andδ is the deltadistribution at zero. The Laplace transform off ∈ A is well definedand it is given byf (s) =
∫
∞
0 e−stfa(t)dt + f0, for s ∈ C+.
causal Wiener class
The setA = {f | f ∈ A} of Laplace transforms of the functions inAis contained in the Hardy spaceH∞ and it is known in the literature asthecausal Wiener class.
Wiener class
TheWiener class of infinite-dimensional transfer functions isW =
{
g ∈ L∞(iR,C) | g(i·) = g1(i·) + g2(i·), g1, g∼2 ∈ A}
,
whereL∞ is the space of functions bounded almost everywhere onthe imaginary axis (iR).
The Wiener classConsider the setA of functionsf with the representation
f (t) =
{
fa(t) + f0δ(t), t ≥ 00, t < 0
wherefa(t) andf0 are inC,∫
∞
0 |fa(t)|dt < ∞, andδ is the deltadistribution at zero. The Laplace transform off ∈ A is well definedand it is given byf (s) =
∫
∞
0 e−stfa(t)dt + f0, for s ∈ C+.
causal Wiener class
The setA = {f | f ∈ A} of Laplace transforms of the functions inAis contained in the Hardy spaceH∞ and it is known in the literature asthecausal Wiener class.
Wiener class
TheWiener class of infinite-dimensional transfer functions isW =
{
g ∈ L∞(iR,C) | g(i·) = g1(i·) + g2(i·), g1, g∼2 ∈ A}
,
whereL∞ is the space of functions bounded almost everywhere onthe imaginary axis (iR).
Standard and canonical factorizations
Definition
The matrix valued functionZ ∈ Wn×n is said to admit a(right-)standard factorization relative to the imaginary axis if Z can bedecomposed as
Z = Z−DZ+, (1)
with Z+,Z∼− ∈ GAn×n, andD a diagonal matrix function
D (s) = diag
[
(
s − s+,1
s − s−,1
)k1
, ...,
(
s − s+,n
s − s−,n
)kn]
, s ∈ iR, (2)
with s+,i ∈ C−, s−,i ∈ C+, ki ∈ Z andk1 ≥ ... ≥ kn. The integerski
are called(the right-) partial indices of the factorization. In the casek1 = ... = kn = 0, so that,Z = Z−Z+, thenZ is said to admit a(right-) canonical factorization relative to the imaginary axis.
Standard and canonical factorizations
Theorem
Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization
Z = Z−Z+.2 Each of the equations
X − P((I − Z)X) = I (3)
Y − Q(Y(I − Z)) = I (4)
is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations
X − P((I − Z)X) = F (5)
Y − Q(Y(I − Z)) = G (6)
are uniquely solvable inWn×n.
Standard and canonical factorizations
Theorem
Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization
Z = Z−Z+.2 Each of the equations
X − P((I − Z)X) = I (3)
Y − Q(Y(I − Z)) = I (4)
is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations
X − P((I − Z)X) = F (5)
Y − Q(Y(I − Z)) = G (6)
are uniquely solvable inWn×n.
Standard and canonical factorizations
Theorem
Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization
Z = Z−Z+.2 Each of the equations
X − P((I − Z)X) = I (3)
Y − Q(Y(I − Z)) = I (4)
is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations
X − P((I − Z)X) = F (5)
Y − Q(Y(I − Z)) = G (6)
are uniquely solvable inWn×n.
Standard and canonical factorizations
Theorem
Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization
Z = Z−Z+.2 Each of the equations
X − P((I − Z)X) = I (3)
Y − Q(Y(I − Z)) = I (4)
is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations
X − P((I − Z)X) = F (5)
Y − Q(Y(I − Z)) = G (6)
are uniquely solvable inWn×n.
Standard and canonical factorizations
Theorem
Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization
Z = Z−Z+.2 Each of the equations
X − P((I − Z)X) = I (3)
Y − Q(Y(I − Z)) = I (4)
is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations
X − P((I − Z)X) = F (5)
Y − Q(Y(I − Z)) = G (6)
are uniquely solvable inWn×n.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Find a matrix-valued functionA ∈ Wn×n which admits acanonical factorization and a rational matrix-valued function Rinvertible overRLn×n
∞ such that
Z = AR.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Find a matrix-valued functionA ∈ Wn×n which admits acanonical factorization and a rational matrix-valued function Rinvertible overRLn×n
∞ such that
Z = AR.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs, the unique solutions of
X − P((I − A)X) = I, and
Y − Q(Y(I − A)) = I.
WriteZ = A−A+R,
wereA− = Y−1s andA+ = X−1
s .
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs, the unique solutions of
X − P((I − A)X) = I, and
Y − Q(Y(I − A)) = I.
WriteZ = A−A+R,
wereA− = Y−1s andA+ = X−1
s .
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs, the unique solutions of
X − P((I − A)X) = I, and
Y − Q(Y(I − A)) = I.
WriteZ = A−A+R,
wereA− = Y−1s andA+ = X−1
s .
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find a matrix-valued functionΘ+ ∈ GAn×n and a rationalmatrix-valued functionΛ invertible overRLn×n
∞ such thatA+R = ΛΘ+. Write
Z = A−ΛΘ+.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find a matrix-valued functionΘ+ ∈ GAn×n and a rationalmatrix-valued functionΛ invertible overRLn×n
∞ such thatA+R = ΛΘ+. Write
Z = A−ΛΘ+.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find a matrix-valued functionΘ+ ∈ GAn×n and a rationalmatrix-valued functionΛ invertible overRLn×n
∞ such thatA+R = ΛΘ+. Write
Z = A−ΛΘ+.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find a matrix-valued functionΘ+ ∈ GAn×n and a rationalmatrix-valued functionΛ invertible overRLn×n
∞ such thatA+R = ΛΘ+. Write
Z = A−ΛΘ+.
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ
Λ = Λ−DΛ+,
whereΛ+,Λ∼− ∈ GAn×n andD as in (2).
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ
Λ = Λ−DΛ+,
whereΛ+,Λ∼− ∈ GAn×n andD as in (2).
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ
Λ = Λ−DΛ+,
whereΛ+,Λ∼− ∈ GAn×n andD as in (2).
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ
Λ = Λ−DΛ+,
whereΛ+,Λ∼− ∈ GAn×n andD as in (2).
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ
Λ = Λ−DΛ+,
whereΛ+,Λ∼− ∈ GAn×n andD as in (2).
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorizationΛ = Λ−DΛ+.
Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorizationΛ = Λ−DΛ+.
Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorizationΛ = Λ−DΛ+.
Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorizationΛ = Λ−DΛ+.
Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.
Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n
∞ .
Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and
A+ = X−1s .
Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.
Step 4: Find a standard factorizationΛ = Λ−DΛ+.
Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
Algorithmic construction
Theorem
ConsiderZ ∈ GWn×n. Then a standard factorization forZ is
Z = (A−Λ−)D(Λ+Θ+)
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
Algorithmic construction
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1
s .
(6) Factorize the matrixZ∼−Z−1
+ asZ∼−Z−1
+ = U∼JU.
(7) Finally, one has theJ-spectral factorization
Z = V∼JV
where theJ-spectral factor is given by
V := UZ+ = UX−1s .
Algorithmic construction
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1
s .
(6) Factorize the matrixZ∼−Z−1
+ asZ∼−Z−1
+ = U∼JU.
(7) Finally, one has theJ-spectral factorization
Z = V∼JV
where theJ-spectral factor is given by
V := UZ+ = UX−1s .
Algorithmic construction
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1
s .
(6) Factorize the matrixZ∼−Z−1
+ asZ∼−Z−1
+ = U∼JU.
(7) Finally, one has theJ-spectral factorization
Z = V∼JV
where theJ-spectral factor is given by
V := UZ+ = UX−1s .
Algorithmic construction
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1
s .
(6) Factorize the matrixZ∼−Z−1
+ asZ∼−Z−1
+ = U∼JU.
(7) Finally, one has theJ-spectral factorization
Z = V∼JV
where theJ-spectral factor is given by
V := UZ+ = UX−1s .
Algorithmic construction
Theorem
If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±
can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:
(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1
s .
(6) Factorize the matrixZ∼−Z−1
+ asZ∼−Z−1
+ = U∼JU.
(7) Finally, one has theJ-spectral factorization
Z = V∼JV
where theJ-spectral factor is given by
V := UZ+ = UX−1s .
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Lemma 3.2
Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n
∞ such thatZ = AR andA has a canonicalfactorization.
Lines of the proof
Let R be a rational matrix-valued function invertible overRLn×n∞
Z = AR for A = I − (R − Z)R−1.
‖R − Z‖ < ε1 and‖R−1‖ < ‖Z−1‖+ ε2.
A has a canonical factorization.
Proof: based on constructive results
Proposition 3.1
Consider a matrix-valued functionZ ∈ Wn×n which admits acanonical factorization. Then each of the equations
X − P((I − Z)X) = I, and (7)
Y − Q(Y(I − Z)) = I (8)
is uniquely solvable inWn×n with Xs andYs the unique solutions ofthe equations (7) and (8). Then a canonical factorization for Z is givenby Z = Y−1
s X−1s .
Lines of the proof
The solvability of the equations (7) and (8) and the uniqueness ofXs andYs are obtained from Clancey and Gohberg.
Prove thatZ = Y−1s X−1
s is a canonical factorization forZ.
Use the decompositionW = A ⊕ A∼0 .
Proof: based on constructive results
Proposition 3.1
Consider a matrix-valued functionZ ∈ Wn×n which admits acanonical factorization. Then each of the equations
X − P((I − Z)X) = I, and (7)
Y − Q(Y(I − Z)) = I (8)
is uniquely solvable inWn×n with Xs andYs the unique solutions ofthe equations (7) and (8). Then a canonical factorization for Z is givenby Z = Y−1
s X−1s .
Lines of the proof
The solvability of the equations (7) and (8) and the uniqueness ofXs andYs are obtained from Clancey and Gohberg.
Prove thatZ = Y−1s X−1
s is a canonical factorization forZ.
Use the decompositionW = A ⊕ A∼0 .
Proof: based on constructive results
Proposition 3.1
Consider a matrix-valued functionZ ∈ Wn×n which admits acanonical factorization. Then each of the equations
X − P((I − Z)X) = I, and (7)
Y − Q(Y(I − Z)) = I (8)
is uniquely solvable inWn×n with Xs andYs the unique solutions ofthe equations (7) and (8). Then a canonical factorization for Z is givenby Z = Y−1
s X−1s .
Lines of the proof
The solvability of the equations (7) and (8) and the uniqueness ofXs andYs are obtained from Clancey and Gohberg.
Prove thatZ = Y−1s X−1
s is a canonical factorization forZ.
Use the decompositionW = A ⊕ A∼0 .
Proof: based on constructive results
Proposition 3.1
Consider a matrix-valued functionZ ∈ Wn×n which admits acanonical factorization. Then each of the equations
X − P((I − Z)X) = I, and (7)
Y − Q(Y(I − Z)) = I (8)
is uniquely solvable inWn×n with Xs andYs the unique solutions ofthe equations (7) and (8). Then a canonical factorization for Z is givenby Z = Y−1
s X−1s .
Lines of the proof
The solvability of the equations (7) and (8) and the uniqueness ofXs andYs are obtained from Clancey and Gohberg.
Prove thatZ = Y−1s X−1
s is a canonical factorization forZ.
Use the decompositionW = A ⊕ A∼0 .
Proof: based on constructive results
Proposition 3.1
Consider a matrix-valued functionZ ∈ Wn×n which admits acanonical factorization. Then each of the equations
X − P((I − Z)X) = I, and (7)
Y − Q(Y(I − Z)) = I (8)
is uniquely solvable inWn×n with Xs andYs the unique solutions ofthe equations (7) and (8). Then a canonical factorization for Z is givenby Z = Y−1
s X−1s .
Lines of the proof
The solvability of the equations (7) and (8) and the uniqueness ofXs andYs are obtained from Clancey and Gohberg.
Prove thatZ = Y−1s X−1
s is a canonical factorization forZ.
Use the decompositionW = A ⊕ A∼0 .
Proof: based on constructive results
Lemma 3.4
Let A+ ∈ GAn×n andR a rational matrix-valued function invertibleoverRLn×n
∞ .Then there exist a matrix-valued functionΘ+ ∈ GAn×n
and a rational matrix-valued functionΛ invertible overRLn×n∞ such
thatA+R = ΛΘ+.
Lines of the proof
Direct constructive procedure.
An alternative (constructive) procedure is by using the languageof zero and pole chains (J.A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, Operator theory:advances and applications, Birkhauser, 1990.)
Proof: based on constructive results
Lemma 3.4
Let A+ ∈ GAn×n andR a rational matrix-valued function invertibleoverRLn×n
∞ .Then there exist a matrix-valued functionΘ+ ∈ GAn×n
and a rational matrix-valued functionΛ invertible overRLn×n∞ such
thatA+R = ΛΘ+.
Lines of the proof
Direct constructive procedure.
An alternative (constructive) procedure is by using the languageof zero and pole chains (J.A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, Operator theory:advances and applications, Birkhauser, 1990.)
Proof: based on constructive results
Lemma 3.4
Let A+ ∈ GAn×n andR a rational matrix-valued function invertibleoverRLn×n
∞ .Then there exist a matrix-valued functionΘ+ ∈ GAn×n
and a rational matrix-valued functionΛ invertible overRLn×n∞ such
thatA+R = ΛΘ+.
Lines of the proof
Direct constructive procedure.
An alternative (constructive) procedure is by using the languageof zero and pole chains (J.A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, Operator theory:advances and applications, Birkhauser, 1990.)
Proof: based on constructive results
Lemma 3.4
Let A+ ∈ GAn×n andR a rational matrix-valued function invertibleoverRLn×n
∞ .Then there exist a matrix-valued functionΘ+ ∈ GAn×n
and a rational matrix-valued functionΛ invertible overRLn×n∞ such
thatA+R = ΛΘ+.
Lines of the proof
Direct constructive procedure.
An alternative (constructive) procedure is by using the languageof zero and pole chains (J.A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, Operator theory:advances and applications, Birkhauser, 1990.)
Proof: based on constructive results
If Z has aJ-spectral factorization, it follows thatD = I.
One can write now
Z = A−Λ−Λ+Θ+ = Z−Z+,
whereZ− = A−Λ− andZ+ = Λ+Θ+.
Notice thatZ∼+Z∼
− = Z∼ = Z = Z−Z+. ThenZ∼−Z−1
+ = (Z∼+)
−1Z−, inwhich the right-hand side and the∼ of the left-hand side are inA.ConsequentlyZ∼
−Z−1+ is a invertible constant matrix such that
Z∼−Z−1
+ = (Z∼−Z−1
+ )∗. Therefore,Z∼−Z−1
+ = U∼JU for some unitarymatrix U.
From the previous step one hasZ− = Z∼+U∼JU which gives
Z = Z−Z+ = Z∼+U∼JUZ+.
The choiceV := UZ+ = UΛ+Θ+ gives aJ-spectral factor forZ.
Proof: based on constructive results
If Z has aJ-spectral factorization, it follows thatD = I.
One can write now
Z = A−Λ−Λ+Θ+ = Z−Z+,
whereZ− = A−Λ− andZ+ = Λ+Θ+.
Notice thatZ∼+Z∼
− = Z∼ = Z = Z−Z+. ThenZ∼−Z−1
+ = (Z∼+)
−1Z−, inwhich the right-hand side and the∼ of the left-hand side are inA.ConsequentlyZ∼
−Z−1+ is a invertible constant matrix such that
Z∼−Z−1
+ = (Z∼−Z−1
+ )∗. Therefore,Z∼−Z−1
+ = U∼JU for some unitarymatrix U.
From the previous step one hasZ− = Z∼+U∼JU which gives
Z = Z−Z+ = Z∼+U∼JUZ+.
The choiceV := UZ+ = UΛ+Θ+ gives aJ-spectral factor forZ.
Proof: based on constructive results
If Z has aJ-spectral factorization, it follows thatD = I.
One can write now
Z = A−Λ−Λ+Θ+ = Z−Z+,
whereZ− = A−Λ− andZ+ = Λ+Θ+.
Notice thatZ∼+Z∼
− = Z∼ = Z = Z−Z+. ThenZ∼−Z−1
+ = (Z∼+)
−1Z−, inwhich the right-hand side and the∼ of the left-hand side are inA.ConsequentlyZ∼
−Z−1+ is a invertible constant matrix such that
Z∼−Z−1
+ = (Z∼−Z−1
+ )∗. Therefore,Z∼−Z−1
+ = U∼JU for some unitarymatrix U.
From the previous step one hasZ− = Z∼+U∼JU which gives
Z = Z−Z+ = Z∼+U∼JUZ+.
The choiceV := UZ+ = UΛ+Θ+ gives aJ-spectral factor forZ.
Proof: based on constructive results
Proposition 3.5
Let Z = Z∼ ∈ Wn×n and suppose thatV ∈ GAn×n is aJ-spectralfactor forZ. ThenW ∈ GAn×n satisfies
W∼JW = Z = V∼JV
on the imaginary axis if and only if
W = QV,
whereQ is a constant matrix satisfying
Q∼JQ = J.
Remarks
Other classes of matrix-valued functions for which the standardandJ-spectral factorizations are possible are: Wiener algebraonthe unit circle (Theorem 6.1, page 59, ClaGoh81), algebras ofHolder continuous functions (Theorem 6.2, page 61, ClaGoh81),Functions analytic on a contour (Corollary 6.1, page 61,ClaGoh81) and The Wiener algebra on the real line (Theorem6.3, page 63, ClaGoh81).
These examples in ClaGoh81 address only the existence astandard factorization.
The steps (5)-(7) define a particularJ-spectral factor asV = UZ+. TheJ-spectral factorV is unique up to multiplicationby a constantJ-unitary matrixQ.
Remarks
Other classes of matrix-valued functions for which the standardandJ-spectral factorizations are possible are: Wiener algebraonthe unit circle (Theorem 6.1, page 59, ClaGoh81), algebras ofHolder continuous functions (Theorem 6.2, page 61, ClaGoh81),Functions analytic on a contour (Corollary 6.1, page 61,ClaGoh81) and The Wiener algebra on the real line (Theorem6.3, page 63, ClaGoh81).
These examples in ClaGoh81 address only the existence astandard factorization.
The steps (5)-(7) define a particularJ-spectral factor asV = UZ+. TheJ-spectral factorV is unique up to multiplicationby a constantJ-unitary matrixQ.
Remarks
Other classes of matrix-valued functions for which the standardandJ-spectral factorizations are possible are: Wiener algebraonthe unit circle (Theorem 6.1, page 59, ClaGoh81), algebras ofHolder continuous functions (Theorem 6.2, page 61, ClaGoh81),Functions analytic on a contour (Corollary 6.1, page 61,ClaGoh81) and The Wiener algebra on the real line (Theorem6.3, page 63, ClaGoh81).
These examples in ClaGoh81 address only the existence astandard factorization.
The steps (5)-(7) define a particularJ-spectral factor asV = UZ+. TheJ-spectral factorV is unique up to multiplicationby a constantJ-unitary matrixQ.
Remarks
Other classes of matrix-valued functions for which the standardandJ-spectral factorizations are possible are: Wiener algebraonthe unit circle (Theorem 6.1, page 59, ClaGoh81), algebras ofHolder continuous functions (Theorem 6.2, page 61, ClaGoh81),Functions analytic on a contour (Corollary 6.1, page 61,ClaGoh81) and The Wiener algebra on the real line (Theorem6.3, page 63, ClaGoh81).
These examples in ClaGoh81 address only the existence astandard factorization.
The steps (5)-(7) define a particularJ-spectral factor asV = UZ+. TheJ-spectral factorV is unique up to multiplicationby a constantJ-unitary matrixQ.
Remarks
Assume now that it is known thatZ has aJ-spectral factorizationand one would apply the whole Theorem toZ. TakeR = I inStep(1) of the algorithm, thenA = Z. In this case, the set ofequations in Step(2) amounts to finding a canonicalfactorization (as in ClaGoh). In Step(3), takeΘ+ = A+ = X−1
sandΛ = I. Take nowΛ = Λ± = I in Step(4). The rest is thestandard procedure to convert a canonical factorization ofZ to aJ-spectral factorization of Z in caseZ = Z∼.
If one takesR 6= I, operator equations forA := ZR−1 need to besolved in Step(2), which may be in some cases moreconvenient. However, the price to be paid is that Step(3) andStep (4) in the theorem should also be also performed.
Remarks
Assume now that it is known thatZ has aJ-spectral factorizationand one would apply the whole Theorem toZ. TakeR = I inStep(1) of the algorithm, thenA = Z. In this case, the set ofequations in Step(2) amounts to finding a canonicalfactorization (as in ClaGoh). In Step(3), takeΘ+ = A+ = X−1
sandΛ = I. Take nowΛ = Λ± = I in Step(4). The rest is thestandard procedure to convert a canonical factorization ofZ to aJ-spectral factorization of Z in caseZ = Z∼.
If one takesR 6= I, operator equations forA := ZR−1 need to besolved in Step(2), which may be in some cases moreconvenient. However, the price to be paid is that Step(3) andStep (4) in the theorem should also be also performed.
Remarks
Assume now that it is known thatZ has aJ-spectral factorizationand one would apply the whole Theorem toZ. TakeR = I inStep(1) of the algorithm, thenA = Z. In this case, the set ofequations in Step(2) amounts to finding a canonicalfactorization (as in ClaGoh). In Step(3), takeΘ+ = A+ = X−1
sandΛ = I. Take nowΛ = Λ± = I in Step(4). The rest is thestandard procedure to convert a canonical factorization ofZ to aJ-spectral factorization of Z in caseZ = Z∼.
If one takesR 6= I, operator equations forA := ZR−1 need to besolved in Step(2), which may be in some cases moreconvenient. However, the price to be paid is that Step(3) andStep (4) in the theorem should also be also performed.
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Theorem
Assume thatZ,Zk ∈ Wn×n, k ∈ N admitJ-spectral factorizations andsatisfyZk → Z in theW-norm ask → ∞. ConsiderV,Vk theparticularJ-spectral factors associated toZ,Zk, respectively, obtainedas in the constructive procedure. Then there exist constants c1, c2 > 0such that
‖Vk − V‖W
≤ c1‖Zk − Z‖W, and
‖V−1k − V−1‖
W≤ c2‖Zk − Z‖
W.
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Theorem
Assume thatZ,Zk ∈ Wn×n, k ∈ N admitJ-spectral factorizations andsatisfyZk → Z in theW-norm ask → ∞. ConsiderV,Vk theparticularJ-spectral factors associated toZ,Zk, respectively, obtainedas in the constructive procedure. Then there exist constants c1, c2 > 0such that
‖Vk − V‖W
≤ c1‖Zk − Z‖W, and
‖V−1k − V−1‖
W≤ c2‖Zk − Z‖
W.
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Approximation of theJ-spectral factorization
In practice,J-spectral factors for irrational function are approximatedusing rational functions. For the scalar case, it is known(JacWinZwa99) that the spectral factor depends continuously on thespectral density in the Wiener norm.
Counterexamples
Anderson (1985, Math. Appl. Comput.)
Jacob, Winkin and Zwart (1998)
Positive results - only for the Spectral Factorization
Spectral Factorization via symmetric extraction Callier andWinkin (1992, Automatica)
For the scalar Spectral Factorization have been found by Jacob,Winkin and Zwart (1999, SCL).
Conclusions
A constructive algorithm.
Results on the approximation.
Thank you!
Questions?
Conclusions
A constructive algorithm.
Results on the approximation.
Thank you!
Questions?
Conclusions
A constructive algorithm.
Results on the approximation.
Thank you!
Questions?
Conclusions
A constructive algorithm.
Results on the approximation.
Thank you!
Questions?
Conclusions
A constructive algorithm.
Results on the approximation.
Thank you!
Questions?
Wiener norm
W is the set of distributionsh which can be represented as
h(t) = ha(t) +n=∞∑
n=−∞
hnδ(t − tn), t ∈ R
wheretn ∈ R, hn ∈ C, t0 = 0, tn > 0, t−n < 0 for n ∈ N, δ(t − tn) isthe Dirac delta distribution centered intn. Then
‖h‖W
= ‖h‖W = ‖ha‖L1(R) +n=∞∑
n=−∞
|hn| < ∞
where
h(jω) =∫
∞
∞
e−jωτha(τ)dτ +
n=∞∑
n=−∞
hne−jωtn , ω ∈ R.
W ⊂ C(jR) and‖f‖C(jR) = supω|f (jω)|, ‖ · ‖C(jR) ≤ ‖ · ‖
W.