Example 2, αIp ≤ Δ(μ) ≤ βIp for some β ≥ α > 0
Tải bản đầy đủ - 0trang
112
H. Dym
12. Spectral densities in the Wiener algebra
If
Δ(μ) = Ip + h(μ) with h ∈ Lp1
(9)
and Δ(μ) > 0 for μ ∈ R, then, since Δ(μ) is continuous on R and, by the Riemann–
Lebesgue lemma, Δ(±∞) = Ip , Δ meets the constraints in (7) for some choice of
β > α > 0. Consequently, in view of (8),
Πa ΔKωa v = Πa (Ip + h)Kωa v
= Kωa v + Πa hKωa v = Zωa v.
Thus, as
a
Kωa (λ) =
eiλs ϕω (s)ds
and Zωa (λ) =
0
1
2π
a
eiλs e−iωs Ip ds,
0
the formula
Kωa v + Πa hKωa v = Zωa v
can be reexpressed in the time domain as
a
h(t − s)ϕω (s)ds =
ϕω (s) +
0
1 −iωs
e
Ip
2π
for s ∈ [0, a].
(10)
If it is also assumed that h(t) is continuous, then the solution of (10) can be
expressed explicitly as
ϕω (t) =
a
1 −iωt
1
e
Ip +
2π
2π
γa (t, s)e−iωs dsIp
0
in which γa (t, s) is the kernel of an integral operator and the RK of Z [0,a] (Δ)
a
Kωa (λ) =
eiλt ϕω (t)dt
0
=
1
2π
a
a
eiλt e−iωt Ip +
0
e−iωs γa (t, s)ds dt.
0
With the help of the Krein–Sobolev formula (see, e.g., [GK85] for a clear discussion
of this formula)
∂
γa (t, s) = γa (t, a)γa (a, s)
∂a
and a variant thereof
∂
γa (a − t, a − s) = γa (a − t, 0)γa (0, a − s)
∂a
it can be checked by brute force calculation that
∂ a
1 a
a
Kω (λ) =
E (λ)E−
(ω)∗
∂a
2π −
where
a
a
E−
(λ) = eiλa Ip +
eiλt γa (t, a)dt.
0
(11)
Twenty Years After
113
Thus, as Kω0 (λ) = 0,
a
Kωa (λ) =
0
∂ s
1
K (λ)ds =
∂s ω
2π
a
(λ)
The p × 2p mvf Ea (λ) = E−
a
0
s
s
E−
(λ)E−
(ω)∗ ds.
(12)
a
E+
(λ) with
a
a
E+
(λ) = Ip +
eiλs γa (s, 0)ds
0
is a de Branges matrix and
∂
I
Et (λ) = iλEt (λ) p
0
∂t
0
γt (t, 0)
0
.
+ Et (λ)
0
0
γt (0, t)
The assumption that h(t) is continuous on R can be relaxed to the weaker
assumption that h(t) is continuous on (−∞, 0) ∪ (0, ∞) with left and right limits
at 0. This is shown in a recent paper of Alpay, Gohberg, Kaashoek, Lerer and
Alexander Sakhnovich [AGKLS10].
If h = 0 in formula (9), then formulas (11) and (12) reduce to
a
(λ) = eiλa Ip
E−
and Kωa (λ) = Zωa (λ) =
1
2π
a
ei(λ−ω)s ds Ip ,
0
respectively.
13. 1993–2011
The formulas referred to in the previous section for Δ(μ) of the form (9) are
attractive and were accessible in 1992. However, this class of spectral densities is
far too restrictive. It does not even include the simple case
Δ(μ) =
1
.
1 + μ2
Thus, it was clear that it was essential to develop analogous projection formulas
for a wider class of spectral densities. This lead us to investigate:
(1) Direct and inverse problems for canonical integral and diﬀerential systems
and Dirac–Krein systems.
(2) Bitangential interpolation and extension problems.
The exploration of these two topics and the interplay between them before we
returned to reconsider multivariate prediction took almost twenty years. The conclusions from these studies were presented in a lengthy series of articles that culminated in due course in the two volumes [ArD08b] and [ArD12]. A small sample
of some of the major themes are surveyed brieﬂy in the remaining sections of
this paper. The focus will be on spectral densities Δ(μ) that meet the constraints
in (3).
114
H. Dym
14. Entire J -inner mvf ’s
A matrix J ∈ Cm×m is said to be a signature matrix, if it is both self-adjoint and
unitary with respect to the standard inner product, i.e., if
J = J∗
and J ∗ J = Im .
The main choices of J are
±Im ,
jpq =
Ip
0
0
,
−Iq
jp = jpp
and Jp =
0
−Ip
−Ip
.
0
The signature matrix jpq is most appropriate for problems concerned with contractive mvf’s, whereas Jp is most appropriate for problems concerned with mvf’s
having a nonnegative real part, since:
if ε ∈ Cp×q , then
Ip − ε∗ ε ≥ 0 ⇐⇒ ε∗
Ip
Ip
0
if ε ∈ Cp×p , then
ε + ε∗ ≥ 0 ⇐⇒ ε∗
Ip
0
−Ip
0
−Iq
−Ip
0
ε
≤ 0;
Ip
ε
Ip
≤ 0.
The signature matrices Jp and jp are unitarily equivalent:
1 −Ip
V= √
2 Ip
Ip
=⇒ VJp V = jp
Ip
and Vjp V = Jp .
An m × m mvf U (λ) is said to belong to the class E ∩ U(J) of entire J-inner
mvf’s with respect to an m × m signature matrix J if
(1) U (λ) is an entire mvf.
(2) J − U (λ)JU (λ) is positive semideﬁnite for every point λ ∈ C+ .
(3) J − U (λ)JU (λ) = 0 for every point λ ∈ R.
The last equality extends by analytic continuation to
U (λ)JU # (λ) = J
for every point λ ∈ C
and thus implies further that
(4) U (λ) is invertible for every point λ ∈ C.
(5) U (λ)−1 = JU # (λ)J for every point λ ∈ C.
(6) J − U (λ)JU (λ) is negative semideﬁnite for every point λ ∈ C− .
15. Canonical systems
A canonical integral system is a system of integral equations of the form
t
u(t, λ) = u(0, λ) + iλ
u(s, λ)dM (s)J,
(13)
0
where M (s) is a continuous nondecreasing m × m mvf on [0, d] or [0, ∞) with
M (0) = 0 and signature matrix J.
Twenty Years After
115
t
In many problems M (t) = 0 H(s)ds with H(s) ≥ 0 a.e. and at least locally
summable. Then, the integral system can be written as
t
u(t, λ) = u(0, λ) + iλ
u(s, λ)H(s)dsJ
0
and the fundamental solution of this system is the m × m continuous solution of
the integral system
t
U (t, λ) = Im + iλ
U (s, λ)H(s)dsJ.
0
Then, by iterating the inequality
t
U (t, λ) ≤ 1 + |λ|
U (s, λ)
H(s) ds,
0
it is readily checked that
t
U (t, λ) ≤ exp |λ|
H(s) ds ,
0
and hence that U (t, λ) is an entire mvf of exponential type in the variable λ.
Moreover,
1
J − U (t, λ)JU (t, ω)∗
=
ρω (λ)
2π
t
U (s, λ)H(s)U (s, ω)∗ ds.
(14)
0
Formula (14) implies that the kernel
⎧
J − U (t, λ)JU (t, ω)∗
⎪
⎨
for λ = ω
ρω (λ)
KωUt (λ) =
⎪
⎩ 1 ∂Ut (ω)
for λ = ω
2πi ∂λ
is positive and hence, by the matrix version of a theorem of Aronszajn (see, e.g.,
Theorem 5.2 in [ArD08b]), there exists exactly one RKHS of m × 1 vvf’s with
KωUt (λ) as its RK. We shall denote this space by H(Ut ).
Formula (14) also implies that
a
J − U (t, λ)JU (t, ω)∗ = −i(λ − ω)
U (s, λ)H(s)U (s, ω)∗ ds
0
and hence that
J − U (t, ω)JU (t, ω)∗ ≥ 0 if ω ∈ C+ with equality if ω ∈ R.
Thus, Ut (λ) = U (t, λ) belongs to the class
E ∩ U ◦ (J)
of entire J-inner mvf’s U with U (0) = Im
(in the variable λ).
Formula (15) also implies that
J − U (t, ω)JU (t, ω)∗ = 0
(15)
116
H. Dym
and hence that Ut (ω) is invertible for every point ω ∈ C.
The spaces H(Ut ) are nested:
H(Ut1 ) ⊆ H(Ut2 )
if 0 ≤ t1 ≤ t2 ,
but the inclusions are not necessarily isometric.
In particular, if At (λ) denotes the fundamental solution of (13) when J = Jp
and
√
t
t
(λ) E+
(λ) = 2 0 Ip At (λ)V,
E−
then
√
2 0 Ip
=
=
√
2 0
Jp − At (λ)Jp At (ω)∗
ρω (λ)
√
2
0
Ip
Jp − At (λ)Vjp VAt (ω)∗
ρω (λ)
Ip
√ 0
2
Ip
t
t
t
t
E+
(λ)E+
(ω)∗ − E−
(λ)E−
(ω)∗
,
ρω (λ)
where
ρω (λ) = −2πi(λ − ω).
The point is that the positivity of the ﬁrst kernel implies the positivity of the
second kernel and
t
t
(0) E+
(0) = Ip
At (0) = Im =⇒ E−
t
(λ)
Thus, E−
Ip .
t
t
t
E+
(λ) is an entire de Branges matrix with E−
(0) = E+
(0) = Ip .
16. Linear fractional transformations
Let
S p×p = {ε : ε is holomorphic in C+ and s(λ) ≤ 1 in C+ },
denote the Schur class and
C p×p = {τ : τ is holomorphic in C+ and
c(λ) ≥ 0 in C+ }
denote the Carath´eodory class.
If W ∈ U(jp ), then the linear fractional transformation
TW [ε] = (w11 ε + w12 )(w21 ε + w22 )−1
maps
ε ∈ S p×p → S p×p ,
whereas, if A ∈ U(Jp ), then
TA [ε] = (a11 ε + a12 )(a21 ε + a22 )−1
when det{a21 ε + a22 } ≡ 0 in C+ .
maps
ε ∈ C p×p → C p×p ,
Twenty Years After
If A ∈ E ∩ U ◦ (Jp ) and B(λ) = A(λ)V =
TB [ε] = (b11 ε + b12 )(b21 ε + b22 )−1
b11
b21
117
b12
, then
b22
maps
ε ∈ S p×p → C p×p ,
(16)
when det{b21 ε + b22 } ≡ 0 in C+ .
17. Subclasses of E ∩ U ◦ (J ) with J = ±Im
A mvf U ∈ E ∩ U ◦ (J) with J = ±Im belongs to the class
US (J)
UrR (J)
UrsR (J)
of singular J-inner mvf’s if it is of minimal exponential type
of right regular J-inner mvf’s if it has no singular right divisors
of strongly right regular J-inner mvf’s if it is
unitarily equivalent to a mvf W ∈ U(jpq ) in the class
UrsR (jpq )
of strongly right regular jpq -inner mvf’s if there exists a mvf
ε ∈ S p×q such that TW [ε] ≤ δ < 1 .
18. A pleasing RK result
A pleasing result that was obtained early in this period (in [ArD97]) is that a mvf
U ∈ E ∩ U(J) with J = ±Im belongs to the class
US (J) ⇐⇒ H(U ) ∩ Lp2 = {0}
UrR (J) ⇐⇒ H(U ) ∩ Lp2 is dense in H(U )
UrsR (J) ⇐⇒ H(U ) ⊂ Lp2 .
Some years later (in [ArD01]) it was discovered that if U ∈ E ∩U(J), J = ±Im
and P± = (Im ± J)/2, then
U ∈ E ∩ UrsR (J)
if and only if the mvf P+ + U (μ)P− U (μ)∗
(17)
satisﬁes the matrix Muckenhoupt (A2 ) condition formulated by Treil and Volberg
in [TV97]. Chapter 10 of [ArD08b] contains characterizations of the class UrsR (J)
of J-inner mvf’s that are not necessarily entire.
This characterization of the class E ∩ UrsR (J) has a nice reformulation
([ArD??]) that rests on the observation that
F(λ) = F− (λ)
F+ (λ) = U (λ)P+ + P−
U (λ)P− + P+ ,
is a de Branges matrix that is related to the mvf in (17) by the formula
F+ (μ)F+ (μ)∗ = P+ + U (μ)P− U (μ)∗ .
118
H. Dym
Moreover,
f ∈ H(U ) ⇐⇒ f ∈ B(F) ⇐⇒ F+−1 f ∈ H2m
and
f
2
H(U)
∞
=
−∞
and (F−−1 f ) ∈ (H2m )⊥
−1
f (μ)∗ {F+ (μ)F+ (μ)∗ }
f (μ)dμ,
which exhibits the role of the mvf P+ + U (μ)P− U (μ)∗ in the calculation of the
norm in H(U ).
19. A simple inverse monodromy problem
The given data for the inverse monodromy problem is a mvf U ∈ E ∩ U(J) with
U (0) = Im .
The objective is to ﬁnd an m × m mvf H(t) on [0, d] such that
([0, d]) and trace H(t) = 1 a.e. on [0, d]
(1) H(t) ≥ 0, H ∈ Lm×m
1
(2) U (λ) = Ud (λ), where
d
Ut (λ) = Im + iλ
Us (λ)H(s)dsJ.
(18)
0
The existence of a solution to this problem is guaranteed by a theorem of
Potapov (see, e.g., pp. 182–184 in [ArD08b] ). Moreover, it follows easily from (18)
that
d
Ud (λ) − Im
J=
Us (λ)H(s)ds
Ut (0) = Im and
iλ
0
and hence that
dUd
d = trace −i
(0)J .
dλ
In general H(t) is not unique unless other constraints are imposed.
If, for example, m = p + q and p = q = 1,
J = j11 ,
W (λ) =
eiλa1
0
0
e−iλa2
with a1 ≥ 0, a2 ≥ 0, a1 + a2 > 0,
then d = a1 + a2 . Thus, if H(t) is a solution of the inverse monodromy problem
for the given W , then the fundamental solution
t
Wt (λ) = Im + iλ
Ws (λ)H(s)dsj11
for t ∈ [0, d],
0
must be of the form
Wt (λ) =
eiλϕ1 (t)
0
0
e
−iλϕ2 (t)
.
Consequently,
−i
∂
ϕ (t)
Wt (0)j11 = 1
0
∂λ
0
=
ϕ2 (t)
t
H(s)ds.
0