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Example 2, αIp ≤ Δ(μ) ≤ βIp for some β ≥ α > 0

# Example 2, αIp ≤ Δ(μ) ≤ βIp for some β ≥ α > 0

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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

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

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 +

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

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

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

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

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)

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

0

and hence that

dUd

d = trace −i

(0)J .

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

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