adaptive orthonormal systems for matrix-valued functionsbernstei/web5/2016_moima_irene.pdf ·...

PreliminariesThe case of matrix-valued functions

The Unit Ball case

Adaptive orthonormal systems for matrix-valuedfunctions

Irene Sabadini

Politecnico di Milano

MOIMA, Hannover, June 20-23, 2016(joint work with D. Alpay, F. Colombo, T. Qian)

Irene Sabadini Adaptive orthonormal systems for matrix-valued functions


The Unit Ball case

Preliminaries

Functions in the Hardy space H2(D) can be decomposed using theso-called Takenaka-Malquist (TM) system

Bn(z) =

√1− |an|2

1− anz

n−1∏k=1

z − ak1− akz

, n = 1, 2, . . .

where an ∈ D.

TM systems have several applications in applied mathematics, e.g. incontrol theory, signal processing. TM systems are subject to the condition

∞∑k=1

(1− |ak |) =∞ (1)

which guarantees that the TM system is dense in Hp(D), 1 ≤ p <∞.



The Unit Ball case

Preliminaries

T. Qian and Y-B Wang proposed a decomposition in which the condition(1) is not requested, thus the obtained subset of the TM system is notcomplete in Hp(D). The goal is:

to obtain a decomposition a given function f in an adaptive way

to have fast convergence



The Unit Ball case

Preliminaries

Algorithm

We consider H and a dictionary D (consisting of unit elements in H).Given f we look for a decomposition

f =∞∑k=1

〈f , uk〉uk , uk ∈ D.

In the case of H2(D) we consider a dictionary D containing functions ofthe form

ea(z) =

√1− |a|2

1− az, a ∈ D.



The Unit Ball case

Preliminaries

Let f ∈ H2(D), f1 = f , a1, ..., an in D.We have Ba1 = ea1 and 〈f , ea1〉 =

√1− |a1|2f (a1).

f (z) = f1(z) = (f1(z)− 〈f1, ea1〉ea1(z)) + 〈f1, ea1〉ea1(z)

= g2(z) + 〈f1, ea1〉ea1(z)

g2(z) is the reminder. Since g2(a1) = 0:

f (z) = 〈f1, ea1〉ea1(z) + f2(z)z − a1

1− a1z,



The Unit Ball case

Preliminaries

Then:

f (z) = 〈f1, ea1〉ea1(z) + f2(z)z − a1

1− a1z,

and

f2(z) = 〈f2, ea2〉ea2(z) + f3(z)z − a2

1− a2z,

then, iterating,

f (z) =n∑

k=1

〈fk , eak 〉eak (z) + fn+1(z)n∏

k=1

z − ak1− akz

=n∑

k=1

〈gk ,Ba1,...,ak 〉Ba1,...,ak (z) + fn+1(z)n∏

k=1

z − ak1− akz

.



The Unit Ball case

Preliminaries

Idea (Qian-Wang): use a non a typical greedy algorithm:

the points ak are not assigned

we use at each step the maximal selection principle



The Unit Ball case

Preliminaries

We have Ba = ea and 〈f ,Ba〉 =√

1− |a|2f (a).

f (z) = f1(z) = (f1(z)− 〈f1, ea1〉ea1(z)) + 〈f1, ea1〉ea1(z)

= g2(z) + 〈f1, ea1〉ea1(z)

g2(z) is the reminder.

Idea

To minimize ‖g2‖2. This is equivalent to maximize |〈f1, ea1〉ea1(z)|2.



The Unit Ball case

Preliminaries

Maximal selection principle

For any g ∈ H2(D) there exists a ∈ D such that

|〈g , ea〉ea(z)|2 = sup{|〈g , ea1〉ea1(z)|2, a1 ∈ D}



The Unit Ball case

Theorem

Given f ∈ H2(D) using the maximal selection principle we have

f =∞∑k=1

〈f ,Bk〉Bk .



The Unit Ball case

The case of matrix-valued functions

Definition

We denote by Hp×q2 the space of p × q matrices with entries in H2(D).

A function F ∈ Hp×q2 if and only if it can be written as

F (z) =∞∑n=0

Fnzn,

where F` ∈ Cp×q, ` = 1, 2, . . ., are such that

∞∑n=0

Tr (F ∗n Fn) <∞.



The Unit Ball case


Definition

Let F (z) =∑∞

n=1 Fnzn and G (z) =

∑∞n=1 Gnz

n in Hp×q2 , we define

[F ,G ] =∞∑n=0

G∗n Fn ∈ Cq×q

and

‖F‖2 = Tr [F ,F ] =∞∑n=0

Tr (F ∗n Fn).



The Unit Ball case


Definition (Matrix-valued Blaschke factors)

They are defined as

Bw ,P(z) = Ip − P + Pbw (z),

where for w ∈ D,bw =

z − w

1− zw,

and P ∈ Cp×p is any orthogonal projection.



The Unit Ball case



Let k0 ∈ {1, . . . , p} and let F ∈ Hp×q2 . There exists w0 ∈ D and an

orthogonal projection P0 of rank k0 such that

(1−|w |2) (Tr [PF (w),F (w)]) ≤ (1−|w0|2) (Tr [P0F (w0),F (w0)]) , (1)

for every choice of w ∈ D and every orthogonal projection P of rank k0.



The Unit Ball case


Remark

Let w0 ∈ D and let P0 ∈ Cp×p be an orthogonal projection. We can write

F (z) = P0F (w0)ew0(z)√

1− |w0|2 + F (z)− P0F (w0)ew0(z)√

1− |w0|2,

where

ew0(z) =

√1− |w0|2

1− zw0.



The Unit Ball case


Proposition

Let w0 ∈ D, let P0 ∈ Cp×p be an orthogonal projection and let

H0(z) = P0F (w0)ew0(z)√

1− |w0|2

H(z) = F (z)− P0F (w0)ew0(z)√

1− |w0|2.

We have that F (z) = H0(z) + H(z), moreover

P0H(w0) = 0

and[F ,F ] = [H0,H0] + [H,H].



The Unit Ball case


Proposition

Let H ∈ Hp×q2 be such that P0H(w0) = 0p×q. Then

H = Bw0,P0G , G ∈ Hp×q2

and[H,H] = [G ,G ].



The Unit Ball case

The Algorithm

For any given w0 ∈ D and any orthogonal projection P0 ∈ Cp×p, thedecomposition F = H0 + H can be rewritten in a unique way as theorthogonal sum

F (z) = M0ew0(z)√

1− |w0|2 + Bw0,P0(z)F1(z),

where M0 = P0F (w0) ∈ Cp×q and F1 ∈ Hp×q2 . We have

M0ew0

√1− |w0|2 ∈

(Hp×q

2 Bw0,P0Hp×q2

)and Bw0,P0F1 ∈ Bw0,P0H

p×q2 .

Finally,[F ,F ] = (1− |w0|2)[P0F (w0),F (w0)] + [F1,F1].



The Unit Ball case

The Algorithm

We can then reiterate and, after fixing k1 ∈ {1, . . . , p}, get adecomposition for F1,

F1(z) = P1F (w1)ew1(z)√

1− |w1|2 + Bw1,P1(z)F2(z), (2)

where w1 is any complex number in the disc, and P1 is any orthogonalprojection of rank k1.



The Unit Ball case

The Algorithm

Thus F admits the orthogonal decomposition (with M1 = P1F (w1))

F (z) = M0ew0(z)√

1− |w0|2+

+ Bw0,P0(z)M1ew1(z)√

1− |w1|2 + Bw0,P0(z)Bw1,P1(z)F2(z)



The Unit Ball case

The Algorithm

Let

B0(z) = P0e0(z) and Bk(z) =

(k−1∏u=0

Bwu,Pu (z)

)Pkewk

(z), k = 1, 2, . . .

setting Mk = Mk

√1− |wk |2 = PkF (wk)

√1− |wk |2, we have

F (z) =N∑

k=0

Bk(z)Mk +

(N∏

u=0

Bwu,Pu (z)

)FN+1(z).



The Unit Ball case

The Algorithm

Theorem

Suppose that at each step one selects wk and Pk according to themaximal selection principle. Then, the algorithm converges, meaning that

limN→N0

Tr [F (z)−N∑

k=0

Bk(z)Mk , F (z)−N∑

k=0

Bk(z)Mk ] = 0,

where N0 can be finite or infinite.



The Unit Ball case

The Unit Ball case

Definition

Let BN be the unit ball in CN . We define

H(BN) =

f (z) =∑α∈NN

0

zαfα : ‖f ‖2H(BN )=∑α∈NN

0

α!

|α|!|fα|2 <∞

.

where we use the notation zα = zα11 · · · z

αN

N , α! = α1! · · ·αN !.It is possible to define H(BN)n×m where fα ∈ Cn×m.

We define

[f , g ](H(BN ))n×m =

∑α∈NN

0

g∗αfα, and (3)

〈f , g〉(H(BN ))n×m = Tr [f , g ](H(BN ))

n×m . (4)



The Unit Ball case

Definition

ea(z) =

√1− ‖a‖2

1− 〈z , a〉and ba(z) =

(1− ‖a‖2)1/2

1− 〈z , a〉(z − a)(IN − a∗a)−1/2.

Note that ba(z) is C1×N = C1×N -valued.



The Unit Ball case

Blaschke factor

A Blaschke factor is an element of the form

B = U

(ba(z) 01×(n−1)

0(n−1)×N In−1

)where U is a unitary matrix in Cn×n. B is Cn×(N+n−1)-valued.A Blaschke product is a product of such elements, of compatible sizes.



The Unit Ball case


Let B be a Cu×n-valued rational function of the variables z1, . . . , zN ,analytic in an neighborhood of the closed unit ball BN , and takingco-isometric values on the unit sphere, let r0 ∈ {1, . . . , n}, and letF ∈ H(BN)n×m. There exists w0 ∈ BN and a Cn×n-valued orthogonalprojection P0 of rank r0 such that

(1− ‖w0‖2) (Tr [B(w0)P0F (w0),B(w0)P0F (w0)]) is maximum.



The Unit Ball case

The Algorithm

Let F ∈ (H(BN))n×m. We choose w0 ∈ BN and r0 ∈ {1, . . . , n}.Use the maximal selection principle and get a decomposition

F (z) = P0F (w0)ew0(z)√

1− ‖w0‖2 + Bw0,P0(z)F1(z),

where F1 ∈ (H(BN))(n+r ′0(N−1))×m.Then select w1 ∈ BN and r1 ∈ {1, . . . , n + r ′0(N − 1)}, and iterate byapplying the m.s. principle to (Bw0,P0(z),F1(z)):

F1(z) = P1F1(w1)ew1(z)√

1− ‖w1‖2 + Bw1,P1(z)F2(z),

where F2 ∈ (H(BN))(n+(r ′0+r ′1)(N−1))×m.



The Unit Ball case

Set:

Bk(z) =

{ √1− ‖w0‖2ew0(z) for k = 0,√1− ‖wk‖2ewk

(z)Bw0,P0(z) · · ·Bwk−1,Pk−1(z) for k ≥ 1.

and Mk = PkFk(wk).

Theorem

If the selection on wk and Pk is done according to the m.s. principle thenthe algorithm converges and

F (z) =∞∑k=0

Bk(z)Mk .



The Unit Ball case

References

D. Alpay, F. Colombo, T. Qian, I. Sabadini, Adaptive orthonormalsystems for matrix-valued functions, preprint.

D. Alpay, F. Colombo, T. Qian, I. Sabadini, Adaptativedecomposition: the case of the Drury-Arveson space, preprint.

T. Qian and Y. B. Wang. Adaptive Decomposition Into BasicSignals of Non-negative Instantaneous Frequencies - A Variation andRealization of Greedy Algorithm. Adv. Comput. Math., 2011.


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