16 1. MATHEMATICAL SHAPES OF UNCERTAINTY

It is a Hilbert space equipped with the norm F

E

= F/E

L2(R)

. If E(z) is of

exponential type then all the functions in the de Branges space BE are of exponential

type not greater then the type of E(z) (see, for example, the last part in the proof

of lemma 3.5 in [42]). A de Branges space is called short (or regular) if together

with every function F (z) it contains (F (z) − F (a))/(z − a) for any a ∈ C.

One of the most important features of de Branges spaces is that they admit

a second, axiomatic, definition. Let H be a Hilbert space of entire functions that

satisfies the following axioms:

• (A1) If F ∈ H, F (λ) = 0, then

F

(z)(z−¯)

λ

z−λ

∈ H with the same norm;

• (A2) For any λ ∈ R, point evaluation at λ is a bounded linear functional

on H;

• (A3) If F ∈ H then F

#

∈ H with the same norm.

Then H = BE for a suitable de Branges function E. This is theorem 23 in [26].

Usually, for a given Hilbert space of entire functions it is not diﬃcult to verify

the above axioms and conclude that the space is a de Branges space. It is however

a challenging problem in many situations to find a generating function E. This

problem can be viewed as a deep and abstract generalization of the inverse spectral

problem for second order differential operators.

Every de Branges function E(z) gives rise to a meromorphic inner function

Θ(z) = E#(z)/E(z) and a model space KΘ that this inner function generates.

There exists a well known isometric isomorphism between BE and KΘ given by

F → F/E.

Conversely, every meromorphic Θ can be represented as Θ(z) = E#(z)/E(z)

for some de Branges function E. As was mentioned above, in this case all Clark

measures σα of Θ are discrete and their point masses can be computed by σα(λ) =

2π/ |Θ (λ)| for λ ∈ {Θ = α}.

We will call the measures |E|2σα, where σα is a Clark measure for Θ(z) =

E#(z)/E(z), spectral measures of the corresponding de Branges space. It is well

known that for any spectral measure ν of a de Branges space BE the natural em-

bedding gives an isometric isomorphism between BE and

L2(ν).

This isomorphism

generalizes the Parseval theorem.

As we mentioned before, on the real line each inner Θ(z) coming from a

de Branges function can be written as Θ(t) = eiθ(t), t ∈ R, where θ(t) is real

analytic strictly increasing function, a continuous branch of the argument of Θ(z)

on R. The phase function of the corresponding de Branges space is defined by

φ(t) = θ(t)/2 and is equal to − arg E.

Function theory discussed in this section will be revisited in slightly more detail

in chapter 7.