ORTHOGONAL POLYNOMIALS, L^{2}-SPACES

AND ENTIRE FUNCTIONS.

Christian Berg^{1} and Antonio J. Duran^{2}

1 Københavns Universitet 2 Universidad de Sevilla

Abstract. We show that for determinate measures μ having moments of every order and finite index of determinacy, (i.e., a polynomial p exists for which the measure |p|^{2}μ is indeterminate) the space L^{2}(μ) consists of entire functions of minimal exponential type in the Cartwright class.

1. Introduction

Let M^{∗} denote the set of positive Borel measures on the real line having moments of every order and infinite support. We are interested in finding conditions on μ ∈ M^{∗} such that L^{2}(μ) consists of entire functions in the following sense: (i) There exists a continuous linear injection E : L^{2}(μ) → H(ℂ), where H(ℂ) denotes the set of entire functions with the topology of compact convergence. (ii) For all f ∈ L^{2}(μ) we have E(f) = f μ-a.e.. We say that E is a realization of L^{2}(μ) as entire functions. In the discussion of this problem we need for μ ∈ M^{∗} the corresponding sequence of orthonormal polynomials (p_{n}). It is uniquely determined by the orthonormality condition and the requirement that p_{n} is a polynomial of degree n with positive leading coefficient. The sequence of orthonormal polynomials depends only on the moments of μ, so if μ is indeterminate, i.e. there are other measures having the same moments as μ, all these measures lead to the same sequence (p_{n}).

When the measure μ is indeterminate, the Fourier expansion of f ∈ L^{2}(μ)

∑_{n=0}^{∞} (∫ f(t)p_{n}(t)dμ(t)) p_{n}(z)  (1.1)

converges in L^{2}(μ) and uniformly on compact subsets of ℂ to an entire function F (f)(z), which is the orthogonal projection of f onto the closure in L^{2}(μ) of the set ℂ[t] of polynomials. We recall that z ↦ (p_{n}(z))_{n} is an entire function with values in the Hilbert space ℓ^{2}, so in particular (p_{n}^{(m)}(z))_{n} ∈ ℓ^{2} for all z ∈ ℂ, m ∈ ℕ, cf. [4]. By a theorem of M. Riesz ([8], [1]) F (f) is of minimal exponential

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 42C05, 44A60

Key words and phrases. Orthogonal polynomials. Index of determinacy. Entire functions.

The work of the second author was supported by DGICYT ref. PB 93-0926.

2 CHRISTIAN BERG^{1} AND ANTONIO J. DURAN^{2}

type. If the indeterminate measure μ is Nevanlinna extremal (N-extremal in short), which means that ℂ[t] is dense in L^{2}(μ), then μ is discrete and F (f)(x) = f(x) for x ∈ supp(μ). This means that F (f) is an extension of f to an entire function of minimal exponential type.

Furthermore f ↦ F (f) is a continuous injection of L^{2}(μ) into H(ℂ). In fact, for any compact set K ⊆ ℂ we find by (1.1) and Parsevals formula

sup_{z∈K} |F (f)(z)| ≤ ∥f∥_{2} sup_{z∈K} ρ(z) ,

where

ρ(z) = (∑_{k=0}^{∞} |p_{k}(z)|^{2})^{\frac{1}{2}}

is continuous. Riesz ([8]) also showed that

∫_{−∞}^{∞} \frac{log ρ(t)}{1 + t^{2}} dt < ∞ ,

and it follows that

∫_{−∞}^{∞} \frac{log^{+} F (f)(t)}{1 + t^{2}} dt < ∞ .

For a survey of the interplay between entire functions and indeterminate moment problems see [2].

In the following we denote by C_{0} the class of entire functions f of minimal exponential type satisfying

∫_{−∞}^{∞} \frac{log^{+} |f(t)|dt}{1 + t^{2}} < ∞ .

It is the functions in the Cartwright class which are of minimal exponential type.

In the case of an N-extremal measure μ we have thus seen that L^{2}(μ) consists of entire function of class C_{0}. The function F (f) given by (1.1) will be called the canonical extension of f.

The purpose of the present paper is to establish that also for certain determinate measures μ ∈ M^{∗} the space L^{2}(μ) consists of entire functions. A determinate measure μ with this property must necessarily be discrete, as we shall see below. It turns out that L^{2}(μ) consists of entire functions of class C_{0}, if μ is a determinate measure of finite index, meaning that there exists a polynomial p such that the measure |p|^{2}μ is indeterminate. If k is the smallest possible degree of a polynomial p such that |p|^{2}μ is indeterminate, then k − 1 is the index of μ. This concept was studied in previous papers of the authors, cf. [4], [5].

In the case of an N-extremal measure μ the canonical extension F (f) of f ∈ L^{2}(μ) has the additional property that F (p)(z) = p(z) for all z ∈ ℂ, when p is a polynomial. We shall see that this property cannot subsist in the determinate case. It will be replaced by a condition which involves discrete differential operators of the form

T = ∑_{l=1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)}, a_{l,j} ∈ ℂ  (1.2)

associated to a system (z_{i}, k_{i}), i = 1, ⋯, N of mutually different points z_{i} ∈ ℂ and multiplicities k_{i} ∈ ℕ. These operators act on entire functions F via the formula

T (F) = ∑_{l=1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}F^{(j)}(z_{l}) .

It is well-known that any T of the form (1.2) has a unique continuous extension from ℂ[t] to L^{2}(μ) if μ is N-extremal. This extension T̃ satisfies

T̃(f) = T (F (f)) , f ∈ L^{2}(μ) ,  (1.3)

where F (f) is the canonical extension of f ∈ L^{2}(μ). In fact, if (q_{n}) ∈ ℂ[t] converges in L^{2}(μ) to f ∈ L^{2}(μ) then q_{n} = F (q_{n}) converges in H(ℂ) to F (f) and hence lim_{n→∞} T (q_{n}) = T (F (f)). We notice that (T (p_{n})) ∈ ℓ^{2}, and if f ∈ L^{2}(μ) has the Fourier expansion ∑ c_{n}p_{n} then

T̃(f) = ∑_{n=0}^{∞} c_{n}T (p_{n}) .  (1.4)

If μ is determinate then T given by (1.2) has a (unique) continuous extension from ℂ[t] to L^{2}(μ) if and only if (T (p_{n})) ∈ ℓ^{2}. Although (p_{n}(z)) ∉ ℓ^{2} for z ∉ supp(μ), it is possible to characterize the differential operators T for which (T (p_{n})) ∈ ℓ^{2}. This was done in [5]. For determinate measures μ of finite index there are “many” of these operators, see below, and we shall prove the following:

Theorem 1.1. Let μ be a determinate measure of finite index. Then L^{2}(μ) consists of entire functions of class C_{0} via a continuous linear injection E : L^{2}(μ) → H(ℂ) with the additional property that

T̃(f) = T (E(f))  (1.5)

for all f ∈ L^{2}(μ) and all operators T of the form (1.2) for which (T (p_{n})) ∈ ℓ^{2}.

A realization f ↦ E(f) satisfying (1.5) is not uniquely determined. We give several different realizations, and to complete the paper, we characterize for given f ∈ L^{2}(μ) the set of entire functions F satisfying

T̃(f) = T (F)

for all operators T such that (T (p_{n})) ∈ ℓ^{2}. All these functions F turn out to be of class C_{0}.

2. Preliminary results

As claimed in the introduction it imposes severe restrictions on a determinate measure μ, if L^{2}(μ) consists of entire functions.

Proposition 2.1. Let μ ∈ M^{∗} be determinate and assume that E : L^{2}(μ) → H(ℂ) is a realization of L^{2}(μ) as entire functions. Then μ is a discrete measure, and for each z ∈ ℂ \ supp(μ) there exists p ∈ ℂ[t] such that p(z) ≠ E(p)(z).

Proof. If the support S of μ is non-discrete we can choose x_{0} ∈ S and a compact subset F ⊆ S \ {x_{0}} having accumulation points. Let f : ℝ → ℝ be a continuous function with compact support vanishing on F and such that f(x_{0}) = 1. The extension E(f) of f to an entire function must necessarily vanish identically, but this is a contradiction.

For a discrete determinate measure μ it is known that ∑ |p_{n}(z)|^{2} = ∞ for all z ∉ supp(μ). Fix z ∉ supp(μ) and let us assume that the realization E has the property E(p)(z) = p(z) for all p ∈ ℂ[t]. We define a sequence S_{n} of continuous linear functionals on ℓ^{2} by

S_{n}(c) = ∑_{k=0}^{n} c_{k}p_{k}(z), c = (c_{n}) ∈ ℓ^{2} .

For any c ∈ ℓ^{2} there exists f ∈ L^{2}(μ) such that

∑_{k=0}^{n} c_{k}p_{k} → f in L^{2}(μ) ,

and hence

S_{n}(c) = E (∑_{k=0}^{n} c_{k}p_{k}) (z) → E(f)(z) .

By the Banach-Steinhaus Theorem this implies that

sup_{n} ∥S_{n}∥ = (∑_{0}^{∞} |p_{k}(z)|^{2})^{\frac{1}{2}} < ∞ ,

which is a contradiction. □

The determinate measures of finite index are discrete, and we shall realize L^{2}(μ) as entire functions for this class of measures.

The index of determinacy of a determinate measure μ was introduced and studied by the authors in [4]. This index checks the determinacy under multiplication by even powers of |t − z| for z a complex number, and it is defined as

ind_{z}(μ) = sup{k ∈ ℕ | |t − z|^{2k}μ is determinate}.  (2.1)

Using the index of determinacy, determinate measures can be classified as follows:

If μ is constructed from an N-extremal measure by removing the mass at k + 1 points in the support, then μ is determinate with

ind_{z}(μ) = \{ \begin{matrix}k, & for z ∉ supp(μ) \\ k + 1, & for z ∈ supp(μ). \end{matrix}  (2.2)

For an arbitrary determinate measure μ the index of determinacy is either infinite for every z, or finite for every z. In the latter case the index has the form (2.2), and μ is derived from an N-extremal measure by removing the mass at k +1 points. Such an N-extremal measure is highly non-unique by a perturbation result of Berg and Christensen, cf. [3, Theorem 8].

Using that the index of determinacy is constant at complex numbers outside of the support of μ, we define the index of determinacy of μ by

ind(μ) := ind_{z}(μ), z ∉ supp(μ).  (2.3)

We stress that a measure μ of finite index is discrete and ind(μ) + 1 is the smallest degree of a polynomial p such that |p|^{2}μ is indeterminate.

To each measure μ which is either N-extremal or determinate of finite index we associate an entire function F_{μ} with simple zeros at the points of supp(μ). We recall from [4] that

F_{μ}(w) = exp (−w ∑_{n=0}^{∞} \frac{1}{x_{n}} ) ∏_{n=0}^{∞} (1 − \frac{w}{x_{n}} ) exp(\frac{w}{x_{n}}),  (2.4)

where {x_{n} : n ∈ ℕ} is the support of μ. This function F_{μ} is the uniquely determined entire function of minimal exponential type having supp(μ) as its set of zeros and satisfying F_{μ}(0) = 1. In the above formulation we tacitly assume 0 ∉ supp(μ). If however 0 ∈ supp(μ), the above expression for F_{μ} shall be multiplied with w and {x_{n} : n ∈ ℕ} = supp(μ) \ {0}.

That F_{μ} is of minimal exponential type follows by a theorem of M. Riesz [8], according to which the entire functions in the Nevanlinna matrix for an indeterminate moment problem are of minimal exponential type. The function F_{μ} is also in the Cartwright class.

Theorem 2.2. Let μ be N-extremal. For each f ∈ L^{2}(μ) we have

F (f)(z) = ∑_{x∈supp(μ)} \frac{F_{μ}(z)}{F′_{μ}(x)(z − x)}f(x) , z ∈ ℂ ,

where the series converges uniformly on compact subsets of ℂ.

Proof. Without loss of generality we may assume that 0 ∈ supp(μ), so F_{μ} is proportional to the function D from the Nevanlinna matrix, cf. [1], and it is well known that

∑_{n=0}^{∞} p_{n}(z)p_{n}(x) = \frac{B(z)D(x) − B(x)D(z)}{z − x} ,

cf. [4], [7], where

B(z) = −1 + z ∑_{n=0}^{∞} q_{n}(0)p_{n}(z) .

Here (q_{n}) denotes the sequence of polynomials of the second kind given by

q_{n}(z) = ∫ \frac{p_{n}(z) − p_{n}(x)}{z − x}dμ(x) .

Since D vanishes on supp(μ) we get

F (f)(z) = ∫ (∑_{n=0}^{∞} p_{n}(z)p_{n}(x)) f(x)dμ(x) = −D(z) ∫ \frac{B(x)f(x)}{z − x}dμ(x) ,

and

\frac{B(x)f(x)}{z − x} = −\frac{f(x)}{z − x} + \frac{xf(x)}{z − x} ∑_{n=0}^{∞} q_{n}(0)p_{n}(x)

belongs to L^{1}(μ) because ∑ q_{n}(0)p_{n}(x) ∈ L^{2}(μ).

The mass at x ∈ supp(μ) is given by ([1, p. 114])

μ(\{x\}) = \frac{−1}{B(x)D′(x)}

showing that

F (f)(z) = ∑_{x∈supp(μ)} \frac{D(z)}{D′(x)(z − x)}f(x)

and the series converges uniformly on compact subsets of ℂ. Since D and F_{μ} are proportional the result follows. □

From Theorem 2.2 it is easy to verify that the realization F (L^{2}(μ)) is a Hilbert space of entire functions in the sense of de Branges, see [6, p. 57]. For details see Corollary 3.3 below.

In [5] we obtained the following result:

Theorem 2.3. Let μ ∈ M^{∗} be determinate and let (p_{n}) be the sequence of orthonormal polynomials corresponding to μ. Let (z_{1}, k_{1}), …, (z_{N}, k_{N}) be given, where the z’s are different complex numbers and the k’s are nonnegative integers. Putting M = ∑_{l=1}^{N}(k_{l} + 1) and

T = {T = ∑_{l=1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)} | a_{l,j} ∈ ℂ}

we have:

(i) If

ind(μ) ≥ ( ∑_{l:μ({z_{l}})>0} k_{l} + ∑_{l:μ({z_{l}})=0}(k_{l} + 1)) − 1,

then the sequence (T (p_{n})) belongs to ℓ^{2} only in the trivial cases, i.e., if and only if T is a linear combination of Dirac deltas evaluated at points z_{l} which are mass points of the measure μ.

(ii) If

0 ≤ ind(μ) ≤ ( ∑_{l:μ({z_{l}})>0} k_{l} + ∑_{l:μ({z_{l}})=0}(k_{l} + 1)) − 2,

then,

dim {T ∈ T | (T (p_{n})) ∈ ℓ^{2}} = M − ind(μ) − 1 ≥ 1.

Furthermore, (T (p_{n})) ∈ ℓ^{2} if and only if T (z^{k}F_{μ}(z)) = 0 for k = 0, 1, …, ind(μ).

Corollary 2.4. Let μ ∈ M^{∗} be a determinate measure of finite index. For an operator T ∈ T we have (T (p_{n})) ∈ ℓ^{2} if and only if T (z^{k}F_{μ}(z)) = 0 for k = 0, 1, ⋯, ind(μ).

Proof. It is enough to consider the case (i), and to prove that the equations T (z^{k}F_{μ}(z)) = 0 for k ≤ ind(μ) imply that T is a linear combination of Dirac deltas at mass points of μ. To simplify the notation we assume that the system is ordered such that there exist positive integers 0 ≤ N_{1} ≤ N_{2} ≤ N for which

μ({z_{l}}) > 0 and k_{l} = 0 for l = 1, ⋯, N_{1}
μ({z_{l}}) > 0 and k_{l} > 0 for l = N_{1} + 1, ⋯, N_{2}
μ({z_{l}}) = 0 for l = N_{2} + 1, ⋯, N .

Using F_{μ}(z_{l}) = 0 for l = 1, ⋯, N_{2}, the equations T (z^{k}F_{μ}(z)) = 0 can be written

∑_{l=N_{1}+1}^{N_{2}} ∑_{j=1}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)}(z^{k}F_{μ}(z)) + ∑_{l=N_{2}+1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)}(z^{k}F_{μ}(z)) = 0 .

This system has

p := ∑_{l=N_{1}+1}^{N_{2}} k_{l} + ∑_{l=N_{2}+1}^{N}(k_{l} + 1)

variables a_{l,j} and ind(μ) + 1 equations, and p ≤ ind(μ) + 1 since we consider the case (i). We claim that the system of equations with k ≤ p − 1 (≤ ind(μ)) has a non-singular matrix, and therefore the variables involved are 0, i.e.

T = ∑_{l=1}^{N_{2}} a_{l,0}δ_{z_{l}} .

The columns of the matrix can be put together in blocks

\{δ_{z_{l}}^{(j)}(z^{k}F_{μ}(z))\}\begin{matrix}k=0,··· ,p−1 \\ j=1,··· ,k_{l} \end{matrix} , l = N_{1} + 1, ⋯, N_{2}

and

\{δ_{z_{l}}^{(j)}(z^{k}F_{μ}(z))\}\begin{matrix}k=0,··· ,p−1 \\ j=0,··· ,k_{l} \end{matrix} , l = N_{2} + 1, ⋯, N .

Since F_{μ}(z_{l}) = 0, F′_{μ}(z_{l}) ≠ 0 for l = N_{1} + 1, ⋯, N_{2} and F_{μ}(z_{l}) ≠ 0 for l = N_{2} + 1, ⋯, N, column operations show that these blocks are equivalent to the blocks

\{δ_{z_{l}}^{(j)}(z^{k})\}\begin{matrix}k=0,··· ,p−1 \\ j=0,··· ,k_{l}−1 \end{matrix} , \{δ_{z_{l}}^{(j)}(z^{k})\}\begin{matrix}k=0,··· ,p−1 \\ j=0,··· ,k_{l} \end{matrix} .

The determinant of the matrix formed by these blocks is a variant of Vandermondes determinant and is non-zero. □

3. The determinate case

For a given measure μ ∈ M^{∗} of finite index of determinacy we denote by D(μ) the set of operators of the form (1.2) for which (T (p_{n})) ∈ ℓ^{2}, allowing the system (z_{i}, k_{i}) and N to vary. It is an infinite dimensional vector space. Any T ∈ D(μ) can be extended from ℂ[t] to a continuous linear operator T̃ in the space L^{2}(μ) via Fourier expansions:

T̃(f) = ∑_{n} (∫_{ℝ} f(t)p_{n}(t)dμ(t)) T (p_{n}), for f ∈ L^{2}(μ).

We choose different real numbers x_{0}, ⋯, x_{ind(μ)} outside of the support of μ and consider the measure

σ = μ + ∑_{i=0}^{ind(μ)} δ_{x_{i}} .  (3.1)

From the above, cf. Theorem 3.9 (1) in [4], it follows that the measure σ is Nextremal.

Given a function f ∈ L^{2}(μ), we extend it to a function f̃ in the space L^{2}(σ) in the following way

f̃(t) = \{ \begin{matrix}f(t), & for t ∈ supp(μ) \\ 0, & for t = x_{i}, i = 0, ⋯, ind(μ). \end{matrix}  (3.2)

Clearly, f ↦ f̃ is a linear isometry of L^{2}(μ) into L^{2}(σ).

Since σ is N-extremal, f̃ has a canonical extension to an entire function of class C_{0} given by

F (f̃)(z) = ∑_{n} (∫_{ℝ} f̃(t)q_{n}(t)dσ(t)) q_{n}(z),  (3.3)

where (q_{n}) is the sequence of orthonormal polynomials with respect to σ. We can now formulate:

Theorem 3.1. Let μ be a determinate measure with finite index of determinacy ind(μ). The mapping E(f) := F (f̃) given by (3.3) is a realization of L^{2}(μ) as entire functions of class C_{0} such that for any operator T ∈ D(μ)

T̃(f) = T (E(f)) , f ∈ L^{2}(μ) .  (3.4)

Proof. It is clear that E(f) = F (f̃) is a realization of L^{2}(μ) as entire functions of class C_{0}.

The set of functions f ∈ L^{2}(μ) for which (3.4) holds is a closed subspace, and therefore it suffices to prove (3.4) for f = χ_{{x}}, x ∈ supp(μ), where χ_{A} denotes the indicator function of the set A. This is a consequence of the following result:

Proposition 3.2. For x ∈ supp(μ) we have

E(χ_{\{x\}})(z) = \frac{F_{μ}(z)p(z)}{F′_{μ}(x)p(x)(z − x)} , z ∈ ℂ ,

where p is the unique monic polynomial of degree ind(μ) + 1 which vanishes at x_{0}, ⋯, x_{ind(μ)}.

The function

\frac{F_{μ}(z)}{F′_{μ}(x)(z − x)}

is an entire function of class C_{0} equal to χ_{{x}} on supp(μ) and we have

T̃(χ_{\{x\}}) = T (E(χ_{\{x\}})) = T (\frac{F_{μ}(z)}{F′_{μ}(x)(z − x)}) for T ∈ D(μ) .

Proof. For f = χ_{{x}} we find

f̃(t) = \{ \begin{matrix}f(t), & if t ∈ supp(μ) \\ 0, & for t = x_{i}, i = 0, ⋯, ind(μ). \end{matrix}
= \{ \begin{matrix}1, & for t = x, \\ 0, & otherwise. \end{matrix}
= χ_{{x}}(t).

For T ∈ D(μ) we denote by T̃ and T̃_{σ} the continuous extensions of T from ℂ[t] to L^{2}(μ) and L^{2}(σ) respectively. We then have T̃(f) = T̃_{σ}(f̃) for f ∈ L^{2}(μ) because ∥f − p∥_{L^{2}(μ)} ≤ ∥f̃− p∥_{L^{2}(σ)} when p ∈ ℂ[t], and in particular T̃(χ_{{x}}) = T̃_{σ}(χ_{{x}}) when x ∈ supp(μ).

By Theorem 2.2 we have

F (f̃)(z) = \frac{F_{σ}(z)}{F′_{σ}(x)(z − x)} = \frac{F_{μ}(z)p(z)}{F′_{μ}(x)p(x)(z − x)} ,

because F_{σ}(z) = βp(z)F_{μ}(z) for a certain constant β, and hence F′_{σ}(x) = βp′(x)F_{μ}(x)+ βp(x)F′_{μ}(x) = βp(x)F′_{μ}(x). This gives by (1.3)

T̃(χ_{\{x\}}) = T (\frac{F_{μ}(z)p(z)}{F′_{μ}(x)p(x)(z − x)}) ,

but since

\frac{F_{μ}(z)p(z)}{F′_{μ}(x)p(x)(z − x)} = \frac{F_{μ}(z)}{F′_{μ}(x)(z − x)} + q(z)F_{μ}(z) ,

where

q(z) = \frac{p(z) − p(x)}{F′_{μ}(x)(z − x)p(x)}

is a polynomial of degree ind(μ), we have T (qF_{μ}) = 0 by Corollary 2.4, and the second assertion follows. □

Corollary 3.3. With the notation above we have

E(f)(z) = ∑_{x∈supp(μ)} \frac{F_{μ}(z)p(z)}{F′_{μ}(x)p(x)(z − x)}f(x) for f ∈ L^{2}(μ) ,  (3.5)

where the series converges uniformly on compact subsets of ℂ.

The realization E(L^{2}(μ)) ⊆ H(ℂ) is a Hilbert space of entire functions in the sense of de Branges.

Proof. Formula (3.5) follows immediately from Theorem 2.2 and Proposition 3.2. To see that E(L^{2}(μ)) is a Hilbert space of entire functions in the sense of de Branges we shall verify the properties (H1)–(H3) from [6, p. 57]. We shall only comment on (H1): If w ∈ ℂ \ ℝ is a zero of E(f) we have

∑_{x∈supp(μ)} \frac{f(x)}{F′_{μ}(x)p(x)(w − x)} = 0 ,

and hence for z ≠ w

E (f(x)\frac{x − \overline{w}}{x − w} ) (z) = F_{μ}(z)p(z) ∑_{x∈supp(μ)} \frac{f(x)}{F′_{μ}(x)p(x)(z − x)} (1 + \frac{w − \overline{w}}{x − w} )
= E(f)(z) + F_{μ}(z)p(z)(w − \overline{w})S(z),

where

S(z) = ∑_{x∈supp(μ)} \frac{f(x)}{F′_{μ}(x)p(x)} (\frac{1}{(z − x)(x − w)} + \frac{1}{(z − w)(w − x)}) .

Therefore we get

E (f(x)\frac{x − \overline{w}}{x − w} ) (z) = E(f)(z)\frac{z − \overline{w}}{z − w} ,

which shows (H1). □

In Theorem 3.1, to get an extension of f ∈ L^{2}(μ) to an entire function, we add mass points to the measure μ until we reach an N-extremal measure σ. We next extend f by zero to a function in L^{2}(σ), and use its canonical extension to an entire function. However, there is a different way to obtain N-extremal measures from a determinate measure μ having finite index of determinacy. We prove that this approach can also be used to find entire extensions of functions in L^{2}(μ), such that (3.4) holds.

For a determinate measure μ with finite index of determinacy ind(μ), we take a polynomial

R(t) = ∏_{l=1}^{N}(t − z_{l})^{k_{l}+1} , with ∑_{l=1}^{N}(k_{l} + 1) = ind(μ) + 1 ,

where z_{l} ∉ supp(μ), l = 1, ⋯, N.

It follows from Lemma 2.1 in [5] that σ = |R|^{2}μ is an indeterminate measure, but the measure |R(t)/(t − z_{1})|^{2} μ is determinate. According to Lemma A in Section 3 of [4], we conclude that the measure σ = |R|^{2}μ is N-extremal.

Given a function f ∈ L^{2}(μ), we define f^{♮} ∈ L^{2}(σ) by f^{♮} = f/R. Since σ is N-extremal, f^{♮} has a canonical extension F (f^{♮}) and we define

E(f)(z) := R(z)F (f^{♮})(z).  (3.6)

Theorem 3.4. Let μ be a determinate measure of finite index and let R be as above. Then L^{2}(μ) is realized as entire functions of class C_{0} via (3.6), and it has the property

T̃(f) = T (E(f)) , f ∈ L^{2}(μ)  (3.7)

for any discrete differential operator T ∈ D(μ).

Proof. The set of functions f ∈ L^{2}(μ) for which (3.7) holds is a closed subspace, and therefore it suffices to prove (3.7) for f = χ_{{x}} , x ∈ supp(μ).

In this case f^{♮}(t) = (1/R(x))χ_{{x}}(t), and since F_{μ} = F_{σ} we get

F (f^{♮})(z) = \frac{F_{μ}(z)}{R(x)F′_{μ}(x)(z − x)} ,

hence

R(z)F (f^{♮})(z) = \frac{F_{μ}(z)}{F′_{μ}(x)(z − x)} + r(z)F_{μ}(z),

where

r(z) = \frac{1}{R(x)F′_{μ}(x)} \frac{R(z) − R(x)}{z − x}

is a polynomial of degree ind(μ). Now formula (3.7) follows from Corollary 2.4 and Proposition 3.2. □

Like in Corollary 3.3 we have

E(f)(z) = ∑_{x∈supp(μ)} \frac{F_{μ}(z)R(z)}{F′_{μ}(x)R(x)(z − x)}f(x) for f ∈ L^{2}(μ) .

The realization E(L^{2}(μ)) is a Hilbert space in the sense of de Branges if R is a real polynomial.

For given f ∈ L^{2}(μ) we shall now describe the set of all entire functions F satisfying

T̃(f) = T (F) for all T ∈ D(μ) .  (3.8)

Theorem 3.5. Let μ be a determinate measure of finite index and let f ∈ L^{2}(μ).

(i) Given (z_{1}, k_{1}), ⋯, (z_{N}, k_{N}), where z_{1}, ⋯, z_{N} are different points of ℂ, k_{1}, ⋯, k_{N} ∈ ℕ, and assume that 0 ≤ N_{2} ≤ N exists such that z_{l} ∈ supp(μ) and k_{l} > 0 for l = 1, ⋯, N_{2} and z_{l} ∉ supp(μ) for l = N_{2} + 1, ⋯, N and that

∑_{l=1}^{N_{2}} k_{l} + ∑_{l=N_{2}+1}^{N}(k_{l} + 1) = ind(μ) + 1 ,  (3.9)

then there exists a unique entire function F satisfying (3.8) and the interpolation conditions

F^{(j)}(z_{l}) = α_{l,j} \{ \begin{matrix}j = 1, ⋯, k_{l} , l = 1, ⋯, N_{2} \\ j = 0, ⋯, k_{l} , l = N_{2} + 1, ⋯, N \end{matrix}  (3.10)

where α_{l,j} are arbitrarily given. This entire function F is of class C_{0}.

(ii) If F is an entire function satisfying (3.8), then F + pF_{μ}, where p is any polynomial of degree not bigger than ind(μ), are the only entire functions satisfying (3.8). All of them are of class C_{0}.

Proof. (i) We first prove the existence. Assume that F is an entire function satisfying (3.8). From the hypothesis on the z_{l}’s and since F_{μ} has simple zeros, we deduce that F′_{μ}(z_{l}) ≠ 0 for l = 1, ⋯, N_{2} and F_{μ}(z_{l}) ≠ 0 for l = N_{2} + 1, ⋯, N. Hence, if p denotes a polynomial, the equations

δ_{z_{l}}^{(j)}(p(z)F_{μ})(z)) = F^{(j)}(z_{l}) − α_{l,j} , \{ \begin{matrix}j = 1, ⋯, k_{l}, l = 1, ⋯, N_{2} \\ j = 0, ⋯, k_{l} , l = N_{2} + 1, ⋯, N \end{matrix}

determine the quantities p^{(j)}(z_{l}) uniquely for j = 0, ⋯, k_{l}−1, l = 1, ⋯, N_{2} and for j = 0, ⋯, k_{l}, l = N_{2} + 1, ⋯, N. The hypothesis (3.9) guarantees that p is uniquely determined as a polynomial of degree ≤ ind(μ). This means that F − pF_{μ} satisfies the interpolation conditions (3.10), and F − pF_{μ} still satisfies (3.8) by Corollary 2.4.

To prove uniqueness, assume that F and G are entire functions satisfying (3.8) and (3.10). We shall prove that F (x) = G(x) for all x ∈ ℂ\(supp(μ)∪{z_{N_{2}+1}, ⋯, z_{N}}). This clearly implies F ≡ G. For x as above we consider the linear system

∑_{l=1}^{N_{2}} ∑_{j=1}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)} (z^{k}F_{μ}(z)) + ∑_{l=N_{2}+1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)} (z^{k}F_{μ}(z)) = x^{k}F_{μ}(x)

where 0 ≤ k ≤ ind(μ). The system is quadratic by (3.9), and it has a unique solution (a_{l,j}), cf. the proof of Corollary 2.4. This means that the operator

T := ∑_{l=1}^{N_{2}} ∑_{j=1}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)} + ∑_{l=N_{2}+1}^{N} ∑_{j=0}^{k_{l}} a_{l,j}δ_{z_{l}}^{(j)} − δ_{x}

belongs to D(μ), so T (F) = T (G) = T̃(f) by (3.8), but since F and G both satisfy (3.10) we conclude that F (x) = G(x).

Since (3.8) has a solution F which is of class C_{0}, the solution F − pF_{μ} from the existence part is again of class C_{0}.

(ii) Let F, G be entire functions satisfying (3.8). The method in (i) shows that it is possible to find a polynomial p of degree ≤ ind(μ) such that G − pF_{μ} satisfies the interpolation conditions

δ_{z_{l}}^{(j)}(G − pF_{μ}) = F^{(j)}(z_{l})

with l, j as in (3.10). By the uniqueness assertion G − pF_{μ} = F. □

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Matematisk Institut, Københavns Universitet, Universitetsparken 5, DK-2100 København Ø, Denmark; Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160. 41080-Sevilla, Spain.