A PROBLEM ON COMPLETENESS OF EXPONENTIALS

A. POLTORATSKI

Abstract. Let μ be a finite positive measure on the real line. For a > 0 denote by E_{a} the family of exponential functions

E_{a} = {e^{ist}| s ∈ [0, a]}. The exponential type of μ is the infimum of all numbers a such that the finite linear combinations of the exponentials from E_{a} are dense in L^{2}(μ). If the set of such a is empty, the exponential type of μ is defined as infinity. The well-known type problem asks to find the exponential type of μ in terms of μ. In this note we present a solution to the type problem and discuss its relations with known results.

1. Introduction

1.1. Completeness of exponentials. Let μ be a finite positive Borel measure on ℝ. Let us consider the family E_{Λ} of exponential functions exp(iλt) on ℝ whose frequencies λ belong to a certain set Λ ⊂ ℂ:

E_{Λ} = {exp(iλt)| λ ∈ Λ}.

One of the classical problems of Harmonic analysis is to find conditions on μ and Λ that ensure completeness, i.e. density of finite linear combinations, of functions from E_{Λ} in L^{2}(μ).

Versions of this problem were considered by many prominent analysts. The case when Λ is a sequence and μ is Lebesgue measure on an interval was solved by Beurling and Malliavin in the early sixties [4, 5]. The so-called Beurling-Malliavin theory, created to treat that problem, is considered to be one of the deepest parts of the 20th century Harmonic Analysis.

Other cases of the problem and its multiple reformulations were studied by Wiener, Levinson, Kolmogorov, Krein and many others. Such an extensive interest is largely due to the fact that it is naturally related to other fields of classical analysis, such as stationary Gaussian processes and prediction theory, spectral problems for differential operators, approximation theory, signal processing, etc. Despite considerable efforts by the analytic community many important cases of the problem remain open.

The author is supported by N.S.F. Grant No. 0800300.

1.2. The type problem. Perhaps the most studied among such open cases is the so-called type problem. Consider a family E_{a} = E_{[0,a]} of exponential functions whose frequencies belong to the interval from 0 to a. If μ is a finite positive measure on ℝ we denote by G_{μ}^{2} its exponential type that is defined as

G_{μ}^{2} = inf{ a > 0 | E_{a} is complete in L^{2}(μ) }  (1.1)

if the set of such a is non-empty and as infinity otherwise. The type problem asks to calculate G_{μ}^{2} in terms of μ.

This question first appears in the work of Wiener, Kolmogorov and Krein in the context of stationary Gaussian processes (see [17, 18] or the book by Dym and McKean [11]). If μ is a spectral measure of a stationary Gaussian process, completeness of E_{a} in L^{2}(μ) is equivalent to the property that the process at any time is determined by the data for the time period from 0 to a. Hence the type of the measure is the minimal length of the period of observation necessary to predict the rest of the process. Since any even measure is a spectral measure of a stationary Gaussian process, and vice versa, this reformulation is practically equivalent.

The type problem can also be restated in terms of the Bernstein weighted approximation, see for instance the book by Koosis [15]. Important connections with spectral theory of second order differential operators were studied by Gelfand and Levitan [12] and Krein [18, 19].

Closely related to spectral problems for differential operators is Krein – de Branges’ theory of Hilbert spaces of entire functions, see [7]. One of the deep results of the theory says that for any positive finite (or more generally Poisson-finite) measure μ on ℝ there is a unique nested regular chain of de Branges’ spaces of entire functions isometrically embedded in L^{2}(μ). An important characteristic of such a chain is the supremum S_{μ} of the exponential type taken over all entire functions contained in the embedded spaces. For instance, if such a chain corresponds to a regular Schrödinger operator on an interval, i.e. if μ is the spectral measure of such an operator, then S_{μ} is equal to the length of the interval and all spaces of the chain can be parametrized by their exponential type. It is well-known, and not difficult to show, that the problem of finding the value of S_{μ} is equivalent to the type problem, i.e. S_{μ} = G_{μ}^{2}.

For more on the history and connections of the type problem see, for instance, a note by Dym [10] or a recent paper by Borichev and Sodin [6].

1.3. General case p ≠ 2. The family E_{a} is incomplete in L^{2}(μ) if and only if there exists a function f ∈ L^{2}(μ) orthogonal to all elements of E_{a}. Expanding to other 1 ⩽ p ⩽ ∞ we define

G_{μ}^{p} = sup{ a | ∃ f ∈ L^{p}(μ), ∫ f(x)e^{iλx}dμ(x) = 0, ∀ λ ∈ [0, a] }. (1.2)

We put G_{μ}^{p} = 0 if the set in (1.2) is empty. By duality, for 1 < p ⩽ ∞, G_{μ}^{p} can still be defined as the infimum of a such that E_{a} is complete in L^{q}(μ), \frac{1}{p} + \frac{1}{q} = 1. Cases p ≠ 2 were considered in several papers, see for instance articles by Koosis [16] or Levin [21] for the case p = ∞ or [29] for p = 1.

Since μ is a finite measure we have

G_{μ}^{p} ⩽ G_{μ}^{q} for p ⩾ q.  (1.3)

Apart from this obvious observation, the problems of finding G_{μ}^{p} for different p were generally considered non-equivalent. One of the consequences of theorem 2, section 3.1, is that, in some sense, there are only two significantly different cases, p = 1 (the gap problem) and 1 < p ⩽ ∞ (the general type problem).

In this paper we restrict our attention to the class of finite measures. The formal reason for that is the fact that μ has to be finite for exponentials to belong to L^{2}(μ). This obstacle can be easily overcome if instead of E_{a} one considers E_{a}, the set of Fourier transforms of smooth functions supported on [0, a]. All elements of E_{a} decay fast at infinity and one one can ask about the density of E_{a} in L^{p}(μ) for wider sets of μ, see for instance [6]. One of such traditional sets is the class of Poisson-finite measures satisfying

∫ \frac{d|μ|(x)}{1 + x^{2}} < ∞.

However, due to the reasons similar to lemma 1 below (note that if μ is Poisson-finite then μ/(1 + x^{2}) is finite and vice versa), considering such a wider set of measures will not change the problem and all of the statements will remain the same or analogous.

1.4. The gap problem. One of the important cases is the so-called gap problem, p = 1. Here one can reformulate the question as follows.

Let X be a closed subset of the real line. Denote

G_{X} = sup{ a | ∃ μ ≠ 0, supp μ ⊂ X, μ̂ = 0 on [0, a] }.

Here and in the rest of the paper μ̂ denotes the (inverse) Fourier transform of a finite measure μ on ℝ:

μ̂(z) = ∫_{ℝ} e^{izt}dμ(t).

As was shown in [29], for any finite measure μ on ℝ, G_{μ}^{1}, as defined in the previous section, depends only on its support:

G_{μ}^{1} = G_{X}, X = supp μ.

This property separates the gap problem from all the cases p > 1.

For a long time both the gap problem and the type problem were considered by experts to be ”transcendental,” i.e. not having a closed form solution.

Following an approach developed in [23] and [24], a solution to the gap problem was recently suggested in [29], see section 2.3. Some of definitions and results from [29] are used in the present paper.

1.5. Known examples. We say that a function f on ℝ is Poisson-summable if it is summable with respect to the Poisson measure Π,

dΠ = dx/(1 + x^{2}).

We say that a sequence of real numbers A = {a_{n}} is discrete if it does not have finite accumulation points. We always assume that a discrete sequence is enumerated in the natural increasing order: a_{n} ⩾ a_{n−1}. Since the sequences considered here have ±∞ as their density points, the indices run over ℤ. In most of our statements and definitions, the sequences do not have multiple points. We call a discrete sequence {a_{n}} ⊂ ℝ separated if |a_{n} − a_{k}| > c for some c > 0 and any n ≠ k.

A classical result by Krein [17] says that if dμ = w(x)dx and log w is Poissonsummable then G_{μ}^{p} = ∞ for all p, 1 ⩽ p ⩽ ∞. A partial inverse, proved by Levinson and McKean, holds for even monotone w, see section 4.5.

A theorem by Duffin and Schaeffer [9] implies that if μ is a measure such that for any x ∈ ℝ

μ([x − L, x + L]) > d

for some L, d > 0 then G_{μ}^{2} ⩾ 2π/L, see section 4.7.

For discrete measures, in the case supp μ = ℤ, a deep result by Koosis shows an analogue of Krein’s result: if μ = ∑ w(n)δ_{n}, where

∑ \frac{log w(n)}{1 + n^{2}} > −∞,

then G_{μ}^{p} = 2π for all p, 1 ⩽ p ⩽ ∞ [16]. Not much was known about supports other than ℤ besides a recent result from [25], which implies that if

μ = ∑ \frac{δ_{a_{n}}}{1 + a_{n}^{2}}

for a separated sequence A = {a_{n}} ⊂ ℝ then G_{μ}^{p} = 2πD_{∗}(A), where D_{∗} is the interior Beurling-Malliavin density of A, see section 2.3 for the definition. We generalize these results in section 3.2.

In addition to these few examples, classical theorems by Levinson-McKean, Beurling and de Branges show that if a measure has long gaps in its support or decays too fast, then G_{μ}^{p} = 0, see section 4. Examples of measures of positive type can be constructed using the results by Benedicks [2], see section 4.8. The most significant recent development, that allows one to modify existing examples, is the result by Borichev and Sodin [6], which says that ”exponentially small” changes in weight or support do not change the type of a measure, see section 4.6.

1.6. Approach and goals of the paper. The problems discussed above belong to the area often called the Uncertainty Principle in Harmonic Analysis [13]. A new approach developed by N. Makarov and the author in [23, 24] allows one to study this area with modern tools of analytic function theory and singular integrals. Together with traditional methods, such as de Branges’ theory of Hilbert spaces of entire functions or the Beurling-Malliavin theorems, these techniques have produced some new ideas and developments. Among them is an extension of the Beurling-Malliavin theory [24], a solution to the Pólya-Levinson problem on sampling sets for entire functions of zero type [25] and a solution to the gap problem [29]. In the present paper we continue to apply the same approach.

We focus on the type problem, the problem of finding G_{μ}^{2} in terms of μ. Our main results are theorem 2 and its corollaries contained in section 3. In most of our statements, treating p > 1, p ≠ 2 did not require any additional efforts, and hence they were formulated for general p > 1. The case p = 1, studied in [29], provided us with some useful definitions and statements, see section 2.3.

Acknowledgements. I am grateful to Nikolai Makarov whose deep mathematical insight and intuition led to the development of the methods used in this paper. I would also like to thank Misha Sodin for getting me interested in the gap and type problems and for numerous invaluable discussions.

1.7. Contents. The paper is organized as follows:

• Section 2 contains preliminary material, including the basics of the so-called Clark theory, definitions of Beurling-Malliavin densities and a short discussion of the gap problem.

• In section 3 we state the main results of the paper.

• Section 4 discusses connections of our results with classical theorems by Beurling, de Branges, Duffin and Schaeffer, Krein, Levinson and McKean as well as more recent results by Benedicks, Borichev and Sodin.

• Section 5 contains several lemmas needed for the main proofs.

• In section 6 we give the proofs of the main results.

2. Preliminaries

2.1. Clark theory. By H^{2} we denote the Hardy space in the upper halfplane ℂ_{+}. We say that an inner function θ(z) in ℂ_{+} is meromorphic if it allows a meromorphic extension to the whole complex plane. The meromorphic extension to the lower half-plane ℂ_{−} is given by

θ(z) = \frac{1}{θ^{#}(z)}

where θ^{#}(z) = θ̄(z̄).

Each inner function θ(z) determines a model subspace

K_{θ} = H^{2} ⊖ θH^{2}

of the Hardy space H^{2}(ℂ_{+}). These subspaces play an important role in complex and harmonic analysis, as well as in operator theory, see [27].

For each inner function θ(z) one can consider a positive harmonic function

ℜ\frac{1 + θ(z)}{1 − θ(z)}

and, by the Herglotz representation, a positive measure μ such that

ℜ\frac{1 + θ(z)}{1 − θ(z)} = py + \frac{1}{π} ∫ \frac{ydμ(t)}{(x − t)^{2} + y^{2}} ,  z = x + iy,  (2.1)

for some p ⩾ 0. The number p can be viewed as a point mass at infinity. The measure μ is Poisson-finite, singular and supported on the set where non-tangential limits of θ are equal to 1. The measure μ+pδ_{∞} on ℝ̂ is called the Clark measure for θ(z).

Following standard notations, we will sometimes denote the Clark measure defined in (2.1) by μ_{1}. More generally, if α ∈ ℂ, |α| = 1 then μ_{α} is the measure defined by (2.1) with θ replaced by ᾱθ.

Conversely, for every positive singular Poisson-finite measure μ and a number p ⩾ 0, there exists an inner function θ(z) satisfying (2.1).

Every function f ∈ K_{θ} can be represented by the formula

f(z) = \frac{p}{2πi}(1 − θ(z)) ∫ f(t)\overline{(1 − θ(t))}dt + \frac{1 − θ(z)}{2πi} ∫ \frac{f(t)}{t − z} dμ(t). (2.2)

If the Clark measure does not have a point mass at infinity, the formula is simplified to

f(z) = \frac{1}{2πi}(1 − θ(z))Kfμ

where Kfμ stands for the Cauchy integral

Kfμ(z) = ∫ \frac{f(t)}{t − z} dμ(t).

This gives an isometry of L^{2}(μ) onto K_{θ}. Similar formulas can be written for any μ_{α} corresponding to θ.

In the case of meromorphic θ(z), every function f ∈ K_{θ} also has a meromorphic extension in ℂ, and it is given by the formula (2.2). The corresponding Clark measure is discrete with atoms at the points of {θ = 1} given by

μ(\{x\}) = \frac{2π}{|θ′(x)|}.

If Λ ⊂ ℝ is a given discrete sequence, one can easily construct a meromorphic inner function θ satisfying {θ = 1} = Λ by considering a positive Poissonfinite measure concentrated on Λ and then choosing θ to satisfy (2.1). One can prescribe the derivatives of θ at Λ with a proper choice of pointmasses.

For more details on Clark measures and further references the reader may consult [30].

2.2. Interior and exterior densities. A sequence of disjoint intervals {I_{n}} on the real line is called long (in the sense of Beurling and Malliavin) if

∑_{n} \frac{|I_{n}|^{2}}{1 + dist^{2}(0, I_{n})} = ∞,  (2.3)

where |I_{n}| stands for the length of I_{n}. If the sum is finite, we call {I_{n}} short.

One of the obvious properties of short sequences is that |I_{n}| = o(dist(0, I_{n})) as n → ∞. In particular, dist(0, I_{n}) can be replaced with any x_{n} ∈ I_{n} in (2.3).

Following [5] we say that a discrete sequence Λ ⊂ ℝ is a-regular if for every ϵ > 0 any sequence of disjoint intervals {I_{n}} that satisfies

|\frac{#(Λ ∩ I_{n})}{|I_{n}|} − a| ⩾ ϵ

for all n, is short.

A slightly different a-regularity can be defined in the following way, that is more convenient in some settings. For a discrete sequence Λ ⊂ ℝ we denote by n_{Λ}(x) its counting function, i.e. the step function on ℝ, that is constant between any two points of Λ, jumps up by 1 at each point of Λ and is equal to 0 at 0. We say that Λ is strongly a-regular if

∫ \frac{|n_{Λ}(x) − ax|}{1 + x^{2}} < ∞.

Conditions like this can be found in many related results, see for instance [7] or [15]. Even though a-regularity is not equivalent to strong a-regularity, in the following definitions of densities changing ”a-regular” to ”strongly a-regular” will lead to equivalent definitions.

The interior BM (Beurling-Malliavin) density of a sequence Λ is defined as

D_{∗}(Λ) := sup{ a | ∃ a-regular subsequence Λ′ ⊂ Λ }.  (2.4)

If the set is empty we put D_{∗}(Λ) = 0. Similarly, the exterior BM density is defined as

D^{∗}(Λ) := inf{ a | ∃ a-regular supsequence Λ′ ⊃ Λ }.  (2.5)

If no such sequence exists, D^{∗}(Λ) = ∞.

It is interesting to observe that after the two densities were simultaneously introduced over fifty years ago, the exterior density immediately became one of the staples of harmonic analysis and spectral theory, mostly due to its appearance in the celebrated Beurling-Malliavin theorem, see [5], [13] or [15]. Meanwhile, the interior density remained largely forgotten until its recent comeback in [25] and [29]. It will continue to play an important role in our discussions below.

2.3. The gap problem and d-uniform sequences. Let Λ = {λ_{1}, ..., λ_{n}} be a finite set of points on ℝ. Define

E(Λ) = ∑_{λ_{k},λ_{j}∈Λ} log |λ_{k} − λ_{l}|.  (2.6)

According to the 2D Coulomb law, the quantity E(Λ) can be interpreted as potential energy of the system of ”flat electrons” placed at Λ, see [29]. That observation motivates the term we use for the condition (2.9) below.

The following example is included to illustrate our next definition.

Key example:

Let I ⊂ ℝ be an interval and let Λ = d^{−1}ℤ ∩ I for some d > 0. Then

Δ = #Λ = d|I| + O(1)

and

E = E(Λ) = ∑_{1⩽m⩽Δ} log [d^{−Δ+1}(m − 1)!(Δ − m)!] = Δ^{2} log |I| + O(|I|^{2})

(2.7)

as follows from Stirling’s formula. Here the notation O(·) corresponds to the direction |I| → ∞.

Remark 1. The uniform distribution of points on the interval does not maximize the energy E(Λ) but comes within O(|I|^{2}) from the maximum, which is negligible for our purposes, see the main definition and its discussion below. It is interesting to observe that the maximal energy for k points is achieved when the points are placed at the endpoints of I and the zeros of the Jacobi (1, 1)-polynomial of degree k − 2, see for example [14].

Let

... < a_{−2} < a_{−1} < a_{0} = 0 < a_{1} < a_{2} < ...

be a discrete sequence of real points. We say that the intervals I_{n} = (a_{n}, a_{n+1}] form a short partition of ℝ if |I_{n}| → ∞ as n → ±∞ and the sequence {I_{n}} is short.

Main Definition:

Let Λ = {λ_{n}} be a discrete sequence of real points. We say that Λ is d-uniform if there exists a short partition {I_{n}} such that

Δ_{n} = d|I_{n}| + o(|I_{n}|) for all n (density condition)  (2.8)

as n → ±∞ and

∑_{n} \frac{Δ_{n}^{2} log |I_{n}| − E_{n}}{1 + dist^{2}(0, I_{n})} < ∞ (energy condition)  (2.9)

where Δ_{n} and E_{n} are defined as

Δ_{n} = #(Λ ∩ I_{n}) and E_{n} = E(Λ ∩ I_{n}) = ∑_{λ_{k},λ_{l}∈I_{n}, λ_{k}≠λ_{l}} log |λ_{k} − λ_{l}|.

Remark 2. Note that the series in the energy condition is positive: every term in the sum defining E_{n} is at most log |I_{n}| and there are no more than Δ_{n}^{2} terms.

As follows from the example above, the first term in the numerator of (2.9) is approximately equal to the energy of Δ_{n} electrons spread uniformly over I_{n}. The second term is the energy of electrons placed at Λ ∩ I_{n}. Thus the energy condition is a requirement that the placement of the points of Λ is close to uniform, in the sense that the work needed to spread the points of Λ uniformly on each interval is summable with respect to the Poisson weight. For a more detailed discussion of this definition see [29]

In [29], d-uniform sequences were used to solve the gap problem mentioned in the introduction. Recall that with any closed X ⊂ ℝ one can associate its (spectral) gap characteristic G_{X} defined as in section 1.4. The main result of [29] is the following statement:

Theorem 1. [29] Let X be a closed set on ℝ. Then

G_{X} = sup{ d | X contains a d − uniform sequence }.

Recall that, as was proved in [29], G_{X} = G_{μ}^{1} for any μ such that supp μ = X. The following simple observations will also be useful to us in the future:

Remark 3.

• If Λ is a d-uniform sequence then D_{∗}(Λ) = d, as follows easily from the density condition (2.8).

• Among other things, the energy condition ensures that the points of Λ are not too close to each other. In particular, if Λ is d-uniform for some d > 0 and Λ′ = {λ_{n_{k}}} is a subsequence such that for all k,

λ_{n_{k}+1} − λ_{n_{k}} ⩽ e^{−c|λ_{n_{k}} |}

for some c > 0, then D_{∗}(Λ′) = 0.

• An exponentially small perturbation of a d-uniform sequence contains a d-uniform subsequence. More precisely, if c > 0 and Λ is a duniform sequence then any sequence A = {α_{n}} such that |λ_{n} −α_{n}| ⩽ e^{−c|λ_{n}|} contains a d-uniform subsequence A′ consisting of all α_{n_{k}} such that

λ_{n_{k}+1} − λ_{n_{k}} ⩾ e^{−(c−ε)|λ_{n_{k}} |}.

• As discussed in [29], the energy condition always holds for separated sequences. If Λ is separated then it is d-uniform if and only if D_{∗}(Λ) = d.

2.4. Polynomial decay. In this section we prove a version of the wellknown property that adding or removing polynomial decay cannot change the type of a measure.

Lemma 1. Let μ be a finite positive measure on ℝ and let α > 0. Consider the measure ν satisfying

dν(x) = \frac{dμ(x)}{1 + |x|^{α}} .

Then for any 1 ⩽ p ⩽ ∞

G_{μ}^{p} = G_{ν}^{p}.

Proof. Since dν/dμ ⩽ 1, one only needs to show that G_{μ}^{p} ⩽ G_{ν}^{p}. Suppose that f ∈ L^{p}(μ) is such that f̄μ annihilates all e^{iaz}, a ∈ (0, d). This is equivalent to the property that the Cauchy integral Kfμ is divisible by e^{idz} in ℂ_{+}, i.e. it decays like e^{idz} along the positive imaginary axis iℝ_{+}, see for instance lemma 2 in [25].

Let N ⩾ α be an integer. It is enough to prove the statement for N = 1: the general case will follow by induction.

First let us assume that Kfμ has at least one zero a in ℂ \ ℝ. It is wellknown, and not difficult to verify, that then the measure \frac{f}{x−a}μ satisfies

K (\frac{f}{x − a}μ) = \frac{Kfμ}{z − a}.

Hence the Cauchy integral in the left-hand side still decays like e^{idz} along iℝ_{+} and therefore the measure still annihilates e^{iaz}, a ∈ (0, d). It is left to notice that

f(x)\frac{1 + |x|^{α}}{x − a} ∈ L^{p}(ν).

If Kfμ does not have any zeros outside of ℝ, note that the Cauchy integral of the measure η = e^{−iεx}fμ satisfies

Kη = e^{−iεz}Kfμ

(see for instance theorems 3.3 and 3.4 in [28]) and therefore

K(fμ − cη) = K(1 − ce^{−iεx})fμ

has infinitely many zeros in ℂ \ ℝ for any c, |c| ≠ 1, while still decaying like e^{i(d−ε)z} along iℝ_{+}. □

Let μ be a finite measure on a separated sequence X, with point masses decaying polynomially. Lemma 1 together with elementary estimates imply that then G_{μ}^{2} = G_{μ}^{1}. Hence in this case theorem 2 below becomes the statement from [25] mentioned above:

G_{μ}^{2} = 2πD_{∗}(X).

3. Main Results

3.1. Main Theorem. Let τ be a finite positive measure on the real line. We say that a function W ⩾ 1 on ℝ is a τ-weight if W is lower semi-continuous, tends to ∞ at ±∞ and W ∈ L^{1}(τ).

Theorem 2. Let μ be a finite positive measure on the line. Let 1 < p ⩽ ∞ and a > 0 be constants.

Then G_{μ}^{p} ⩾ a if and only if for any μ-weight W and any 0 < d < a there exists a d-uniform sequence Λ = {λ_{n}} ⊂ supp μ such that

∑ \frac{log W (λ_{n})}{1 + λ_{n}^{2}} < ∞.  (3.1)

We postpone the proof until section 6.

One of the immediate corollaries of the above statement is that the p-type of a measure, G_{μ}^{p} for 1 < p ⩽ ∞, does not depend on p, which may come as a surprise to some of the experts. Further corollaries of theorem 2 and its connections with classical results are discussed in the following sections.

3.2. Discrete case. The conditions of theorem 2 are simplified for many specific classes of measures. In particular, if the measure is discrete, or absolutely continuous with regular enough density, the weight W may be eliminated from the statement. Here we treat the discrete case that is important in spectral theory of differential operators and other adjacent areas. Our results in this section may be viewed as extensions of the result by Koosis mentioned in the introduction.

The following statement gives a simplified formula for the type of a measure supported on a discrete sequence, excluding pathological cases when the counting function of the sequence grows exponentially.

Theorem 3. Let B = {b_{n}} be a discrete sequence of real points. Let

μ = ∑ w(n)δ_{b_{n}}

be a finite positive measure supported on B. Define

D = sup\{ d | ∃ d-uniform B′ ⊂ B, ∑_{λ_{n}∈B′} \frac{log w(n)}{1 + n^{2}} > −∞ \}.

Then for any 1 < p ⩽ ∞,

G_{μ}^{p} ⩾ 2πD.

If the counting function of B satisfies log(|n_{B}| + 1) ∈ L_{Π}^{1} then

G_{μ}^{p} = 2πD.

The proof is given in section 6. The condition log(|n_{B}| + 1) ∈ L_{Π}^{1} in the second part of the statement is sharp. The corresponding examples can be easily constructed using theorem 2 or the result by Borichev and Sodin [6], see theorem 11 below.

In the case when the sequence is separated, the condition can be simplified even further. Note that for p = 1, G_{μ}^{1} = D_{∗}(Λ) for any separated sequence Λ and any measure μ, supp μ = Λ, by theorem 1. For p > 1 we have

Theorem 4. Let Λ = {λ_{n}} be a separated sequence and let

μ = ∑ w(n)δ_{λ_{n}}

be a finite positive measure supported on Λ. Define

D = sup D_{∗}(Λ′),

where the supremum is taken over all subsequences Λ′ ⊂ Λ satisfying

∑_{λ_{n}∈Λ′} \frac{log w(n)}{1 + n^{2}} > −∞.  (3.2)

Then

G_{μ}^{p} = 2πD

for all 1 < p ⩽ ∞.

Proof. Suppose that G_{μ}^{p} > 2πD for some D > 0, p > 1. Define the μ-weight W as W (λ_{n}) = (μ({λ_{n}})(1 + λ_{n}^{2}))^{−1}. Then by theorem 3 there exists a subsequence Λ′ ⊂ Λ such that D_{∗}(B′) > D and (3.2) is satisfied.

In the opposite direction the statement follows directly from theorem 3 and remark 3. □

3.3. A general sufficient condition. As a corollary of theorem 3 we obtain the following sufficient condition for general measures. The condition seems to be reasonably sharp, as it is satisfied by all examples of measures with positive type existing in the literature.

Theorem 5. Let μ be a finite positive measure on ℝ. Let A = {a_{n}} be a d-uniform sequence of real numbers such that

∑ \frac{log μ((a_{n} − ε_{n}, a_{n} + ε_{n}))}{1 + n^{2}} > −∞,  (3.3)

where

ε_{n} = \frac{1}{3} min ((a_{n+1} − a_{n}), (a_{n} − a_{n−1})) .

Then G_{μ}^{∞} ⩾ 2πd.

Proof. For each τ ∈ [0, 1] let us define a discrete measure ν_{τ} as follows. The measure ν_{τ} has exactly one pointmass of the size

μ((a_{n} − ε_{n}, a_{n} + ε_{n}))

in each interval

(a_{n} − ε_{n}, a_{n} + ε_{n})

at the point x_{n}^{τ} chosen as

x_{n}^{τ} = inf{ a | μ((a_{n} − ε_{n}, a)) ⩾ τμ((a_{n} − ε_{n}, a_{n} + ε_{n})) }.

Notice that {x_{n}^{τ}} is a d-uniform sequence. In view of (3.3) and theorem 3, ν_{τ} satisfies

G_{ν_{τ}}^{∞} ⩾ 2πd.

Then

ν = ∫_{0}^{1} ν_{τ} dτ

satisfies dν/dμ ⩽ 1 and therefore

G_{μ}^{∞} ⩾ G_{ν}^{∞} ⩾ 2πd.
□

4. Classical results and further corollaries

The goal of this section is to give examples of applications of theorem 2 and discuss its connections with classical results on the type problem. Due to this reason, we prefer to deduce each statement directly from the results of the last section, rather than obtaining them from each other, even when the latter approach may slightly shorten the proof.

In our estimates we write a(n) ≲ b(n) if a(n) < Cb(n) for some positive constant C, not depending on n, and large enough |n|. Similarly, we write a(n) ≍ b(n) if ca(n) < b(n) < Ca(n) for some C ⩾ c > 0. Some formulas will have other parameters in place of n or no parameters at all.

4.1. Beurling’s Gap Theorem.

Theorem 6 (Beurling [3]). If μ is a finite measure supported on a set with long gaps and the Fourier transform of μ vanishes on an interval, then μ ≡ 0.

Proof. If supp μ has long gaps than for every short partition of ℝ infinitely many intervals of the partition must be contained in the gaps of supp μ. Therefore supp μ does not contain a sequence satisfying the density condition (2.8), i.e. it does not contain a d-uniform sequence for any d > 0. □

4.2. Levinson’s Gap Theorem.

Theorem 7 (Levinson [22]). Let μ be a finite measure on ℝ whose Fourier transform vanishes on an interval. Denote

M(x) = |μ|((x, ∞)).

If log M is not Poisson-summable on ℝ_{+} then μ ≡ 0.

Proof. Suppose that log M is not Poisson-summable on ℝ_{+}. Without loss of generality, M(0) = 1. Let 0 = a_{0} ⩽ a_{1} ⩽ a_{2} ⩽ ... be the sequence of points such that

a_{n} = inf{ a | M(a) ⩽ 3^{−n} }.

Define a |μ|-weight W as 2^{n} on each (a_{n−1}, a_{n}], a_{n−1} < a_{n}.

Since μ̂ vanishes on an interval, by theorem 2 there exists a sequence Λ ⊂ supp μ satisfying the density condition (2.8) with some a > 0 on a short partition I_{n} = (b_{n}, b_{n+1}], such that (3.1) holds. WLOG b_{0} = 0. Notice that log W is an increasing step function on ℝ_{+} satisfying log W ≳ − log M. Also, since {I_{n}} is short, cb_{n+1} ⩽ b_{n} for some 0 < c < 1 and all n > 0. Hence,

∑_{n} \frac{log W (λ_{n})}{1 + λ_{n}^{2}} ≳ ∑_{n=1}^{∞} \frac{log W (b_{n})|I_{n}|}{1 + b_{n}^{2}}

≳ ∑_{n=1}^{∞} \frac{log W (cb_{n+1})|I_{n}|}{1 + b_{n}^{2}} ≳ ∫_{0}^{∞} \frac{− log M(cx)dx}{1 + x^{2}} = ∞.

□

Levinson’s result above was later improved by Beurling [3] who showed that instead of vanishing on an interval μ̂ may vanish on a set of positive Lebesgue measure with the same conclusion. Note that an analogous improvement cannot be made in Beurling’s own gap theorem above, as illustrated by Kargaev’s counterexample, see [15, vol. 1, p. 305].

4.3. A hybrid theorem. Beurling’s and Levinson’s Gap Theorems compliment each other by treating measures with sparse supports and fast decay correspondingly. In this section we suggest a hybrid theorem that combines the features of both statements. In comparison with Beurling’s result it shows that the measure does not have to be zero on a long sequence of intervals, it just has to be small on it. In regard to Levinson’s theorem, our statement says that the measure does not have to decay fast along the whole axis, just along a large enough set. One can show that the statement is sharp in both scales.

Theorem 8. Let μ be a finite measure on ℝ whose Fourier transform vanishes on an interval. Suppose that there exists a sequence of disjoint intervals {I_{n}} such that

∑ \frac{|I_{n}| min (|I_{n}|, log \frac{1}{|μ|(I_{n})})}{1 + dist^{2}(I_{n}, 0)} = ∞.  (4.1)

Then μ ≡ 0.

Proof. We can assume that |I_{n}| → ∞ because any subsequence of intervals with uniformly bounded lengths can be deleted from {I_{n}} without affecting (4.1). Define the |μ|-weight W as

W = [|μ|(I_{n})(1 + dist^{2}(I_{n}, 0))]^{−1}

on each I_{n}. If μ̂ vanishes on an interval then for some d > 0 there exists a d-uniform sequence Λ ⊂ supp μ satisfying (3.1). Let

N = \{ n | #(Λ ∩ I_{n}) > \frac{d}{2}|I_{n}|\}.

Note that the sequence {I_{n}}_{n∉N} cannot be long because otherwise Λ will not satisfy the density condition (2.8) on any short partition. Therefore the part of the sum in (4.1) corresponding to n ∉ N is finite and

∑_{n∈ℤ} \frac{log W (λ_{n})}{1 + λ_{n}^{2}} ⩾ ∑_{λ_{n}∈∪_{k∈N} I_{k}} \frac{log W (λ_{n})}{1 + λ_{n}^{2}} ≳ ∑_{n∈N} \frac{|I_{n}| log \frac{1}{|μ|(I_{n})}}{1 + λ_{n}^{2}} = ∞.

□

4.4. De Branges’ Gap Theorem.

Theorem 9 (de Branges, theorem 63 [7]). Let K(x) be a continuous function on ℝ such that K(x) ⩾ 1, log K is uniformly continuous and Poissonunsummable. Then there is no nonzero finite measure μ on ℝ such that

∫_{−∞}^{∞} Kd|μ| < ∞  (4.2)

and μ̂ vanishes on an interval.

Proof. Suppose that μ satisfies (4.2) and its Fourier spectrum has a gap. Since K is a μ-weight there must exist a d > 0 and a d-uniform sequence Λ ⊂ supp μ satisfying (3.1) with K in place of W. Since Λ has positive interior density and log K is uniformly continuous, (3.1) implies that log K is Poisson-summable. □

4.5. A theorem by Krein, Levinson and McKean. Our next statement combines results by Krein (part I in the statement below, case p = 2) and by Levinson and McKean (part II, p = 2).

Theorem 10 (Krein [17], Levinson-McKean [11]). Let μ be a finite measure on ℝ, μ = w(x)dx, where w(x) ⩾ 0. Then

I) If log w is Poisson-summable then for any 1 ⩽ p ⩽ ∞, G_{μ}^{p} = ∞.

II) If log w is monotone and Poisson-unsummable on a half-axis (−∞, x) or (x, ∞) for some x ∈ ℝ then for any 1 < p ⩽ ∞, G_{μ}^{p} = 0.

Proof. If log w is Poisson-summable, denote by H(z) the outer function in ℂ_{+} satisfying |H| = w on ℝ. Then for any a > 0 the measure η = e^{−iax}H̄(x)dx annihilates all exponentials with frequencies from [0, a). (Here we use the fact that the integral over ℝ for any function from H^{1}(ℂ_{+}) is 0.) Since |η| = μ, it follows that G_{μ}^{p} = ∞ for any 1 ⩽ p ⩽ ∞.

In the opposite direction, suppose that log w is Poisson-unsummable and monotone on ℝ_{+}. Consider a μ-weight W (x) = (w(x)(1 + x^{2}))^{−1}. If G_{μ}^{p} > 2πd > 0, there exists a d-uniform sequence Λ satisfying (3.1). Suppose that Λ satisfies (2.8) on a short partition I_{n} = (b_{n}, b_{n+1}], b_{0} = 0. Then, similarly to the proof of theorem 7, for some 0 < c < 1, cb_{n+1} < b_{n}. Together with monotonicity of log w = − log W − log(1 + x^{2}), we obtain

∑ \frac{log W (λ_{n})}{1 + λ_{n}^{2}} ≳ ∑_{n=0}^{∞} \frac{log W (b_{n})|I_{n}|}{1 + b_{n}^{2}} + const

≳ ∑_{n=0}^{∞} \frac{log W (cb_{n+1})|I_{n}|}{1 + b_{n}^{2}} + const ≳ ∫_{0}^{∞} \frac{− log w(cx)dx}{1 + x^{2}} + const = ∞.

□

4.6. A result by Borichev and Sodin on stability of type. If I ⊂ ℝ is an interval and D > 0 is a constant we denote by DI the interval concentric with I of length D|I|. Following [6], for δ > 0 and x ∈ ℝ, we denote

I_{x,δ} = [x − e^{−δ|x|}, x + e^{−δ|x|}].

If μ and ν are two finite positive measures on ℝ we write μ ≼ ν if there exist constants δ > 0, C > 0, and l > 0, such that, for all x ∈ R,

μ(I_{x,δ}) ⩽ C(1 + |x|)^{l} (ν(2I_{x,δ}) + e^{−2δ|x|}) .

Instead of finite measures [6] deals with a wider class of polynomially growing measures and uses the corresponding definition of type. As was mentioned in the introduction, in view of the statements like lemma 1 above, such differences are not essential for the type problem and the corresponding results are equivalent.

Theorem 11. [6] If μ ≼ ν then G_{μ}^{2} ⩽ G_{ν}^{2}.

Proof. Let {a_{n}}_{n∈ℤ} be a strictly increasing discrete sequence of real points satisfying a_{−n} = −a_{n} and

a_{n+1} − a_{n} = 2e^{−δb_{n}}, b_{n} = \frac{a_{n+1} + a_{n}}{2} for all n ∈ ℕ,

where δ > 0 is the constant from the definition of the relation μ ≼ ν. Denote I_{n} = (a_{n}, a_{n+1}]. Let W be a ν-weight. Then the step-function W^{∗} defined as

W^{∗}(x) = 1 + (1 + |b_{n}|)^{−l} [\frac{1}{ν(2I_{n}) + e^{−2δb_{n}}} ∫_{2I_{n}} W dν] on each I_{n}

is a μ-weight, as follows from the condition μ ≼ ν. Assume that G_{μ}^{2} = 2πd > 0. Then there exists an (d − ε)-uniform sequence Λ ∈ supp μ, that satisfies (3.1) with W^{∗}. Our goal is to modify Λ into an (d − ε)-uniform sequence in supp ν satisfying (3.1) with W.

Notice that WLOG we can assume that each interval 2I_{n} contains at most one point of Λ, see remark 3. Choose k_{n} so that λ_{n} ∈ 2I_{k_{n}}.

Now for each λ_{n} ∈ 2I_{k_{n}} choose a point α_{n} ∈ 2I_{k_{n}} ∩ supp ν such that

W (α_{n}) ⩽ \frac{1}{ν(2I_{n})} ∫_{2I_{n}} W dν.

WLOG

∫_{2I_{n}} W dν ⩾ e^{− \frac{3}{2} δb_{n}}

for all n: otherwise we can increase the weight W to satisfy this condition and it will still remain ν-summable. If W is such a weight, then the interior density of the subsequence of Λ that falls in the intervals I_{n} satisfying ν(2I_{n}) ⩽ e^{−2δb_{n}} must be zero: otherwise the sum (3.1) for Λ and W^{∗} would diverge. We can assume that Λ does not have such points. Then

log W (α_{n}) ⩽ log W^{∗}(λ_{n}) + 2l log(1 + |λ_{n}|)

and therefore A = {α_{n}} satisfies (3.1) with W. By remark 3, A has a (d − ε)-uniform subsequence. Hence G_{ν}^{2} ⩾ G_{μ}^{2} − 2πε. □

Notice that our proof is p-independent, i.e. G^{2} can be replaced with G^{p} for any 1 < p ⩽ ∞ in the Borichev-Sodin result.

4.7. A sufficient condition by Duffin and Schaeffer. Our next statement is formulated in [9, 6] for Poisson-finite measures. Here we present an equivalent finite version.

Theorem 12. Let μ be a finite positive measure on ℝ such that for any x ∈ ℝ

μ([x − L, x + L]) > c(1 + x^{2})^{−1}

for some L, c > 0. Then G_{μ}^{2} ⩾ π/L.

Proof. If ε > 0 consider a_{n} = n(2L+ε). Then in every interval (a_{n} −L, a_{n} + L) there exists a subinterval I_{n} of the length ε satisfying

μ(I_{n}) ⩾ \frac{dε}{L(1 + a_{n}^{2})}.

It is left to apply theorem 5 to the sequence of centers of I_{n}. □

4.8. Benedicks’ result on unions of intervals. The following reslut contained in [2] provides non-trivial examples of measures with positive type. Until now, only a few examples of this kind existed in the literature.

Theorem 13. [2] Let ... < a_{−1} < a_{0} = 0 < a_{1} < a_{2} < ... be a discrete sequence of points and let I_{n} = (a_{n}, a_{n+1}] be the corresponding partition of ℝ. Suppose that there exist positive constants C_{1}, C_{2}, C_{3} such that

1) if

C_{1}^{−1}a_{2n+1} < a_{2k+1} < C_{1}a_{2n+1},

for some n, k, then

C_{2}^{−1}|I_{2n+1}| < |I_{2k+1}| < C_{2}|I_{2n+1}|;

2) for all n

C_{1}^{−1}|a_{2n+1}| < |a_{2n−1}| < C_{1}|a_{2n+1}|;

3) for all n

|I_{2n+1}| > C_{3} max(|I_{2n}|, 1);

4)

∑ \frac{|I_{2n+1}|^{2}}{1 + a_{2n+1}^{2}} [log_{+} \frac{|I_{2n+1}|}{|I_{2n}|} + 1] < ∞.  (4.3)

Then for any real number A > 0 and 1 ⩽ p < ∞ there exists a nonzero function

f ∈ L^{1}(ℝ) ∩ L^{p}(ℝ) ∩ C^{∞}(ℝ), supp f ⊂ ∪I_{2n},

such that f̂ = 0 on [0, A].

Here we will not concern ourselves with the condition f ∈ C^{∞}. The rest of the statement, i.e. the existence of f ∈ L^{1}(ℝ) ∩ L^{p}(ℝ), follows from theorem 2. Moreover, conditions 1 and 2 prove to be redundant.

Proof. Let {b_{n}}_{n∈ℤ} be a sequence of positive integers, monotonically increasing to ∞ as n → ∞ and as n → −∞, such that if one replaces |I_{2n+1}| in (4.3) with b_{n}|I_{2n+1}| the series still converges. Consider the sup-partition of {I_{n}} defined in the following way. Let

n_{0} = 0, n_{k+1} − n_{k} = b_{n_{k}}

for k > 0 and

n_{k+1} − n_{k} = b_{n_{k+1}}

for n < 0. Define J_{k} = (a_{2n_{k}}, a_{2n_{k+1}}]. By 3, the new partition satisfies the property |J_{n}| → ∞ and, because of monotonicity of b_{n},

∑ \frac{|J_{n}|^{2}}{1 + dist^{2}(0, J_{n})} [log_{+} \frac{|J_{n}|}{|J_{n} ∩ (∪I_{2k}) |} + 1] < ∞.  (4.4)

In particular {J_{n}} is short.

Let C be a large positive number. By [·] we will denote the integer part of a real number. Define a sequence Λ as follows. On each J_{k} = (a_{2n_{k}}, a_{2n_{k+1}}] place N = [C|J_{k}|] points of Λ inside J_{k} ∩ (∪I_{2n}) so that

λ_{m_{k}} < λ_{m_{k+1}} < ... < λ_{m_{k}+N}

and

| (∪I_{2n}) ∩ (a_{2n_{k}}, λ_{m_{k}}]| = | (∪I_{2n}) ∩ (λ_{m_{k}+N}, a_{2n_{k+1}}]| = | (∪I_{2n}) ∩ (λ_{l}, λ_{l+1}]|,

for all l, m_{k} ⩽ l < m_{k} + N − 1.

Then conditions 3 and 4 of the theorem imply that Λ satisfies the energy condition (2.9) on J_{n} and that D_{∗}(Λ) = C. Also the measure

ν = χ_{∪I_{2n}}Π

and Λ satisfy conditions of theorem 5. Therefore G_{ν}^{p} ⩾ 2πC for any 1 ⩽ p ⩽ ∞ which implies the existence of the desired function f satisfying f̂ = 0 on (0, 2πC).

□

Notice that our proof actually produces f ∈ L^{∞}. If, in addition to the conditions of the theorem, |I_{2n}| > const > 0, then the remaining property f ∈ C^{∞} can be added with little effort. One would need to construct f supported on ∪\frac{1}{2}I_{2n} and then consider a convolution f ∗ ϕ with a C^{∞}- function ϕ with small support. In the general case f can be ”smoothed out” using functions with exponentially decreasing size of support and involving arguments like theorem 11.

5. Proofs: Auxiliary Statements

This section contains the results that will be needed to prove theorems 2 and 3.

5.1. A measure with positive type. The following lemma is essentially proved, but not explicitly stated in [29].

Lemma 2. Let A = {a_{n}} be a discrete sequence of distinct real numbers that has bounded gaps, i.e. a_{n+1} − a_{n} < C for some 0 < C < ∞. Denote by b_{n} the middle of the interval (a_{n}, a_{n+1}), b_{n} = (a_{n} + a_{n+1})/2. Suppose that the sequence A is d-uniform for some d > 0. Then there exists a finite positive measure supported on B = {b_{n}},

μ = ∑ β_{n}δ_{b_{n}},

satisfying

0 < β_{n} ⩽ \frac{\sqrt{a_{n+1} − a_{n}}}{1 + a_{n}^{2}} ,  (5.1)

such that G_{μ}^{∞} ⩾ 2πd.

Proof. Let θ be the meromorphic inner function constructed for the sequence A as in lemma 5 from [29]. By construction, the Clark measure ν = μ_{−1} corresponding to θ is supported on B and satisfies

ν({b_{n}}) ≲ a_{n+1} − a_{n},  (5.2)

see the estimate (7.3) in [29].

Let c = d − ε. As was proved in [29], if θ satisfies the conditions of lemma 5, [29], and A is d-uniform, then there exists f ∈ K_{θ} that is divisible by e^{icz} in ℂ_{+}. (This is one of the main steps in the proof of theorem 2, [29]. See the part from the fourth line before claim 1 to the end of part I of the proof.)

Then, by the Clark representation, 2πif = (1 + θ)Kfν. Since 1 + θ is outer, Kfν is divisible by e^{icz} in C_{+}. Because ε is arbitrary, by lemma 1, the measure μ = |f|ν/(1 + x^{2}) satisfies G_{μ}^{∞} ⩾ 2πd. Since f ∈ L^{2}(ν) and ν satisfies (5.2), considering a constant multiple of μ if necessary, we obtain (5.1). □

5.2. Construction of an auxiliary sequence. To apply our previous lemma in the main proofs we will need the following

Lemma 3. Let B = {b_{n}} be a d-uniform sequence satisfying (2.8) and (2.9) on a short partition {I_{n}}. Let w(n) be a positive bounded function on ℤ such that

∑ \frac{log w(n)}{1 + n^{2}} > −∞.  (5.3)

Then for any ε > 0 there exists a discrete sequence A = {a_{n}} satisfying:

1) a_{n+1} − a_{n} < 1/ε.

2) Define the sequence C = \{c_{k}\} as c_{k} = \frac{a_{k+1}+a_{k}}{2} . Then the sequence B′ = B ∩ C satisfies

# (B′ ∩ I_{n}) ⩾ (d − e)|I_{n}|

for large enough |n|.

3) If b_{n} = c_{k}, i.e. b_{n} is the middle of (a_{k}, a_{k+1}), then a_{k+1} − a_{k} ⩽ w(n).

4) A is 2d-uniform

5) D^{∗}(C \ B) ⩽ d + ε.

Proof. Denote

l_{n} = min(b_{n+1} − b_{n}, b_{n} − b_{n−1}, w(n)).

Consider the sequence P = {p_{n}} defined as

p_{2n} = b_{n} − \frac{1}{3}l_{n}, p_{2n+1} = b_{n} + \frac{1}{3}l_{n}.

Choose a large L >> 1/ε. Define the sequence Q as follows: if p_{2n+2} − p_{2n+1} > L, insert M = [(p_{2n+2} − p_{2n+1})/L] points of Q into the interval (p_{2n+1}, p_{2n+2}) uniformly, i.e. at the points

p_{2n+1} + k \frac{p_{2n+1} − p_{2n+2}}{M + 1} , k = 1, 2, ...M.

Now put A = P ∪ Q.

By our construction the sequence A satisfies

2#(B ∩ I_{n}) − 2 ⩽ #(A ∩ I_{n}) ⩽ 2#(B ∩ I_{n}) + ε|I_{n}|.

To make A satisfy the more precise density condition (2.8) with 2d we may need to delete some points of B on each interval I_{n} and consider a smaller sequence B′ in place of B in the above construction. Note that we would have to delete at most ε|I_{n}| points from B on each I_{n} and that B′ will satisfy the energy condition (2.9) as a subsequence of B. After such an adjustment, A will satisfy 1), 2), 3) and the density condition (2.8) with 2d.

Note that A satisfies the energy condition on {I_{n}}. Indeed, let us denote Δ_{n} = #(P ∩ I_{n}) and Γ_{n} = #(Q ∩ I_{n}). Then

#(A ∩ I_{n})^{2} log |I_{n}| − ∑_{a_{n},a_{k}∈A∩I_{n}} log |a_{n} − a_{k}| =

(Δ_{n}^{2} log |I_{n}| − ∑_{a_{n},a_{k}∈P ∩I_{n}} log |a_{n} − a_{k}|) +

(Γ_{n}^{2} log |I_{n}| − ∑_{a_{n},a_{k}∈Q∩I_{n}} log |a_{n} − a_{k}|) +

2 (Δ_{n}Γ_{n} log |I_{n}| − ∑_{a_{n}∈P ∩I_{n},a_{k}∈Q∩I_{n}} log |a_{n} − a_{k}|) =

I + II + III.

To estimate I notice that for any p_{2k} ∈ P ∩ I_{n},

− log(p_{2k+1} − p_{2k}) ⩽ − log w(k),

by our choice of points p_{2k}, p_{2k+1}. The rest of the terms in I can be estimated by the similar terms for B′, i.e.

I ≲ (#(B′ ∩ I_{n})^{2} log |I_{n}| − ∑_{b_{n},b_{k}∈B′∩I_{n}} log |b_{n} − b_{k}|)

− ∑_{p_{2k}∈P ∩I_{n}} log w(k) + O(|I_{n}|^{2}).

Since B′ satisfies the energy condition and because of (5.3) and shortness of the partition, I will give finite contribution to the energy sum in (2.9).

To estimate II notice that points in Q are at a distance at least L/2 from each other. Therefore

II ≲ (Γ_{n}^{2} log |I_{n}| − ∑_{0⩽n,k⩽Γ_{n}} log |n − k|) + O(Γ_{n}^{2}) =

Γ_{n}^{2} log \frac{|I_{n}|}{Γ_{n}} + O(Γ_{n}^{2})

after estimating the sum via Stirling’s formula. Notice that since Γ_{n} < |I_{n}| and

log \frac{|I_{n}|}{Γ_{n}} < \frac{|I_{n}|}{Γ_{n}},

the last quantity will also give finite contribution to (2.9).

Finally, III can be estimated similarly to II. Just notice that any point a_{j} in P is at a distance at least L/2 from Q and therefore

Γ_{n} log |I_{n}| − ∑_{a_{k}∈Q∩I_{n}} log |a_{j} − a_{k}| ≲ Γ_{n} log \frac{|I_{n}|}{Γ_{n}} + O(|I_{n}|^{2}).

Summing over all a_{j} ∈ P ∩ I_{n} and recalling that #(P ∩ I_{n}) = Δ_{n} ≲ |I_{n}| we again get a finite quantity in (2.9).

To prove 5), let us split C into two subsequences:

C_{1} = {(a_{n} + a_{n+1})/2 | a_{n}, a_{n+1} ∈ P } and C_{2} = C \ C_{1}.

Notice that C_{1} \ B′ has at most one point between each two points of B′. Therefore,

D^{∗}(C_{1} \ B) ⩽ D^{∗}(B) ⩽ d + ε.

Also, if 2/L << ε then D^{∗}(C_{2}) < ε, because any two points of C_{2} are at a distance at least L/2 from each other. □

5.3. Existence of extremal measure with a spectral gap. The lemma in this section can be viewed as a version of de Branges’ theorem 66 from [7]. The last section of [29] contains a discussion of that theorem and its equivalent reformulations.

Here and throughout the rest of the paper we will use the standard notation S(z) = e^{iz} for the exponential inner function in the upper half-plane. In general, S^{a}(z) = e^{iaz} is inner in ℂ_{+} if a > 0 and inner in ℂ_{−} if a < 0.

Lemma 4. Let μ be a finite complex measure such that μ̂ ≡ 0 on [0, a]. Let W be a |μ|-weight. Then there exists a finite measure ν = ∑ α_{n}δ_{λ_{n}} concentrated on a discrete sequence Λ = {λ_{n}} such that

1) Λ ⊂ supp μ;

2) W is a |ν|-weight;

3) ν̂ ≡ 0 on [0, a];

4) The Cauchy integral Kν has no zeros in ℂ, Kν/S^{a} is outer in ℂ_{+} and Kν is outer in ℂ_{−}.

Proof. It will be more convenient for us to assume that μ̂ ≡ 0 on a symmetric interval [−a, a]. Then μ̄ has the same property. Hence we can assume that the measure is real (otherwise consider μ ± μ̂).

Consider the following set of measures on supp μ:

M_{W} = { ν | ∫ W d|ν| ⩽ 1, ν̂ = 0 on [−a, a], supp ν ⊂ supp μ, ν = ν̄}.

Notice that the set is non-empty, because μ ∈ M_{W}, and convex. It is also ∗-weakly closed in the space of all finite measures on supp μ. Therefore by the Krein-Milman theorem it has an extreme point. Let ν be such a point. We claim that it is the desired measure.

First, let us note that ν̂ ≡ 0 on [−a, a]. It is well-known that this property is equivalent to the property that ν annihilates the Payley-Wiener class P W_{a}, i.e. that for any bounded f ∈ P W_{a},

∫ fdν = 0,

see for instance the last section of [29].

Next, let us show that the set of real L^{∞}(|ν|)-functions h, such that ĥν ≡ 0 on [−a, a], is one-dimensional and therefore h = c ∈ ℝ. (This is equivalent to the statement that the closure of P W_{a} in L^{1}(|ν|) has deficiency 1, i.e. the space of its annihilators is one dimensional.)

Let there be a bounded real h such that ĥν ≡ 0 on [−a, a]. WLOG h ⩾ 0, since one can add constants, and ∫ W |h|d|ν| = 1. Choose 0 < α < 1 so that 0 ⩽ αh < 1. Consider the measures ν_{1} = hν and ν_{2} = (1 − α)^{−1}(ν − αν_{1}).

Then both of them belong to M_{W} and ν = αν_{1} +(1−α)ν_{2} which contradicts the extremality of ν.

Now let us show that ν is discrete. Let g be a continuous compactly supported real function on ℝ such that ∫ gd|ν| = 0. By the previous part, there exists a sequence f_{n} ∈ P W_{a}, f_{n} → g in L^{1}(|ν|). Indeed, otherwise there would exist a function h ∈ L^{∞}(|ν|) annihilating all f ∈ P W_{a} ∩ L^{1}(|ν|) and such that ∫ hgd|ν| = 1. Since ∫ gd|ν| = 0, h ≠ const and we would obtain a contradiction with the property that the space of annihilators is one-dimensional.

Since ν annihilates P W_{a} and (f_{n}(z) − f_{n}(w))/(z − w) ∈ P W_{a} for every fixed w ∈ ℂ \ ℝ,

0 = ∫ \frac{f_{n}(z) − f_{n}(w)}{z − w}dν(z) = Kf_{n}ν̄(w) − f_{n}(w)Kν(w)

and therefore

f_{n}(w) = \frac{Kf_{n}ν}{Kν} (w).

Taking the limit,

f = lim f_{n} = lim \frac{Kf_{n}ν}{Kν} = \frac{Kgν}{Kν} .

Since all of f_{n} are entire, one can show that the limit function f is also entire. Indeed, first notice that there exists a positive function V ∈ L^{1}(|ν|) such that f_{n_{k}}/V → g/V in L^{∞}(|ν|), for some subsequence {f_{n_{k}}}. To find such a V first choose f_{n_{k}} so that ||f_{n_{k}} − g||_{L^{1}(|ν|)} < 3^{−k} and then put

V = 1 + ∑ 2^{k}|f_{n_{k}} − g|.

Denote F_{k} = f_{n_{k}}/V and η = V |ν|. Then F_{k} converge in L^{2}(η) and by the Clark theorem (1 − I)KF_{k}η converge in H^{2}(ℂ_{+}), where I is the inner function whose Clark measure is η. Notice that

f_{n_{k}} = \frac{Kf_{n_{k}}ν}{Kν} = \frac{KF_{k}η}{Kν} = \frac{(1 − I)KF_{k}η}{(1 − I)Kν} .

Now let T be a large circle in ℂ such that |(1 − I)Kν| > const > 0 on T. Denote T_{±} = T ∩ ℂ_{±} and let m_{T} be the Lebesgue measure on T. Since (1 − I)KF_{k}η converge in H^{2}(ℂ_{+}), f_{n_{k}} converge in L^{1}(T_{+}, m_{T}). Similarly, f_{n_{k}} converge in L^{1}(T_{−}, m_{T}). By the Cauchy formula it follows that f_{n_{k}} converge normally inside T and therefore f is analytic inside T. Since such a circle T can be chosen to surround any bounded subset of ℂ, f is entire.

Since the numerator in the representation

f = \frac{Kgν̄}{Kν̄}

is analytic outside the compact support of g, the measure in the denominator must be singular outside of that support: Cauchy integrals of non-singular measures have jumps at the real line on the support of the a.c. part, which would contradict the property that f is entire. Choosing two different functions g with disjoint supports we conclude that ν is singular.

Moreover, since f is entire, the zero set of f has to be discrete. Since ν is singular, Kν tends to ∞ nontangentially in ℂ_{+} at ν-a.e. point and f = 0 at ν-a.e. point outside of the support of g. Again, by choosing two different g with disjoint supports, we can see that ν is concentrated on a discrete set.

It remains to verify 4). Since we chose to deal with the symmetric interval [−a, a], we need to show that Kν/S^{±a} are outer in ℂ_{±} correspondingly.

Let J be the inner function corresponding to |ν| (|ν| is the Clark measure for J). Denote

G = \frac{1}{2πi}(1 − J)Kν ∈ K_{J}.

As was mentioned in section 2.1, G has non-tangential boundary values |ν|-a.e. and

ν = G|ν|.

Since Kν is divisible by S^{a} in ℂ_{+}, G is divisible by S^{a} in ℂ_{+}. Suppose that G = S^{a}UH for some inner U. Since the measure ν is real, Ḡ = G, |ν|-a.e.

Let F ∈ K_{J} be the function such that J̄G = F̄. Since J = 1, |ν|-a.e., F = Ḡ = G, |ν|-a.e. Since functions in K_{J} are uniquely determined by their traces on the support of the Clark measure |ν|, F = G = S^{a}UH. Notice that the function h = S^{a}(1 + U)^{2}H also belongs to K_{J}:

J̄h = J̄S^{a}(1 + U)^{2}H = (J̄G)Ū(1 + U)^{2} = ḠŪ(1 + U)^{2}
= \overline{S^{a}(1 + U)^{2}H} = \overline{h} ∈ \overline{H^{2}}(ℂ_{+}),

because Ū(1 + U)^{2} is real a.e. on ℝ. Denote by γ the measure from the Clark representation of h, i.e.

γ = h|ν|, h = \frac{1}{2πi}(1 − J)Kγ.

Then

γ = h|ν| = Ū(1 + U)^{2}G|ν| = Ū(1 + U)^{2}ν.

The Cauchy integral of γ is divisible by S^{a} in ℂ_{+} because h is divisible by S^{a} in ℂ_{+}. Since Ū(1 + U)^{2} is real, a constant multiple of γ belongs to M_{W}. Since U is non-constant and |ν| is the Clark measure for J, γ is not a constant multiple of ν. Again we obtain a contradiction with the property that the space of annihilators is one-dimensional.

Thus G/S^{a} ∈ K_{J} is outer in C_{+}. Since JḠ = Ḡ, the pseudocontinuation of G does not have an inner factor except S^{−a} in ℂ_{−} as well. Hence Kν/S^{±a} is outer in ℂ_{±}.

If G has a zero at x = a ∈ ℝ outside of supp ν then

\frac{G}{x − a} ∈ K_{J}

and the measure

γ = \frac{G}{x − a}|ν|

leads to a similar contradiction with the property that the space of annihilators is one-dimensional, since (x − a)^{−1} is bounded and real on the support of ν. Since G = \frac{1}{2πi}(1 − J)Kν, Kν does not have any zeros on ℝ. □

Remark 4. A statement similar to lemma 9 from [29], where S^{a} was replaced with an arbitrary inner function, can also be formulated in the case of lemma 4.

5.4. Estimates of log |θ| for a meromorphic inner function.

Lemma 5. Consider a short partition {I_{n}} of ℝ. Consider the set of circles T_{n} = {z | |z − ξ_{n}| = 2|I_{n}|} where ξ_{n} ∈ I_{n}. Let θ be a meromorphic inner function such that

∑_{n} \frac{# (\{θ = 1\} ∩ 10I_{n}) |I_{n}|}{1 + dist^{2}(0, I_{n})} < ∞.  (5.4)

Then the integrals

p_{n} = ∫_{T_{n}} |log |θ(z)|| d|z|

satisfy

∑_{n} \frac{p_{n}}{1 + dist^{2}(0, I_{n})} < ∞.  (5.5)

Proof. Suppose that θ = S^{a}B for some a > 0 and some Blaschke product B, {B = 0} = {a_{n}} ⊂ ℂ_{+}. Then

log |θ| = log |S^{a}| + log |B|.

The integrals of | log |S^{a}|| are summable because

| log |S^{a}|| ≲ |I_{n}|

on T_{n} and the sequence {I_{n}} is short. To estimate the integral of | log |B|| notice that

| log |B(z)|| = ∑_{a_{k}∈D_{n}} |log \frac{|z − a_{k}|}{|z − ā_{k}|}| + ∑_{a_{k}∉D_{n}} |log \frac{|z − a_{k}|}{|z − ā_{k}|}| ,

where D_{n} is the disk, D_{n} = {z | |z − ξ_{n}| ⩽ 3|I_{n}|}. Elementary estimates show that for any a_{n} ∈ D_{n}

∫_{T_{n}} |log \frac{|z − a_{k}|}{|z − ā_{k}|}| d|z| ≲ |I_{n}|.

Also, since for each a_{k} ∈ D_{n} the argument of \frac{z−a_{k}}{z−ā_{k}} increases by at least π on the diameter of D_{n}, that is contained in 10I_{n}, the number of points a_{k} ∈ D_{n} is ≲ # ({θ = 1} ∩ 10I_{n}). Hence, because of (5.4), such integrals will give a finite contribution to the sum in (5.5).

For each a_{k} ∉ D_{n} one can show that

∫_{T_{n}} |log \frac{|z − a_{k}|}{|z − ā_{k}|}| d|z| ≲ ∫_{I_{n}} \frac{|I_{n}|ℑa_{k}dx}{(ℜa_{k} − x)^{2} + (ℑa_{k})^{2}} .

Notice that

∑_{k} ∫_{I_{n}} \frac{ℑa_{k}dx}{(ℜa_{k} − x)^{2} + (ℑa_{k})^{2}} = ∫_{I_{n}}(arg B)′ ⩽ 2π · # (\{θ = 1\} ∩ I_{n}\}) + const.

Again, because of (5.4), the integrals for a_{k} ∉ D_{n} will give a finite contribution in (5.5) □

5.5. A version of the first BM theorem. The following lemma is essentially a version of the so-called first Beurling-Malliavin theorem, see also [24].

Lemma 6. Let {I_{n}} be a long sequence of intervals and let c be a positive constant. Denote by I′_{n} and I′′_{n} the intervals of the length c|I_{n}| adjacent to I_{n} from the left and from the right correspondingly. Let u be a real function on ℝ such that

Δ_{n} = sup_{I′′_{n}} u − inf_{I′_{n}} u ⩾ d|I_{n}|

for all n and for some d > 0. Then u is not a harmonic conjugate of a Poisson-summable function.

Proof. Note that if ũ ∈ L_{Π}^{1} then f = e^{−iu+ũ} is an outer function in the Smirnov class in ℂ_{+}. Moreover, f belongs to the kernel N^{+}[e^{iu}] of the Toeplitz operator with the symbol e^{iu} in the Smirnov class. This contradicts a Toeplitz version of the first BM theorem, see section 4.4 of [24]. □

5.6. An estimate for an extremal discrete measure of positive type. In this section we show that that a discrete measure of positive type, like in the statement of lemma 4, must have log-summable pointmasses. We start with the following elementary statement that can be easily verified.

Lemma 7. Let {I_{n}} be a short sequence of intervals and let C > 1. Denote

l_{n} = ∑_{I_{m}∩CI_{k}≠∅} |I_{m}|.

Then

∑ \frac{l_{n}|I_{n}|}{1 + dist^{2}(0, I_{n})} < ∞.

Our main statement in this section is Lemma 8. Let ν be a finite measure

ν = ∑ α_{n}δ_{λ_{n}}

on a discrete sequence Λ = {λ_{n}}, such that α_{n} ≠ 0, ν̂ ≡ 0 on [0, 2πd], Kν does not have any zeros in ℂ, Kν/S^{d} is outer in ℂ_{+} and Kν is outer in ℂ_{−}. Then for any ε > 0, Λ contains a (d − ε)-uniform subsequence and

∑ \frac{log |α_{n}|}{1 + n^{2}} > −∞.  (5.6)

Proof. The statement that Λ contains a (d − ε)-uniform subsequence follows from the property that G_{Λ} ⩾ d and theorem 1.

To establish (5.6), let us first show that there exists a short partition {I_{k}} of ℝ such that Λ satisfies (2.8) with d on that partition.

Let J be the inner function whose Clark measure is |ν|. Then by the Clark theorem the function

Q(z) = (1 − J)Kν

belongs to K_{J}. It follows from the properties of Kν that Q = S^{d}O in ℂ_{+} for some outer O and J̄Q = Ō. Therefore the argument of O satisfies u = 2 arg O = arg J − dx. Notice that arg J is a growing function that is equal, up to a bounded term, to the counting function of Λ. Also, since O ∈ H^{2}, ũ ∈ L_{Π}^{1}. If the desired short partition {I_{k}}, where Λ satisfies (2.8), does not exist then there exists a long sequence of intervals {J_{k}} such that

|#(Λ ∪ J_{k}) − d|J_{k}|| ⩾ c_{1}|J_{k}|  (5.7)

for each k and for some c_{1} > 0. First, let us assume that the difference in the left-hand side is positive for a long subsequence of {J_{k}}. Let J′_{k}, J′′_{k} denote the intervals of the length c_{2}|J_{k}|, 0 < c_{2} << c_{1}, adjacent to J_{k} from the left and from the right correspondingly. Since u′ is bounded from below we get that

Δ_{k} = inf_{J′′_{k}} u − sup_{J′_{k}} u > c_{3}|J_{k}|

for some c_{3} > 0 on a long subsequence of {J_{k}}, if c_{2} is small enough. By lemma 6, this contradicts the property that ũ ∈ L_{Π}^{1}. If the difference in (5.7) is negative for a long subsequence of {J_{k}} then lemma 6 can be applied to −u and the intervals J′_{k}, J′′_{k} chosen so that J′_{k}, J′′_{k} ⊂ J_{k}, |J′_{k}| = |J′′_{k}| = c_{2}|J_{k}|, J′_{k} shares its left endpoint with J_{k} and J′′_{k} shares its right endpoint with J_{k}, to arrive at the same contradiction. Hence a short partition where Λ satisfies (2.8) with d does exist.

Let {I_{k}} be such a partition. Let λ_{n_{k}} ∈ I_{k} be such that

log_{−} α_{n_{k}} = max_{λ_{n}∈I_{k}} log_{−} α_{n}.

Suppose that (5.6) is not satisfied. Then

∑_{k} #(Λ ∩ I_{k})\frac{log_{−} α_{n_{k}}}{1 + n_{k}^{2}} ≍ ∑_{k} |I_{k}|\frac{log_{−} α_{n_{k}}}{1 + n_{k}^{2}} = ∞.  (5.8)

Consider μ = ∑_{n} δ_{λ_{n}} the counting measure of Λ. Since Λ satisfies (2.8), μ is Poisson-finite. Let θ be the inner function such that μ is its Clark measure. Since ν is finite, it can be represented as ν = fμ with f ∈ L^{2}(μ). Hence, by Clark theory,

F = \frac{1}{2πi}(1 − θ)Kν ∈ K_{θ}

with F (λ_{n}) = f(λ_{n}) = α_{n}.

For each k consider the disk

D_{k} = {z | |z − λ_{n_{k}}| < 2|I_{k}|}

and its boundary circle T_{k} = ∂D_{k}. Notice that for each k, F does not have zeros in D_{k}. It does have poles at the points ā_{n} ∈ ℂ_{−}, where A = {a_{n}} are the zeros of θ in ℂ_{+}. Hence in D_{k} the function F admits factorization F = H_{k}/B_{k}, where B_{k} is the finite Blaschke product in D_{k} (|B_{k}| = 1 on T_{k}) with zeros at Ā ∩ D_{k}, and H_{k} is analytic without zeros in D_{k}.

Notice that

− ∫_{T_{k}} log_{−} |F (z)|d|z| ⩽ ∫_{T_{k}} log |F (z)|d|z| = ∫_{T_{k}} log |H_{k}(z)|d|z| ≲ |I_{k}| log α_{n_{k}}

(5.9)

by Jensen’s inequality, because F has only poles and no zeros in D_{k}. At the same time, since F ∈ K_{θ}, it belongs to H^{2}(ℂ_{+}) and is equal to θḠ, G ∈ H^{2}(ℂ_{+}) in ℂ_{−}. Denote by T_{k}^{±} the upper and lower halves of T_{k}. Since the absolute value of an H^{2} function is bounded by

const + const |y|^{−1/2}

inside the half-plane, we have

∫_{T_{k}} log_{+} |F (z)|d|z| ⩽ ∫_{T_{k}^{+}} log_{+} |F (z)|d|z| + ∫_{T_{k}^{−}} log_{+} |G(z)|d|z|

+ ∫_{T_{k}^{−}} log_{+} |θ(z)|d|z| ≲ |I_{k}| + v_{k},  (5.10)

where ∑ v_{k}/(1 + a_{n_{k}}^{2}) < ∞ by lemma 5, because

#({θ = 1} ∩ 10I_{k}) ≲ ∑_{I_{m}∩10I_{k}≠∅} |I_{m}|

and (5.4) is satisfied by lemma 7.

Since H_{k} ≠ 0 in D_{k}, log |H| is harmonic in D_{k}. Hence its values on I_{k} can be recovered from the values of log |H_{k}| = log |F | on T_{k} via the Poisson formula.

By (5.10), the Poisson integral of log_{+} |F | will deliver a small contribution, i.e. on each I_{k} it will be equal to a function h_{k}^{+} such that

∑ ∫_{I_{k}} h_{k}^{+}(x)dΠ(x) < ∞.

On the other hand, the Poisson integral of log_{−} |F | in D_{k}, restricted to I_{k}, will be equal to h_{k}^{−}, where h_{k}^{−}(x) ≍ log α_{n_{k}} for all x ∈ I_{k} by (5.9). Hence by (5.8)

∑ \frac{∫_{I_{k}} log |H_{k}|dx}{1 + dist^{2}(0, I_{k})} = −∞.

Furthermore, similarly to the proof of lemma 5,

deg B_{k} ≲ #({θ = 1} ∩ 5I_{k}) ≲ ∑_{I_{m}∩5I_{k}≠∅} |I_{m}|.

Therefore by lemma 7

∑_{k} \frac{|I_{k}| deg B_{k}}{1 + dist^{2}(0, I_{k})} < ∞.

Thus

∫_{ℝ} log |F (x)|dΠ ≍ ∑_{k} \frac{∫_{I_{k}} (log |B_{k}(x)| + log |H_{k}(x)|) dx}{1 + dist^{2}(0, I_{k})} ≲

∑_{k} \frac{|I_{k}| deg B_{k} + ∫_{I_{k}} log |H_{k}(x)|dx}{1 + dist^{2}(0, I_{k})} = −∞

and we obtain a contradiction. □

Remark 5. Using results of [29] one can prove a slightly stronger statement that Λ itself is d-uniform.

5.7. Equivalence of completeness in L^{p} and C_{W}. The theorem we discuss in this section relates the type problem to Bernstein’s study of weighted uniform approximation, see [15] or [6].

Consider a weight W, i.e. a lower semicontinuous function W : ℝ → [1, ∞) that tends to ∞ as x → ±∞. We define C_{W} to be the space of all continuous functions on ℝ satisfying

lim_{x→±∞} \frac{f(x)}{W (x)} = 0.

We define the norm in C_{W} as

||f|| = ||fW^{−1}||_{∞}.

The following is a well-known result by A. Bakan. For reader’s convenience we supply a short proof.

Theorem 14 ([1]). Let μ be a finite positive measure on ℝ. Then the system of exponentials E_{d} is complete in L^{p}(μ) for some 1 ⩽ p ⩽ ∞ if and only if there exists a μ-weight W ∈ L^{p}(μ) such that E_{d} is complete in C_{W}.

Proof. If E_{d} is complete in C_{W} for some μ-weight W ∈ L^{p}(μ) then for any bounded continuous function f there exists a sequence {S_{n}} of finite linear combinations of exponentials from E_{d} such that S_{n}/W converges to f/W uniformly. Then S_{n} converges to f in L^{p}(μ). Hence E_{d} is complete in L^{p}(μ).

Suppose that E_{d} is complete in L^{p}(μ). Let {f_{n}}_{n∈ℕ} be a set of bounded continuous functions on ℝ, that is dense in C_{W}. Let {S_{n,k}}_{n,k∈ℕ} be a family of finite linear combinations of exponentials from E_{d} such that

||f_{n} − S_{n,k}||_{L^{p}(μ)} < 4^{−(n+k)}.

Denote

W = 1 + ∑_{n,k∈ℕ} 2^{n+k}|f_{n} − S_{n,k}|.

Notice that then W ∈ L^{p}(μ) and S_{n,k}/W → f_{n}/W uniformly as k → ∞. Since {f_{n}} is dense in C_{W}, E_{d} is complete in C_{W}. □

6. Proofs of main results

6.1. Proof of theorem 3.

To prove that G_{μ}^{p} ⩾ 2πD, WLOG we can assume that B itself is a d-uniform sequence for some d > 0 and that w satisfies (3.2).

Fix a small ε > 0. Let C = {c_{n}} be the sequence provided by lemma 3. Then by lemma 2 (applied to C and w^{2}) there exists a finite positive measure ν = ∑ σ_{n}δ_{c_{n}} concentrated on C, satisfying

0 < σ_{n} < w(k) for c_{n} = b_{k} and G_{ν}^{∞} ⩾ 2π(2d).

Let θ be the Clark inner function corresponding to ν. Then there exists a function in K_{θ} divisible by S^{2π(2d−ε)} in the upper half-plane, i.e. S^{2π(2d−ε)}h ∈ K_{θ} for some h ∈ H^{2}: if ϕ̂ν = 0 on [0, 2π(2d − ε)] for some ϕ ∈ L^{∞}(ν), put

h = \frac{1}{2πi}(1 − θ)Kϕν.

By lemma 3, D^{∗}(C \ B) < d + ε. Let J be an inner function such that {J = 1} = C \ B. By a version of the Beurling-Malliavin theorem, see [23] section 4.6, the kernel of the Toeplitz operator with the symbol S^{2π(−d−ε)}J in H^{∞} is non-empty, i.e. there exists a function g ∈ H^{∞}(ℂ_{+}) such that

S^{2π(−d−ε)}Jg ∈ H̄^{∞}.

Since

θ̄S^{2π(−d−ε)}JgS^{2π(2d−ε)}h = θ̄S^{2π(d−2ε)}Jgh ∈ H̄^{2},

we have

S^{2π(d−2ε)}Jgh ∈ K_{θ}.

Since K_{θ} is closed under division by inner components, S^{2π(d−2ε)}gh ∈ K_{θ} and therefore

p = S^{2π(d−2ε)}Jgh − S^{2π(d−2ε)}gh = S^{2π(d−2ε)}(J − 1)gh ∈ K_{θ}.

By the Clark representation formula, p = \frac{1}{2πi}(1 − θ)Kpν, and since 1 − θ is outer, Kpν is divisible by S^{2π(d−2ε)} in ℂ_{+}. Notice that p = (1−J)gh = 0 on C \ B = {J = 1} and p ∈ L^{∞}(ν) on B ∩ C. Therefore, if η is the restriction of ν on B ∩ C, the existence of such p implies

G_{η}^{∞} ⩾ 2π(d − 2ε).

For any ε > 0, the measure η constructed as above will have a bounded density with respect to μ. Hence G_{μ}^{∞} ⩾ 2πd.

To prove the second part of the statement suppose that log(|n_{B}| + 1) ∈ L_{Π}^{1} but G_{μ}^{p} > 2πA > 2πD for some A > D. WLOG assume that the counting function n_{B} is non-zero outside of [−1, 1] and define

W (b_{k}) = \begin{cases}\frac{1}{2^{n}n_{B}(2^{n})w(k)} + 1 if b_{k} ∈ (2^{n−1}, 2^{n}] for some n ∈ ℕ \\ −\frac{1}{2^{n}n_{B}(−2^{n})w(k)} + 1 if b_{k} ∈ (−2^{n}, −2^{n−1}] for some n ∈ ℕ \\ 1 if b_{k} ∈ (−1, 1] \end{cases} .

Then W is a μ-weight and by lemmas 4 and 8 there exists a measure ν = ∑ α_{k}δ_{b_{n_{k}}} supported on B′ = {b_{n_{k}}} ⊂ B such that W is a ν-weight, B′ is an A-uniform sequence and α_{k} satisfy (5.6). Since W is a ν-weight, |α_{k}| ≤ C/W (b_{n_{k}}). Since α_{k} satisfy (5.6), the definition of W implies that

∑_{k} \frac{log w(n_{k})}{1 + n_{k}^{2}} > −∞.

Hence D ⩾ A and we obtain a contradiction. □

6.2. Proof of theorem 2.

I) First, suppose that G_{μ}^{p} ⩾ a for some 1 < p ⩽ ∞. Then for any d > 0, 2πd < a, there exists f ∈ L^{p}(μ) such that f̂μ = 0 on [0, 2πd]. Let W be a μ-weight. Denote V = W^{1/q} where \frac{1}{p} + \frac{1}{q} = 1. Then

∫ V |f|dμ < ∞.

Therefore by lemma 4 there exists a discrete measure ν = ∑ α_{n}δ_{λ_{n}}, Λ = {λ_{n}} ⊂ supp μ such that ν̂ = 0 on [0, 2πd], V is a |ν|-weight and ν satisfies the rest of the conditions of lemma 8. Then by lemma 8, Λ contains a duniform subsequence Λ′ and α_{n} satisfy (5.6). Since V is a ν-weight, V (λ_{n}) < C/|α_{n}| for all n. It is left to notice that log W (λ_{n}) = q log V (λ_{n}) and therefore

∑_{λ_{n}∈Λ′} \frac{log W (λ_{n})}{1 + λ_{n}^{2}} < ∞.

II) Now suppose that G_{μ}^{p} < d < a for some 1 < p ⩽ ∞.

Since G_{μ}^{p} < d, by theorem 14 there exists a μ-weight W such that finite linear combinations of exponentials from E_{d−ε} are dense in C_{W} for some ε > 0. Suppose that there exists a d-uniform sequence Λ = {λ_{n}} ⊂ supp μ, satisfying (3.1). Then by theorem 3 there exists a measure ν = ∑ α_{n}δ_{λ_{n}} such that |α_{n}| ⩽ W^{−1}(λ_{n})/(1 + λ_{n}^{2}) and ν̂ = 0 on [0, d − ε]. Then the finite measure W ν annihilates all functions e^{ict}/W, c ∈ [0, d−ε]. This contradicts completeness of E_{d−ε} in C_{W}. □

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Texas A&M University, Department of Mathematics, College Station, TX 77843, USA

E-mail address: alexeip@math.tamu.edu