Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials Aleksei Kulikov Alexander Ulanovskii Ilya Zlotnikov

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Completeness of Certain Exponential Systems

and Zeros of Lacunary Polynomials

Aleksei Kulikov Alexander Ulanovskii Ilya Zlotnikov

October 4, 2022

Abstract

Let Γ be a subset of {0,1,2, ...}. We show that if Γ has ‘gaps’ then the

completeness and frame properties of the system {tke2πint :n∈Z, k ∈Γ}

diﬀer from those of the classical exponential systems. This phenomenon is

closely connected with the existence of certain uniqueness sets for lacunary

polynomials.

Keywords: completeness, frame, totally positive matrix, generalized Vandermonde

matrix, uniqueness set, lacunary polynomials

1 Introduction

Let Λ be a separated set of real numbers. Denote by

E(Λ) := {e2πiλt, λ ∈Λ}

the corresponding exponential system.

Approximation and representation properties of exponential systems in dif-

ferent function spaces is a classical subject of investigation. In particular, the

completeness and the frame problems of E(Λ) for the space L2(a, b) can be

stated as follows: Determine if

(a) (Completeness property of E(Λ)) every function Fin L2(a, b) can be ap-

proximated arbitrarily well in L2-norm by ﬁnite linear combinations of

exponential functions from E(Λ);

(b) (Frame property of E(Λ)) there exist two positive constants Aand Bsuch

that for every F∈L2(a, b) we have

AkFk2

2≤X

λ∈Λ

|hF, e2πiλti|2≤BkFk2

where h·,·i is the usual inner product in L2(a, b).

arXiv:2210.00504v1 [math.CA] 2 Oct 2022

Note that the notion of frame is very important and can be deﬁned in similar

manner for an arbitrary system of elements E={eλ}in a Hilbert space H. If

Eis a frame in H, then every element ffrom Hadmits a (maybe, non-unique)

representation

f=X

eλ∈E

cλeλ,

for some l2−sequence of complex numbers cλ(see e.g. [3]).

It is easy to check that the completeness property of E(Λ) is translation-

invariant: If E(Λ) is complete in L2(a, b), then it is complete in L2(a+c, b +c),

for every c∈R. As a ‘measure of completeness’, one may introduce the so-called

completeness radius of E(Λ):

CR(Λ) = sup{a≥0 : E(Λ) is complete in L2(−a, a)}.

Similarly, the frame property of E(Λ) is also translation-invariant, and one may

introduce the frame radius as

F R(Λ) = sup{a≥0 : E(Λ) is a frame in L2(−a, a)}.

Both radii above can be expressed in terms of certain densities:

(A) The celebrated Beurling–Malliavin theorem [1] states that CR(Λ) =

D∗(Λ). Here D∗is the so-called upper (or external) Beurling–Malliavin density.

(B) It follows from the classical ‘Beurling Sampling Theorem’ [2] (see also a

detailed discussion in [7]) that F R(Λ) = D−(Λ), where Λ is a separated (also

called uniformly discrete) set and D−(Λ) is the lower uniform density of Λ.

We refer the reader to [8] or [11] for a complete description of exponential

frames for the space L2(a, b). It is not given in terms of a density of Λ.

Observe that the proofs of (A) and (B) use techniques from the complex

analysis.

The density D∗can be deﬁned and the Beurling–Malliavin formula for the

completeness radius remains valid for the multisets (Λ,Γ(λ)), where Λ ⊂Rand

Γ(λ) = {0, ..., n(λ)−1}, i.e. for the systems

E(Λ,Γ(λ)) := {tke2πiλt :λ∈Λ, t = 0, ..., n(λ)−1}.(1)

Here n(λ) is the multiplicity (number of occurrences) of the element λ∈Λ.

The same is true for the frame radius, see [4]. In particular, if Λ = Zand

Γ(λ)=ΓN:= {0, ..., N −1},λ∈Λ, then one has

CR(Z,ΓN) = F R(Z,ΓN) = N/2 = #ΓN/2,(2)

where #Γ is the number of elements of Γ, CR(Z,ΓN) and F R(Z,ΓN) are the

completeness and frame radius of E(Z,ΓN), respectively.

One may consider the completeness property of systems in (1) in Lp(a, b)

and C([a, b]). For each of these spaces, the completeness property is translation-

invariant. Clearly, the completeness in C([−a, a]) implies the completeness in

Lp(−a, a) for every 1 ≤p < ∞.Observe that if E(Λ,Γ(Λ)) is not complete in

C([−a, a]), its deﬁciency in C([−a, a]) is at most 1, i.e. by adding to the system

an exponential function e2πiat, a 6∈ Λ,the new lager system becomes complete

in C([−a, a]) (see e.g. discussion in [10]). It easily follows that every system in

(1) has the same completeness radius for every space considered above.

2 Statement of Problem and Results

Let us now introduce somewhat more general systems. Assume that Λ ⊂Ris

a discrete set and that to every λ∈Λ there corresponds a ﬁnite or inﬁnite set

Γ(λ)⊂N0:= {0,1,2,3, ...}. Set

E(Λ,Γ(λ)) = {tγe2πiλt :λ∈Λ, γ ∈Γ(λ)}.

Inspired by a recent work of H. Hedenmalm [5], we ask: What are the com-

pleteness and frame properties of E(Λ,Γ(λ))? In this note we restrict ourselves

to the case Λ = Zand Γ(n) = Γ ⊂N0, n ∈Z,is a ﬁxed set. That is, we will

consider the completeness and frame properties of the system

E(Z,Γ) := {tγe2πint :n∈Z, γ ∈Γ},Γ⊂N0.

Let us now introduce the formal analogues of the completeness and frame

radius:

CR(Z,Γ) := sup{a≥0 : E(Z,Γ) is complete in L2(−a, a)},

F R(Z,Γ) := sup{a≥0 : E(Z,Γ) is a frame in L2(−a, a)}.

We also deﬁne the completeness radius CRC(Z,Γ) in the spaces of continuous

functions:

CRC(Z,Γ) := sup{a≥0 : E(Z,Γ) is complete in C([−a, a])}.

In what follows, to exclude trivial remarks, we will always assume that 0 ∈Γ.

Set

Γeven = Γ ∩2Zand Γodd = Γ ∩(2Z+ 1),

and introduce the following number

r(Γ) := 









#Γodd +1

2,if #Γodd <#Γeven,

#Γeven,if #Γodd ≥#Γeven.

Observe that r(Γ) <#Γ/2 unless #Γeven = #Γodd or #Γeven = #Γodd + 1.

It turns out that the completeness and frame properties of E(Z,Γ) may diﬀer

from the ones for the systems considered above. In particular, we have

Theorem 1. Given any ﬁnite or inﬁnite set Γ⊂N0satisfying 0∈Γ.Then

(i) CR(Z,Γ) = #Γ/2;

(ii) CRC(Z,Γ) = F R(Z,Γ) = r(Γ).

Below we prove more precise results.

Theorem 1 shows that property (2) is no longer true for the systems E(Z,Γ).

The proof of part (i) uses mainly basic linear algebra. We will see that

the completeness property of E(Z,Γ) in L2(a, b) is translation invariant, and so

CR(Z,Γ) still can be viewed as a ‘measure of completeness’ of E(Z,Γ).

On the other hand, neither the frame property in L2(a, b) nor the com-

pleteness property in C([a, b]) is translation invariant in the sense that both

of them depend on the length of the interval (a, b) and also on its position.

This phenomenon is intimately connected with the solvability of certain sys-

tems of linear equations and also with the existence of certain uniqueness sets

for lacunary polynomials, see Theorem 2 below.

Given any ﬁnite set M⊂N0, let P(M) denote the set of real polynomials

with exponents in M:

P(M) := {P(x) = X

mj∈M

cjxmj:cj∈R}.

If M⊂N0consists of nelements (shortly, #M=n), then clearly no set

X⊂Rsatisfying #X≤n−1 is a uniqueness set for P(M), i.e. there is a

non-trivial polynomial P∈P(M) which vanishes on X. This is no longer true

if #X=n. Moreover, there exist real uniqueness sets X, #X=n, that are

uniqueness sets for every space P(M),#M=n. Indeed, by Descartes’ rule of

signs, each P∈P(M) may have at most n−1 distinct positive zeros, and so

every set of npositive points is a uniqueness set for P(M). Here we present a

less trivial example of such sets. Given Ndistinct real numbers t1, . . . , tN,set

S(t1, . . . , tN) := {(−1)ktk}N

k=1.(3)

Theorem 2. Assume 0< t1< t2<· · · < tN. Then both sets ±S(t1, . . . , tN)

are uniqueness sets for every space P(M), M ⊂N0,#M=N.

The rest of the paper is organized as follows: In Section 3 several auxiliary

results are proved. Theorem 2 is proved in Section 4. We consider the com-

pleteness property of E(Z,Γ) in L2(a, b) and in C([a, b]) in Sections 5 and 6,

respectively. Finally, in Section 7 we consider the frame property of E(Z,Γ)

and also present some remarks.

3 Auxiliary Lemmas

Given N∈N,x={x0, . . . , xN−1} ⊂ R, and Γ = {γ0, γ1, . . . , γN−1} ⊂ Nwe

denote by V(x,Γ) a generalized N×NVandermonde matrix,

V(x; Γ) :=







xγ0

0xγ0

1xγ0

2. . . xγ0

N−1

xγ1

0xγ1

1xγ1

2. . . xγ1

N−1

. . . . . . . . . . . . . . .

xγN−1

0xγN−1

1xγN−1

2. . . xγN−1

N−1







.(4)

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CompletenessofCertainExponentialSystemsandZerosofLacunaryPolynomialsAlekseiKulikovAlexanderUlanovskiiIlyaZlotnikovOctober4,2022AbstractLetbeasubsetoff0;1;2;:::g.Weshowthatifhas`gaps'thenthecompletenessandframepropertiesofthesystemftke2int:n2Z;k2gdierfromthoseoftheclassicalexponentialsystems.Thisph...

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Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials Aleksei Kulikov Alexander Ulanovskii Ilya Zlotnikov

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