Riesz bases of exponentials for convex polytopes with symmetric faces

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$. The result is new in all dimensions $d$ greater than one.

1. Introduction 1.1.Let Ω ⊂ R d be a bounded, measurable set of positive measure.When is it possible to find a countable set Λ ⊂ R d such that the system of exponential functions constitute a basis in the space L 2 (Ω)?
The answer depends on what we mean by a "basis".The best one can hope for is to have an orthogonal basis of exponentials.The problem of which domains admit an orthogonal basis E(Λ) has been extensively studied and goes back to Fuglede [Fug74], and it is well-known that many reasonable domains do not have such a basis.For example, it was recently proved in [LM19] that a convex domain Ω ⊂ R d admits an orthogonal basis of exponential functions if and only if Ω is a convex polytope which can tile the space by translations (meaning that one can fill the whole space by translated copies of Ω with non-overlapping interiors).In particular, a disk or a triangle in the plane does not have an orthogonal basis E(Λ) (this was shown already in [Fug74]).
If a domain Ω ⊂ R d does not have an orthogonal basis of exponentials, then a Riesz basis is the next best thing one can hope for.A Riesz basis can be defined as the image of an orthonormal basis under a bounded and invertible linear map (there are also other, equivalent definitions, see Section 2) and it shares many of the qualities of an orthonormal basis.In particular, if E(Λ) is a Riesz basis in L 2 (Ω) then any function f from the space has a unique and stable Fourier series expansion f = λ∈Λ c λ e λ .
In sharp contrast to the situation with orthogonal bases, no single example is known of a set Ω ⊂ R d which does not have a Riesz basis of exponentials.At the same time, for most domains it remains unknown whether one can construct a Riesz basis of this type.One of the relatively few results obtained in this direction says that any finite union of intervals in R has a Riesz basis of exponentials [KN15] (see also [KN16] for a multi-dimensional version of this result).However it is still an open problem of whether, say, a disk or a triangle in the plane admits a Riesz basis E(Λ).
1.2.The present paper is concerned with the existence of Riesz bases of exponentials for convex polytopes in R d .Our main result can be stated as follows: Theorem 1.1.Let Ω ⊂ R d be a convex polytope which is centrally symmetric and all of whose faces of all dimensions are also centrally symmetric.Then there is a set Λ ⊂ R d such that the system of exponential functions E(Λ) is a Riesz basis in L 2 (Ω).
The result is new in all dimensions d greater than one.
In [LR00], Lyubarskii and Rashkovskii established the existence of a Riesz basis of exponentials for convex, centrally symmetric polygons in R 2 such that all the vertices of the polygon lie on the integer lattice Z 2 (one may alternatively assume that the vertices lie on some other lattice, due to the invariance of the problem under affine transformations).1 The approach in [LR00] involves methods from the theory of entire functions of two complex variables.The paper [LR00] contains also a weaker result for convex, centrally symmetric polygons whose vertices fail to lie on a lattice, but in this case the result does not amount to the construction of a Riesz basis of exponentials.
A similar result in higher dimensions was obtained in [GL14, Corollary 3], where the existence of a Riesz basis of exponentials was established for centrally symmetric polytopes in R d with centrally symmetric facets, such that all the vertices of the polytope lie on the lattice Z d .The proof is based on the fact that such a polytope multi-tiles the space by lattice translates (in connection with this result, see also [Kol15], [GL18]).
In this paper, our goal is to prove the existence of a Riesz basis of exponentials for convex, centrally symmetric polytopes with centrally symmetric faces, without imposing any extra constraints.This is the content of our main result, Theorem 1.1.1.3.Our approach to the proof of Theorem 1.1 is inspired by the paper [Wal17] due to Walnut.In that paper, the author applies a technique outlined in [BFW06] in order to construct a system of exponentials E(Λ) that is shown to be complete in the space L 2 (Ω), where Ω ⊂ R 2 is a convex, centrally symmetric polygon.The set Λ constructed in [Wal17] is the union of a finite number of shifted lattices in R 2 .It is shown that if the convex polygon satisfies certain extra arithmetic constraints given in [Wal17, Theorem 4.2], then E(Λ) is not only a complete system, but is in fact a Riesz basis, in L 2 (Ω).
In [Wal17] the author does not provide any transparent description as to which convex, centrally symmetric polygons satisfy the extra constraints imposed in [Wal17, Theorem 4.2].One can verify though that these constraints are satisfied if and only if, possibly after applying an affine transformation, all the vertices of the polygon lie in Z 2 .Hence the class of planar convex polygons for which a Riesz basis E(Λ) is constructed in [Wal17] coincides with the class covered by the result in [LR00].
In this paper we will extend the technique from [Wal17] to all dimensions, and also combine it with the Paley-Wiener theorem about the stability of Riesz bases under small perturbations (see Section 2).This will allow us to construct a Riesz basis E(Λ) for any convex, centrally symmetric polytope in R d with centrally symmetric faces, without imposing any extra arithmetic constraints.The set of frequencies Λ in our construction will no longer be a union of finitely many shifted lattices, but it will rather have a less regular structure.

Preliminaries
2.1.Riesz bases.Let H be a separable Hilbert space.A system of vectors {f n } ⊂ H is called a Riesz basis if it is the image of an orthonormal basis under a bounded and invertible linear map.If {f n } is a Riesz basis then any f ∈ H admits a unique expansion in a series f = c n f n , and the coefficients {c n } satisfy A f 2 |c n | 2 B f 2 for some positive constants A, B which do not depend on f .In fact, the latter condition for the system {f n } can serve as an equivalent definition of a Riesz basis.
There are also several other ways to characterize Riesz bases in a separable Hilbert space H.The following characterization, see [You01, Section 4.4, Theorem 8], will be used in the present paper: Proposition 2.1.A system {f n } ⊂ H is a Riesz basis if and only if it satisfies the following three conditions: For a discussion about the various properties and characterizations of Riesz bases we refer the reader to [You01].

Paley-Wiener spaces.
Let Ω ⊂ R d be a bounded, measurable set of positive measure.The Paley-Wiener space P W (Ω) consists of all functions F ∈ L 2 (R d ) which are Fourier transforms of functions from L 2 (Ω), namely, (2.1) A set Λ ⊂ R d is called a set of uniqueness for the space P W (Ω) if whenever a function F from the space satisfies F (λ) = 0, λ ∈ Λ, then F is identically zero.This means that the functions from the space P W (Ω) are uniquely determined by their values on Λ.
We say that Λ is a set of interpolation for P W (Ω) if for any {c(λ)} ∈ ℓ 2 (Λ) there exists at least one F ∈ P W (Ω) satisfying F (λ) = c(λ), λ ∈ Λ.Such a function F is said to solve the interpolation problem with the set of nodes Λ and with the values {c(λ)}.
The Fourier transform is a unitary map from the space L 2 (Ω) onto P W (Ω).This allows to reformulate the uniqueness and interpolation properties of a set Λ ⊂ R d with respect to the space P W (Ω), in terms of properties of the system of exponential functions E(Λ) in the space L 2 (Ω).Thus Λ is a set of uniqueness for P W (Ω) if and only if E(Λ) is a complete system in L 2 (Ω); while Λ is a set of interpolation for P W (Ω) if and only if the system of equations f, e λ = c(λ), λ ∈ Λ, admits at least one solution f ∈ L 2 (Ω) whenever the scalars {c(λ)} belong to ℓ 2 (Λ).
2.3.Uniformly discrete sets.A set Λ ⊂ R d is said to be uniformly discrete if there is δ > 0 such that |λ ′ − λ| δ for any two distinct points λ, λ ′ in Λ.The following proposition may be found for instance in [OU16, Proposition 2.7].
Proposition 2.2.Let Ω ⊂ R d be a bounded, measurable set of positive measure, and let Λ ⊂ R d be a uniformly discrete set.Then there is a constant for every function It is well-known that if Ω ⊂ R d is a bounded, measurable set of positive measure, and if Λ ⊂ R d is a set of interpolation for the space P W (Ω), then Λ must be a uniformly discrete set, see e.g.[OU16, Section 4.2.1].Due to Propositions 2.1 and 2.2, this implies the following characterization of Riesz bases of exponentials in the space L 2 (Ω): Proposition 2.3.Let Ω ⊂ R d be a bounded, measurable set of positive measure, and let Λ ⊂ R d .The system of exponentials E(Λ) is a Riesz basis in L 2 (Ω) if and only if Λ is a set of both uniqueness and interpolation for the space P W (Ω).
2.4.Stability.We will need the following result which goes back to Paley and Wiener [PW34] about the stability of Riesz bases of exponentials under small perturbations: Proposition 2.4.Let Ω ⊂ R d be a bounded, measurable set of positive measure, and let Λ = {λ n } be a sequence of points in R d such that the system E(Λ) is a Riesz basis in L 2 (Ω).Then there is a constant η For a proof of this result the reader may consult [OU16, Section 4.3].

Convex polytopes.
A set Ω ⊂ R d is called a convex polytope if Ω is the convex hull of a finite number of points.Equivalently, a convex polytope is a bounded set which can be represented as the intersection of finitely many closed halfspaces.
A convex polytope Ω ⊂ R d is said to be centrally symmetric if the set −Ω is a translate of Ω.In this case, there exists a unique point x ∈ R d such that −Ω + x = Ω − x, and we say that Ω is symmetric with respect to the point x.
A zonotope in R d is a set Ω which can be represented as the Minkowski sum of a finite number of line segments, that is, Ω = S 1 + S 2 + • • • + S n where each one of the sets S 1 , S 2 , . . ., S n is a line segment in R d .A zonotope in R d can be equivalently defined as the image of a cube in R n under an affine map from R n to R d .
A zonotope is a convex, centrally symmetric polytope, and all its faces of all dimensions are also zonotopes.In particular, all the faces of a zonotope are centrally symmetric.The following proposition shows that the converse is also true: Proposition 2.5 (see [Sch14, Theorem 3.5.2]).Let Ω be a convex polytope in R d .Then the following conditions are equivalent: (i) Ω is a zonotope (i.e.Ω is the Minkowski sum of finitely many line segments); (ii) Ω is centrally symmetric and all its faces of all dimensions are also centrally symmetric; (iii) all the two-dimensional faces of Ω are centrally symmetric.
Thus, for example, any convex centrally symmetric polygon Ω ⊂ R 2 is a zonotope.

Cylindric sets
3.1.We denote a point in R d = R d−1 × R as (x, y) where x ∈ R d−1 and y ∈ R.
Definition 3.1.A bounded, measurable set Σ ⊂ R d will be called a cylindric set if there exists a bounded, measurable set Π ⊂ R d−1 and a bounded, measurable function In other words, Σ is a cylindric set if Σ ⊂ Π × R and for every x ∈ Π, the set {y : (x, y) ∈ Σ} is an interval of length exactly 1, where the position of this interval is allowed to depend on x.The set Π will be called the base of the cylindric set Σ.
In the special case when the function ϕ in (3.1) is constant, the cylindric set Σ has a cartesian product structure, namely Σ = Π × I where I is an interval of length 1.In this case it is well-known that if there is a set Γ ⊂ R d−1 such that the system E(Γ) is a Riesz basis in L 2 (Π), then the system E(Γ × Z) is a Riesz basis in L 2 (Σ).
We will use the fact that the latter assertion remains valid for arbitrary cylindric sets of the form (3.1), not only for those with a cartesian product structure: Lemma 3.2.Let Σ ⊂ R d be a cylindric set with base Π, and suppose that there is a set The proof of this lemma for arbitrary cylindric sets is similar to the one for cartesian products, which should be well-known.We were not able to find the proof in the literature though, so we include it below for completeness.
3.2.For the proof of Lemma 3.2 we will use the following characterization of the exponential systems E(Λ) that form a Riesz basis in the space L 2 (Ω).
Proposition 3.3.Let Ω ⊂ R d be a bounded, measurable set of positive measure, and let Λ ⊂ R d .The system of exponentials E(Λ) is a Riesz basis in L 2 (Ω) if and only if there exists a constant M such that the following two conditions hold: L 2 (Ω) whenever {c(λ)}, λ ∈ Λ, is a sequence of scalars with only finitely many nonzero elements; We note that condition (i) holds if and only if Λ is a set of interpolation for the space P W (Ω), see [OU16, Sections 4.1, 4.2].It follows from (i) that Λ must be a uniformly discrete set.
If the condition (ii) holds, then Λ is said to be a set of stable sampling for the space P W (Ω), see [OU16, Section 2.5].This condition implies in particular that Λ is a set of uniqueness for the space.
3.3.Proof of Lemma 3.2.We suppose that Σ ⊂ R d is a cylindric set of the form (3.1), and that Γ ⊂ R d−1 is a set such that the system of exponentials E(Γ) is a Riesz basis in L 2 (Π).We must prove that the system E(Γ × Z) is a Riesz basis in L 2 (Σ).We will show that the two conditions (i) and (ii) in Proposition 3.3 are satisfied.
We will use γ to denote an element of Γ, and n to denote an element of Z.
First we check that (i) holds.Let {c(γ, n)} be a sequence of scalars with only finitely many nonzero elements.We must show that γ , where f (x, y) := since E(Z) is an orthonormal basis in L 2 (I(x)) for every x.In turn, this implies where the last inequality holds for a certain constant M = M(Π, Γ) since the system E(Γ) is a Riesz basis in L 2 (Π).This confirms that condition (i) is indeed satisfied.
Next we check that (ii) holds.Let f ∈ L 2 (Σ), then we have f (x, y) e n (y) dy.
This is due to the fact that E(Z) is an orthonormal basis in L 2 (I(x)).It follows that where we denote e γ,n (x, y) := e γ (x)e n (y).Observe that the inequality in (3.2) holds since E(Γ) is a Riesz basis in L 2 (Π).This establishes (ii) and so the lemma is proved.
4. Decomposition of functions with a zonotope spectrum 4.1.Suppose that we are given n vectors u 1 , u 2 , . . ., u n in R d .The origin-symmetric zonotope generated by these vectors is the set This set is the Minkowski sum of the line segments [− 1 2 u j , 1 2 u j ], 1 j n, and so it is indeed a zonotope in R d .
We observe that if the linear span of the vectors u 1 , u 2 , . . ., u n is the whole R d , then the zonotope Ω n has nonempty interior; while if these vectors do not span the whole R d then Ω n is contained in some hyperplane, and it is then a set of measure zero.
Lemma 4.1.Assume that the first n − 1 vectors u 1 , u 2 , . . ., u n−1 span the whole R d , and that we have u n = (0, 0, . . ., 0, 1).Then there is a cylindric set Σ n ⊂ R d whose base Π n is a zonotope in R d−1 , such that the following holds: for some G ∈ P W (Σ n ) and some H ∈ P W (Ω n−1 ).Here we denote by Ω n−1 the origin-symmetric zonotope in R d generated by the vectors u 1 , u 2 , . . ., u n−1 .
(ii) Conversely, for any two functions G ∈ P W (Σ n ) and H ∈ P W (Ω n−1 ), the function F defined by (4.2) belongs to the space P W (Ω n ).
This is an extension to all dimensions of [Wal17, Lemma 3.1] where the result was proved for a convex, centrally symmetric polygon Ω n in two dimensions.In the twodimensional case, the set Σ n was taken to be a parallelogram that shares with Ω n its two edges parallel to the vector u n .In dimensions greater than two, we cannot in general take the set Σ n in Lemma 4.1 to be a parallelepiped, nor any other type of convex polytope inscribed in Ω n in a similar way.Instead, the role of the parallelogram will be played in higher dimensions by the cylindric sets introduced in Definition 3.1.
The assumption in Lemma 4.1 that the first n − 1 vectors u 1 , u 2 , . . ., u n−1 span the whole R d is made so as to ensure that the zonotope Ω n−1 has nonempty interior.4.2.Proof of Lemma 4.1.By identifying the elements F , G and H of the spaces P W (Ω n ), P W (Σ n ) and P W (Ω n−1 ) with the Fourier transforms of functions f , g and h from L 2 (Ω n ), L 2 (Σ n ) and L 2 (Ω n−1 ) respectively, the conditions (i) and (ii) can be reformulated as follows: any function f ∈ L 2 (Ω n ) admits a representation of the form for some g ∈ L 2 (Σ n ) and some h ∈ L 2 (Ω n−1 ); and conversely, for any two functions g ∈ L 2 (Σ n ) and h ∈ L 2 (Ω n−1 ), the function f defined by (4.3) belongs to L 2 (Ω n ).
(Notice that we think of a function from the space L 2 (Ω n ), L 2 (Σ n ) or L 2 (Ω n−1 ) as a function on the whole space R d which is assumed to vanish a.e.outside the set Ω n , Σ n or Ω n−1 respectively.)Let Π n ⊂ R d−1 be the image of Ω n−1 under the map (x, y) → x.If we denote u j = (v j , w j ) where v j ∈ R d−1 and w j ∈ R, then Π n is the origin-symmetric zonotope generated by the vectors v 1 , v 2 , . . ., v n−1 .These vectors span the whole space R d−1 and hence the zonotope Π n has nonempty interior.The zonotope Ω n is the Minkowski sum of Ω n−1 and the line segment [− 1 2 u n , 1 2 u n ].Since we have assumed that u n = (0, 0, . . ., 0, 1) this implies that and in particular we have Let ϕ : Π n → R be any bounded, measurable function satisfying a(x) ϕ(x) b(x), x ∈ Π n .(4.7) (For example, one may take ϕ(x) := a(x).)We define a cylindric set Σ n by It follows from (4.5), (4.7) and (4.8) that Σ n is a subset of Ω n (see Figure 4.1).
Suppose now that two functions g ∈ L 2 (Σ n ) and h ∈ L 2 (Ω n−1 ) are given, and let f be the function defined by (4.3).Then f is supported on the set Conversely, suppose that we are given a function f ∈ L 2 (Ω n ).We will show that f admits a representation of the form (4.3) where g ∈ L 2 (Σ n ) and h ∈ L 2 (Ω n−1 ).

First we define the function h by
and h(x, y) := 0 if x / ∈ Π n .(Notice that it is not necessary to define the values of h on the set of points (x, y) such that x ∈ Π n , y = ϕ(x), as this is a set of measure zero.) The various sets involved in the statement and proof of Lemma 4.1 are illustrated.The larger polygon represents Ω n , the smaller polygon is Ω n−1 , while the shaded region is the cylindric set Σ n .
We observe that there is a constant M such that the nonzero terms in the sum in either (4.9) or (4.10) correspond only to values of k that are not greater than M.This is due to the assumption that f is supported on Ω n .Indeed, one can check using (4.5) and (4.7) that it suffices to take M := max (b(x) − a(x)), x ∈ Π n .This shows that h is a well-defined function in L 2 (R d ).It also follows from (4.5) that h(x, y) = 0 whenever x ∈ Π n and y / ∈ [a(x), b(x)].We conclude that h is supported on Ω n−1 , so h ∈ L 2 (Ω n−1 ).
and hence there is H ∈ P W (Ω n−1 ) such that H(x, y) = c(x, y) − G(x, y) sin(πy) , (x, y) ∈ Λ ′ n−1 . (5.6) Using the fact that sin(πy) = 0 for every (x, y) ∈ Γ n × Z, we can now conclude that the function F defined by (4.2) solves the interpolation problem F (λ) = c(λ), λ ∈ Λ n , and this function belongs to P W (Ω n ).We thus obtain that the set Λ n is also a set of interpolation for P W (Ω n ).This completes the proof of Theorem 5.1.

For
each x ∈ Π n we let S(x) := {y ∈ R : (x, y) ∈ Ω n−1 }. (4.4)By the definition of Π n the set S(x) is nonempty.Since Ω n−1 is closed and convex, the set S(x) is a closed interval.We may therefore denote S(x) := [a(x), b(x)].Since Ω n−1 is a convex polytope, each one of a(x), b(x) is a continuous, piecewise linear function on Π n .(This can be deduced from the representation of Ω n−1 as the intersection of finitely many closed halfspaces.) (4.11) Then g is a function in L 2 (R d ) and (4.3) is satisfied.It remains only to show that g(x, y) = 0 a.e.(x, y) ∈ R d \ Σ n .