Bases of complex exponentials with restricted supports

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arXiv:2210.05707v1 [math.CA] 11 Oct 2022

Dae Gwan Leea,∗, G¨otz E. Pfandera, David Walnutb

aMathematical Institute for Machine Learning and Data Science (MIDS), Katholische

Universit¨at Eichst¨att–Ingolstadt, Goldknopfgasse 7, 85049 Ingolstadt, Germany

bDepartment of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

Abstract

The complex exponentials with integer frequencies form a basis for the space of

square integrable functions on the unit interval. We analyze whether the basis

property is maintained if the support of the complex exponentials is restricted

to possibly overlapping subsets of the unit interval. We show, for example, that

if S1,...,SK⊂[0,1] are ﬁnite unions of intervals with rational endpoints that

cover the unit interval, then there exists a partition of Zinto sets Λ1,...,ΛK

such that SK

k=1{e2πiλ(·)χSk:λ∈Λk}is a Riesz basis for L2[0,1]. Here, χS

denotes the characteristic function of S.

Keywords: complex exponentials, Riesz bases, support restriction, spectrum

2000 MSC: 42C15

1. Introduction and Main Results

For a measurable set S⊂Rand a discrete set Λ ⊂R, we deﬁne E(S, Λ) :=

{e2πiλ(·)χS:λ∈Λ}which is the set of all exponential functions with frequencies

in Λ considered on the domain S. A Riesz basis for a separable Hilbert space H

is a sequence of the form {U en}n∈Z, where {en}n∈Zis an orthonormal basis for

Hand U:H → H is a bijective bounded operator. Equivalently, Riesz bases

are characterized as complete Riesz sequences1(see, e.g., [3, Chapter 3.6]).

Most of the results on exponential bases deal with exponential functions

that are deﬁned on the full domain of the involved space. For instance, if one

speaks of exponential functions in the space L2(S), then functions of the form

t7→ e2πiλt with λ∈R, deﬁned on the full domain S, are usually considered.

In this paper, we consider the situation where exponential functions in L2(S)

are restricted to some diﬀerent subsets of S. More precisely, we are inter-

ested in conditions on S1,...,SK⊂Rand Λ1,...,ΛK⊂Rwhich are neces-

sary/suﬃcient for SK

k=1 E(Sk,Λk) to be a Riesz basis for L2(SK

k=1 Sk).

∗Corresponding author

Email addresses: daegwans@gmail.com (Dae Gwan Lee), pfander@ku.de (G¨otz

E. Pfander), dwalnut@gmu.edu (David Walnut)

1A sequence {fn}n∈Z⊂ H is called a Riesz sequence if there exist constants 0 < A ≤B <

∞such that Akck2

ℓ2≤ k Pn∈Zcnfnk2≤Bkck2

ℓ2for all c={cn}n∈Z∈ℓ2(Z).

Preprint submitted to Elsevier October 13, 2022

As our ﬁrst main result, we prove that if each Sk⊂[0,1) is a ﬁnite union

of intervals with rational endpoints, then one can ﬁnd pairwise disjoint sets

Λ1,...,ΛK⊂Zsuch that SK

k=1 E(Sk,Λk) is a Riesz basis for L2(SK

k=1 Sk).

Theorem 1. Let K, N ∈Nwith K≤N. Let I1,...,IKbe distinct intervals

from [ℓ

N,ℓ+1

N),ℓ= 0,...,N−1, and for each k, let Skbe a union of subcollection

of I1,...,IK, including Ik. Then there exists a permutation2ρ∈SKsuch that

[

k=1

E(Sk, NZ+ρ(k)) =

[

k=1 e2πiλ (·)χSk:λ∈NZ+ρ(k)

is a Riesz basis for L2(SK

k=1 Sk).

We point out that if, for instance, S1= [0,1), S2=··· =SN= [0,1

N),

then necessarily the system SN

k=1 E(Sk, NZ+ρ(k)) with ρ∈SNcannot be a

Riesz basis for L2(SN

k=1 Sk). See Section 2 for some necessary conditions for

k=1 E(Sk,Λk) to be a Riesz basis for L2(SK

k=1 Sk).

Corollary 2. Let S1,...,SK⊂[0,1) be ﬁnite unions of intervals with rational

endpoints. There exist pairwise disjoint sets Λ1,...,ΛK⊂Zsuch that

[

k=1

E(Sk,Λk) =

[

k=1 e2πiλ (·)χSk:λ∈Λk

is a Riesz basis for L2(SK

k=1 Sk). In particular, if SK

k=1 Sk= [0,1), then

Λ1,...,ΛKform a partition of Z.

Theorem 1 relies on the following two auxiliary results.

Lemma 3. For any A∈CK×Kand a binary matrix M∈ {0,1}K×K, there

exists a permutation ρ∈SKwith

det((PρA)⊙M)≥R·det(A)

K!,

where Pρis the permutation matrix associated with ρ(more precisely, Pρ=

[pk,ℓ]K

k,ℓ=1 with pk,ℓ = 1 if ℓ=ρ(k)and pk,ℓ = 0 otherwise), the symbol ⊙

denotes the entrywise multiplication (Hadamard product), and Rdenotes the

number of distinct generalized diagonals3of Mthat contain no zeros.

Consequently, if A∈CK×Kis nonsingular and if M∈ {0,1}K×Khas at

least one generalized diagonal of ones, then there exists a permutation matrix P

with det((P A)⊙M)6= 0.

2We denote by SK=S({1, . . . , K}) the symmetric group of {1, . . . , K}, which consists of

all the K! permutations of 1, . . . , K.

3A generalized diagonal of a K×Kmatrix is a selection of Kcells from the K×Kcells,

such that exactly one cell is selected from each row and each column. For instance, the usual

diagonal is a generalized diagonal.

In the proof of Theorem 1, we will apply Lemma 3 to square submatrices A

of the Fourier matrix WN= [e−2πikℓ/N ]k,ℓ∈ZN.

Remark. Lemma 3 implies that for any invertible matrix A∈CN×Nand a

mask M∈ {0,1}N×Nwith at least one generalized diagonal of ones, there exists

a row permutation of M, say f

M, such that A⊙f

Mis invertible.

The following proposition provides a simple characterization for SK

k=1 E(Sk,Λk)

to be a Riesz basis for L2(SK

k=1 Sk) when Λkare N-periodic sets in Rand Skare

some unions of [ ℓ

N,ℓ+1

N), ℓ∈ZN:= {0,...,N−1}. For notational convenience,

we will use indices k= 0,...,K−1, instead of k= 1,...,K.

Proposition 4. Fix any N∈N. Let c0,...,cK−1∈[0, N )be distinct real

numbers and let Lk⊂ZN,k= 0,...,K−1. Deﬁne L:= SK−1

k=0 Lk,L:= |L|,

and W(c0,L0),...,(cK−1,LK−1):= [wk,ℓ]k=0,...,K−1, ℓ∈L ∈CK×Lwith

wk,ℓ =(e−2πickℓ/N if ℓ∈ Lk,

0if ℓ∈ L\Lk.

Also, deﬁne Sk:= Sℓ∈Lk[ℓ

N,ℓ+1

N)for k= 0,...,K−1, and S:= SK−1

k=0 Sk=

Sℓ∈L [ℓ

N,ℓ+1

N). Then

K−1

[

k=0

E(Sk, NZ+ck) =

K−1

[

k=0 e2πiλ (·)χSk:λ∈NZ+ck(1)

•a frame for L2(S)if and only if the mapping x7→ W(c0,L0),...,(cK−1,LK−1)x

is injective, i.e., the matrix W(c0,L0),...,(cK−1,LK−1)has full rank and K≥

•a Riesz sequence in L2(S)if and only if the mapping x7→ W(c0,L0),...,(cK−1,LK−1)x

is surjective, i.e., the matrix W(c0,L0),...,(cK−1,LK−1)has full rank and K≤

•a Riesz basis for L2(S)if and only if the mapping x7→ W(c0,L0),...,(cK−1,LK−1)x

is bijective, i.e., the matrix W(c0,L0),...,(cK−1,LK−1)is invertible (and K=

L).

In any of the above cases, the optimal lower frame/Riesz bound is given by

Nσ2

min(W(c0,L0),...,(cK−1,LK−1)). Furthermore, if K=L=Nand if the matrix

W(c0,L0),...,(cK−1,LK−1)is invertible, then the dual Riesz basis of (1) in L2[0,1)

is given by

N−1

[

k=0 N

N−1

j=0

e−2πickj/N zj,k χ[j

N,j+1

N)· E([0,1), NZ+ck)

N−1

[

k=0 nN

N−1

j=0

e−2πickj/N zj,k χ[j

N,j+1

N)(·)e2πiλ (·):λ∈NZ+cko,

(2)

where zj,kj,k∈ZN=W(c0,L0),...,(cN−1,LN−1)−1.

Unfortunately, if some endpoints of Skare irrational, there is no convenient

characterization for SK

k=1 E(Sk,Λk) to be a Riesz basis for L2(SK

k=1 Sk). Nev-

ertheless, we are able to prove the following as our second main result.

Theorem 5. Let I1, I2, I3be intervals partitioning [0,1) and let Λ1,Λ2,Λ3be

a partition of Zsuch that for each k= 1,2,3, the system E(Ik,Λk)is a Riesz

basis for L2(Ik). Here, the intervals are allowed to be empty sets, and we set

Λk=∅in the case that Ik=∅. Then

[

k=1

E(Sk,Λk) =

[

k=1 e2πiλ(·)χSk:λ∈Λk

is a Riesz basis for L2(S3

k=1 Sk) = L2[0,1) whenever Sk=Sn∈LkInwith k∈

Lk⊂ {1,2,3}for all k= 1,2,3.

Note that the condition k∈ Lkis equivalent to having Ik⊂Sk. This

condition is actually not necessary but is a convenient assumption. We will

discuss this in more detail in Section 2.

It is easily seen that Theorem 5 does not hold for more than three intervals

(see Example 8 below).

We would like to highlight that the statements of Theorems 1 and 5 have

completely diﬀerent quantiﬁers. Given sets S1,...,SK⊂[0,1) with certain

property, Theorem 1 ﬁnds some pairwise disjoint sets Λ1,...,ΛK⊂Zsuch

that SK

k=1 E(Sk,Λk) is a Riesz basis for L2(SK

k=1 Sk). In contrast, Theorem 5

assumes the sets I1,...,IK⊂[0,1) and Λ1,...,ΛK⊂Zto satisfy that E(Ik,Λk)

is a Riesz basis for L2(Ik), k= 1,...,K, and then shows that SK

k=1 E(Sk,Λk)

is a Riesz basis for L2(SK

k=1 Sk) = L2[0,1) if each Skis a union of subcollection

of I1,...,IK, including Ik.

1.1. Related work

Kozma and Nitzan [5] proved that for any ﬁnite union Sof disjoint inter-

vals in [0,1), there exists a set Λ ⊂Zsuch that E(S, Λ) is a Riesz basis for

L2(S). By adapting the proof technique of [5], Caragea and Lee [2] showed that

if Ik= [ak, bk), k= 1,...,K, are disjoint intervals in [0,1) with the numbers

1, a1,...,aK, b1,...,bKbeing linearly independent over Q, then there exist pair-

wise disjoint sets Λk⊂Z,k= 1,...,K, such that for every K ⊂ {1,...,K}, the

system E(∪k∈K Ik,∪k∈K Λk) is a Riesz basis for L2(∪k∈K Ik). This extends the

result of Kozma and Nitzan by requiring the exponential Riesz basis to possess

a hierarchical Riesz bases property so that every subcollections of I1,...,IK

admit Riesz bases with the corresponding frequency sets, when the endpoints

of I1,...,IKare rationally independent. Conjecture 2 below states a similar

hierarchical result for the case where all the intervals I1,...,IKhave rational

endpoints.

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摘要：

arXiv:2210.05707v1[math.CA]11Oct2022BasesofcomplexexponentialswithrestrictedsupportsDaeGwanLeea,∗,G¨otzE.Pfandera,DavidWalnutbaMathematicalInstituteforMachineLearningandDataScience(MIDS),KatholischeUniversit¨atEichst¨att–Ingolstadt,Goldknopfgasse7,85049Ingolstadt,GermanybDepartmentofMathematicalScie...

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Bases of complex exponentials with restricted supports

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