Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems
aa r X i v : . [ c s . I T ] F e b UNIVERSAL SPATIOTEMPORAL SAMPLING SETS FORDISCRETE SPATIALLY INVARIANT EVOLUTIONARY SYSTEMS
SUI TANG
Abstract.
Let ( I, +) be a finite abelian group and A be a circular convolutionoperator on ℓ ( I ). The problem under consideration is how to construct minimalΩ ⊂ I and l i such that Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } is a frame for ℓ ( I ),where { e i : i ∈ I } is the canonical basis of ℓ ( I ). This problem is motivatedby the spatiotemporal sampling problem in discrete spatially invariant evolutionsystems. We will show that the cardinality of Ω should be at least equal to thelargest geometric multiplicity of eigenvalues of A , and we consider the universalspatiotemporal sampling sets (Ω , l i ) for convolution operators A with eigenvaluessubject to the same largest geometric multiplicity. We will give an algebraiccharacterization for such sampling sets and show how this problem is linked withsparse signal processing theory and polynomial interpolation theory. Introduction
Let ( I, +) be a finite abelian group, we denote by ℓ ( I ) the space of square sum-mable complex-valued functions defined on I with the inner product given by h f , g i = X i ∈ I f ( i ) g ( i ) , for f , g ∈ ℓ ( I ) . We denote by { e i : i ∈ I } the canonical basis of ℓ ( I ), where e i is the characteristicfunction of { i } ⊂ I . Definition 1.1.
The operator A on ℓ ( I ) is called a circular convolution operatorif there exists a complex-valued function a defined on I such that ( Af )( k ) := a ∗ f = X i ∈ I a ( i ) f ( k − i ) , ∀ f ∈ ℓ ( I ) and k ∈ I. The function a is said to be the convolution kernel of A . Mathematics Subject Classification.
Primary 94A20, 94A12, 42C15, 15A29.
Key words and phrases.
Spatiotemporal sampling; Finite frames; Discrete Fourier Transform;Interpolation; Full spark matrix.This research was supported in part by NSF Grant DMS-1322099.
Recall that a finite sequence { g j } ⊂ ℓ ( I ) is said to be a frame for ℓ ( I ), if thereexist positive numbers A and B such that A k f k ≤ X j |h f , g j i| ≤ B k f k , ∀ f ∈ ℓ ( I ) . A finite sequence { g j } ⊂ ℓ ( I ) is said to be complete if the vector space spannedby { g j } is ℓ ( I ), i.e, if a function g ∈ ℓ ( I ) satisfies h g , g j i = 0 , ∀ g j , then g = . In the finite dimensional scenario, it is a standard result of finite frametheory that a finite sequence { g j } ⊂ ℓ ( I ) is a frame for ℓ ( I ) if and only if it iscomplete.Let A be a circular convolution operator on ℓ ( I ). We investigate the followingproblem: Problem 1.2.
Let Ω be a proper subset of I . We assign a finite nonnegative integer l i to each i ∈ Ω . Under what conditions on Ω and l i is the sequence (1) Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } a frame for ℓ ( I ) ? In what way does the choice of Ω and l i depend on the operator A ? Problem 1.2 is motivated by the spatiotemporal sampling and reconstruction prob-lem arising in spatially invariant evolution systems [1, 2, 3, 6, 7, 8, 21, 23, 26, 27]. Let f ∈ ℓ ( I ) be an unknown vector that is evolving under the iterated actions of a con-volution operator A , such that at time instance t = n it evolves to be A n f . We callsuch a discrete evolution system spatially invariant . This kind of evolution systemmay arise as a discretization of a physical process; for example, the diffusion processmodeled by the heat equations. We are interested in recovering the initial state f . Inpractice, a large number of sensors are distributed to monitor a physical process suchas pollution, temperature or pressure [5]. The sensor nodes obtain spatiotemporalsamples of physical fields over the region of interest. Increasing the spatial samplingrate is often much more expensive than increasing the temporal sampling rate sincethe cost of the sensor is more expensive than the cost of activating the sensor [21].Given the different costs associated with spatial and temporal sampling, it would bemore economically efficient to recover the initial state f from spatiotemporal samplesby using fewer sensors with more frequent acquisition. Following the notation de-fined in Problem 1.2, Ω will be the locations of sensors. At each i ∈ Ω, f is sampledat time instances t = 0 , · · · , l i . Denote by A ∗ the adjoint operator of A . By Lemma1.2 in [9], f can be recovered stably from these spatiotemporal samples if and onlyif Y = { e i , A ∗ e i , · · · , ( A l i ) ∗ e i : i ∈ Ω } is a frame for ℓ ( I ). Note that A ∗ is alsoa circular convolution operator, and its convolution kernel is ¯ a . Thus, Problem 1.2is equivalent to a signal recovery problem in a discrete spatially invariant evolutionprocess. Contribution.
In this paper, we mainly study the cases I = Z d and I = Z d × Z d , where Z d = { , , · · · , d − } denotes the finite group of integers modulo d for a positive integer d and Z d × Z d is the product group. In the case of I = Z d ,we show that the cardinality of Ω should be at least the largest geometric multi-plicity of eigenvalues of A . We characterize the minimal universal spatiotemporalsampling sets (Ω , l i ) for circular convolution operators with eigenvalues subject tothe same geometric multiplicity. Specifically, we show how full spark frames builtfrom the discrete Fourier matrices correspond to minimal universal constructionsand demonstrate a close connection with the sparse signal processing theory. Inthe case of I = Z d × Z d , we show that finding nontrivial full spark frames from 2DFourier matrices is less favorable than its 1D sibling. In practice, there are manysituations in real applications where the convolution operators have various typesof symmetries in frequency response. For these special cases, we employ ideas frominterpolation theory of multivariate polynomials to provide specific constructions ofminimal sets Ω and l i such that Problem 1.2 is solved. The techniques we developedcan be adapted to the general finite abelian group case; see related discussions inSection 4.1.2. Related Work.
Our work is closely related to [24], in which the authors havestudied the universal spatial sensor locations for discrete bandlimited space; in somesense, finding universal spatiotemporal sampling sets for convolution operators witheigenvalues subject to the same largest geometric multiplicity in our problem, is anal-ogous to finding universal spatial sampling sets for discrete bandlimited space B J that is subject to the same cardinality of J in [24]. However, we do not make spar-sity assumptions on the signal space. Instead, we seek sub-Nyquist spatial samplingrate, but want to compensate the insufficient spatial sampling rate by oversamplingin time.Our work also has similarities with [21, 23, 26, 27]. These works studied the spa-tiotemporal sampling and reconstruction problem in the continuous diffusion field f ( x, t ) = A t f ( x ), where f ( x ) = f ( x,
0) is the initial signal and A t is the time varyingGaussian convolution kernel determined by the diffusion rule. In [6, 7], Aldroubiand his collaborators develop the mathematical framework of Dynamical Sampling to study the spatiotemporal sampling and reconstruction problem in discrete spa-tially invariant evolution processes, which can be viewed as a discrete version ofdiffusion-like processes. Our results can be viewed as an extension of [6, 7] to theirregular setting and the algebraic characterization given in [6] can be viewed as aspecial case of our characterization for the unions of periodic constructions.Other similar works are about the generalized sampling problems [4, 17, 18, 19,25] and the distributed sampling problems [11, 13, 14, 20, 29]. For example, in[17], the authors work in a U -invariant space, and study linear systems { L j : j =1 , · · · , s } such that one can recover any f in the U -invariant space by uniformly SUI TANG downsampling the functions { ( L j f ) : j = 1 , · · · , s } , i.e. taking the generalizedsamples { ( L j f )( M α ) } α ∈ Z d ,j =1 , ··· ,s .1.3. Preliminaries and Notation.
In the following, we use standard notation.By N , we denote the set of all positive integers. For m, n ∈ N , we use gcd ( m, n ) todenote their greatest common divisor. The linear space of all column vectors with M complex components is denoted by C M . The linear space of all complex M × N matrices is denoted by C M × N . For a matrix B ∈ C M × N , its transpose is denoted by B T and its conjugate-transpose by B ∗ . The null space of B is denoted by ker( B ).For convenience, both the index of rows and columns for a matrix B start at 0.If B ∈ C M × N and S ⊂ { , , · · · , N − } , then let B S denote the submatrix of B obtained by selecting row vectors of B corresponding to S . For example, B { } meansthe first row vector of B . We use I M × M to denote the the identity matrix in C M × M .If B is a diagonalizable matrix, we denote by M B the largest geometric multiplicityamong all eigenvalues of B and by N B the number of distinct eigenvalues of B . Wedenote by the null vector in the vector space. For a vector z = ( z i ) ∈ C M , the M × M diagonal matrix built from z is denoted by diag ( z ). The ℓ ∞ norm of z isdefined by k z k ℓ ∞ = max i | z i | .For d ∈ N , we use Z d = { , , · · · , d − } to denote the finite group of integersmodulo d . If d = mJ ( m > m Z d denotes the subgroup of Z d obtained byselecting the elements divisible by m . Let c be an integer between 0 and m − m Z d + c = { c + i : i ∈ m Z d } denotes the translation of m Z d by c units,where “ + ” is addition modulo d . We denote by S the unit circle in the plane.With ω d = e − πid denoting the primitive d th root of unity, we define the normalizeddiscrete Fourier matrix via F d = √ d ( ω jkd ) j,k =0 , ··· d − and denote by ˆf the unnormalizeddiscrete Fourier transform of f . For a set Ω ⊂ I , we use | Ω | to denote its cardinality.If Ω ⊂ { , · · · , M − } , then the subsampling operator S Ω is the linear operator thatmaps z ∈ C M to a vector in C M obtained by modifying the entries of z whose indicesare not in S to be zero. Definition 1.3.
A set Λ ⊂ I is said to be a level set of a ∈ ℓ ( I ) if a ≡ c on Λ andfor all i ∈ I − Λ , a ( i ) = c . Definition 1.4.
Let Ω ⊂ I . Then the set Ω is said to be an admissible set forthe convolution operator A if there exists l i ∈ N for each i ∈ Ω such that Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } is a frame for ℓ ( I ) . Definition 1.5.
Let v ∈ C d and B ∈ C d × d ; then the B -annihilator of v is themonic polynomial of smallest degree among all the monic polynomials p such that p ( B ) v = . Definition 1.6.
For B = ( b ij ) ∈ C M × N and C = ( c ij ) ∈ C L × K , the Kroneckerproduct of B and C is defined as the M L × N K block matrix (2) B ⊗ C = b , C · · · b ,N − C ... ... b M − , C · · · b M − ,N − C . The row indices and column indices of B ⊗ C can be naturally labeled by means ofthe product group Z M × Z N . More precisely, in the row indices ( i , i ), i is intendedto denote on which block row an entry is situated(the row of B ), while the index i denotes more specifically on which position of its block row it is situated (the row of C ). The same rule applies on the column indices. For example, the entry of B ⊗ C at position (( i , i ) , ( j , j )) is b i ,j c i ,j .1.4. Organization.
The rest of paper is as follows: in Section 2, we present mainresults for the case I = Z d . The algebraic characterization stated in Proposition 2.1is the key of solving Problem 1.2. In Section 3, we consider the case I = Z d × Z d . Weinvestigate the similarities and differences with 1D case. A discussion of the generalabelian group case is presented in Section 4. Finally, we summarize our paper inSection 5. 2. Single Variable Case I = Z d In this section, we consider Problem 1.2 for I = Z d . Motivated by the applicationin the spatiotemporal sampling problem, we would like to use the least number ofsensors to save the budget. So the important issues of Problem 1.2 we want toaddress are:(1) Given a circular convolution operator A , what is the minimal cardinality ofΩ such that Y can be a frame for ℓ ( I )?(2) Can we find Ω with minimal cardinality such that Y is a frame? If so, howto determine l i for each i ∈ Ω?In other words, we seek to find minimal admissible sets Ω for A . Suppose that A is given by a convolution kernel a ∈ ℓ ( Z d ). We know that A admits the spectraldecomposition(3) A = F ∗ d diag ( ˆa ) F d . This fact follows from the convolution theorem:(4) c Af = diag ( ˆa )ˆ f , for f ∈ ℓ ( I ) . In Proposition 2.1, we will use an algebraic characterization based on the spec-tral decomposition of A to answer the questions we just proposed. Proposition 2.1explains the frame properties for the examples below: Consider the following twoconvolution kernels on ℓ ( Z ) which define A and A respectively: ˆa = [1 , , , T , ˆa = [1 , , , T . SUI TANG
For the convolution operator A , Proposition 2.1 shows that for any i ∈ Z , Y = { e i , Ae i , A e i , A e i } is a frame for ℓ ( Z ). For the convolution operator A ,Proposition 2.1 implies that the cardinality of Ω should be at least 2. If we chooseΩ = { , } , then Y = { e i , Ae i : i ∈ Ω } is a frame for ℓ ( Z ). However, if we chooseΩ = { , } , then no matter how large l i is, Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } is nevera frame for ℓ ( Z ).Our characterization can be viewed as a special case of Theorem 2.5 in [9], butfor convenience we still state it here and prove it. Proposition 2.1.
Let A = F ∗ d diag ( ˆa ) F d be a circular convolution operator with akernel a ∈ ℓ ( Z d ) , { λ , · · · , λ N A } be the set of distinct eigenvalues of A and { P k : k =1 , · · · N A } be the corresponding eigenspace projections of diag ( ˆa ) . Suppose Ω ⊂ Z d ,we define f i = F d e i . Let Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } .(1) If Y is a frame for ℓ ( Z d ) , then for each k , { P k ( f i ) : i ∈ Ω } is a frame for therange space E k of P k . Hence it is necessary to have | Ω | ≥ max k =1 , ··· ,N A dimE k .(2) For each i ∈ Ω , we define by r i the degree of the A -annihilator of e i . If foreach i ∈ Ω and each k , l i ≥ r i − and { P k ( f i ) : i ∈ Ω } is a frame for therange space E k of P k , then Y is a frame for ℓ ( Z d ) .(3) Let E be the linear span of vectors in Y . If for each i ∈ Ω and each k , A l i +1 e i ∈ E and { P k ( f i ) : i ∈ Ω } is a frame for the range space E k of P k ,then Y is a frame for ℓ ( Z d ) .Proof. Applying F d on Y , we obtain a new set of vectors˜ Y = { f i , diag ( ˆa ) f i , · · · , ( diag ( ˆa )) l i f i : i ∈ Ω } . Since F d is unitary, ˜ Y is a frame for ℓ ( Z d ) if and only if Y is a frame for ℓ ( Z d ).(1) Since we are working with finite dimensional spaces, for each k , to show that { P k ( f i ) : i ∈ Ω } is a frame for E k , it suffices to show it is complete. Nowsuppose that g ∈ E k is orthogonal to vectors in { P k ( f i ) : i ∈ Ω } , then h g , diag ( ˆa ) l f i i = h g , P k diag ( ˆa ) l f i i = λ lj h g , P k ( f i ) i = 0for l = 0 , · · · , l i and all i ∈ Ω, which means that g is orthogonal to ˜ Y .In the above identities, we use the facts that P k g = g , P k = P k and P k diag ( ˆa ) = λ k P k . Since Y is a frame, ˜ Y is a frame and we concludethat g = . Therefore, { P k ( f i ) : i ∈ Ω } is a frame for E k .(2) Given the conditions that the set of vectors { P k ( f i ) : i ∈ Ω } form a framefor E k for k = 1 , · · · , N A and l i ≥ r i − i ∈ Ω, we will show that ˜ Y iscomplete on ℓ ( Z d ). Assume that g ∈ ℓ ( Z d ) is orthogonal to ˜ Y , i.e., h g , diag ( ˆa ) l f i i = for i ∈ Ω and l = 0 , · · · , l i . Since l i ≥ r i − r i is the degree of the A -annihilator of e i for i ∈ Ω, diag ( ˆa ) l f i can be written as a linear combination of vectors in ˜ Y for any l ∈ N and we obtain h g , diag ( ˆa ) l f i i = 0for i ∈ Ω and l = 0 , . . . , N A −
1. Note that N A P k =1 P k is the identity map, wehave the identity(5) h g , diag ( ˆa ) l f i i = N A X k =1 h P j g , diag ( ˆa ) l f i i = N A X k =1 λ lk h g , P k f i i = 0for i ∈ Ω and l = 0 , . . . , N A −
1. Hence we obtain the following system oflinear equations for each i ∈ Ω: · · · λ λ · · · λ N A ... ... ... λ N A − λ N A − · · · λ N A − N A h g , P f i ih g , P f i i ... h g , P N A f i i = . Since the elements of { λ k : k = 1 , · · · , N A } are distinct, we conclude that h g , P k f i i = h P k g , P k f i i = 0for k = 1 , · · · , N A and i ∈ Ω. Since { P k ( f i ) : i ∈ Ω } is a frame for E k , weconclude that P k g = for k = 1 , · · · , N A . Therefore, g = N A P k =1 P k g = . Weprove that ˜ Y is complete, hence it is a frame and Y is a frame.(3) The proof is similar to (2). (cid:3) Proposition 2.1 gives a lower bound of the largest geometric multiplicity of eigen-values of A for the cardinality of an admissible set Ω. It also tells us in what waythe choice of Ω depends on A and how to determine l i . In [6], the authors haveconsidered the case when a is a typical low pass filter, i.e, ˆa is real, symmetric andstrictly decreasing on { , , · · · , d − } . They prove that Y is not a frame if we chooseΩ = m Z d and l i = m −
1, and give explicit formulas to construct Ω ⊂ Z d − Ω suchthat Y ∪ { e i : i ∈ Ω } is a frame. In this case, by counting the largest geometricmultiplicity of eigenvalues of A , we know that the cardinality of an admissible set Ωshould be at least 2. We use Proposition 2.1 to give a complete characterization ofall minimal admissible sets below. Corollary 2.2.
Assume d is odd and a ∈ ℓ ( Z d ) so that ˆa is symmetric and strictlydecreasing on { , , · · · , d − } . Let A be the circular convolution operator with theconvolution kernel a . Suppose that Ω = { i , i } ⊂ Z d , then Y = { e i , Ae i , · · · , A d − e i : i ∈ Ω } SUI TANG is a frame for ℓ ( Z d ) if and only if gcd ( | i − i | , d ) = 1 .Proof. First, we check that the degree of the A -annihilator of e i and e i is d +12 .By the symmetry and monotonicity condition of ˆa , we know that the orthogonaleigenspace projections of diag ( ˆa ) consist of { P j : j = 0 , · · · , d − } , where P j is theorthogonal projection onto the subspace spanned by { e j , e d − j } for j = 1 , · · · , d − and P is the projection onto the span of e . By (2) of Proposition 2.1, it suffices toshow the following 2 × " ω i jd ω i jd ω i ( d − j ) d ω i ( d − j ) d is invertible for j = 1 , · · · , d − . We compute their determinants, and know that theyare invertible if and only if ω ( i − i ) jd = 1 for j = 1 , · · · , d − . Hence, it is equivalentto the condition gcd ( | i − i | , d ) = 1. (cid:3) Recall that we denote by M A the largest geometric multiplicity of eigenvaluesof A . We denote the class of circular convolution operators whose eigenvalues aresubject to the same largest geometric multiplicity L A L = { A ∈ C d × d : Af = a ∗ f for some a ∈ ℓ ( Z d ), M A = L } . By Proposition 2.1, an admissible set Ω ⊂ Z d for any A ∈ A L must contain at least L elements. Since the spectral projections among A ∈ A L could be very different,minimal admissible sets for different A could be different. But can we find a minimaluniversal admissible set Ω with | Ω | = L for any A ∈ A L and then determine l i for i ∈ Ω? It turns out that this question is closely related to full spark frames in thesparse signal processing theory. We will show this connection but first we need thefollowing definition.
Definition 2.3.
Let B ∈ C M × N . Then the spark of B is the size of the smallestlinearly dependent subset of columns, i.e, Spark ( B ) = min {|| f || : Bf = , f = } . If M ≤ N , B is said to be full spark if Spark ( B ) = M + 1 . Equivalently, M × N full spark matrices have the property that every M × M submatrix is invertible. Theorem 2.4.
Let Ω ⊂ Z d with | Ω | = L . Recall that N A is the number of distincteigenvalues of A . Then, for any A ∈ A L , (7) Y = { e i , Ae i , · · · , A N A − e i : i ∈ Ω } is a frame for ℓ ( Z d ) if and only if ( F d ) Ω is a L × d full spark matrix.Proof. For every A ∈ A L , we assume its convolution kernel is a . Recall Definition 1.3,let { Λ j : j = 1 , · · · , N A } be the level sets of ˆa . Then we have max j =1 , ··· ,N A | Λ j | = L .Also let { P Λ j : j = 1 , · · · , N A } be the set of eigenspace projections of diag ( ˆa ), i.e, P Λ j is the projection onto the subspace E j spanned by { e l : l ∈ Λ j } .We first prove the “ if ” part. Let O Ω = { f i = F d e i : i ∈ Ω } . We form a | Λ j | × L matrix with column vectors given by { P Λ j f i : f i ∈ O Ω } . We claim its rank is | Λ j | ,which implies { P Λ j f i : f i ∈ O Ω } is a frame for E j . Using the fact that F d = F Td , itis easy to see that the transpose of the submatrix built from { P Λ j f i : f i ∈ O Ω } is a L × | Λ j | submatrix of ( F d ) Ω . Since ( F d ) Ω is a L × d full spark matrix, any L × | Λ j | submatrix of ( F d ) Ω has rank | Λ j | . The claim follows from this observation. Finally,we compute the degree of the A -annihilator for e i which equals to N A . By (2) ofProposition 2.1, Y defined in (7) is a frame for ℓ ( Z d ).Conversely, suppose that Y defined in (7) is a frame for any A ∈ A L . By (1) ofProposition 2.1, { P Λ j f i : i ∈ Ω } is a frame for E j . Since the level sets { Λ j : j =1 , · · · , N A } of a convolution kernel ˆa for A ∈ A L can have all possibilities of disjointpartitions of Z d satisfying max j | Λ j | = L , and using the same embedding trick withthe “if” part, we know that any L ≤ L column vectors of ( F d ) Ω must be linearlyindependent. In other words, ( F d ) Ω is a full spark matrix. (cid:3) For example, let us consider the universal minimal constructions for the class ofconvolution operators A . Assume Ω = { i , i } ⊂ Z d . It is direct to check that( F d ) Ω is a 2 × d full spark matrix if and only if gcd ( | i − i | , d ) = 1. Thus, we get animmediate corollary which generalizes Corollary 2.2. Corollary 2.5.
Let
Ω = { i , i } ⊂ Z d . Then, for any A ∈ A , Y = { e i , Ae i , · · · , A N A − e i : i ∈ Ω } is a frame for ℓ ( Z d ) if and only if gcd ( | i − i | , d ) = 1 . Note that F d is a Vandemonde matrix, if we choose Ω = { , , · · · L − } , then( F d ) Ω is full spark ([10, Lemma 2]). Thus, we have the following corollary. Corollary 2.6.
Let ≤ L ≤ d be an integer and Ω = { , , · · · , L − } . Then, forany A ∈ A L , Y = { e i , Ae i , · · · , A N A − e i : i ∈ Ω } is a frame for ℓ ( Z d ) . Full spark matrices play an important role in applications like sparse signal pro-cessing, data transmission and phaseless reconstructions. There is a pressing needfor deterministic constructions of full spark matrix ([10, 24] and the related work in[15, 16]) . In [10], the authors have considered the problem of finding Ω such that( F d ) Ω is a full spark matrix. The following useful properties of full spark matricescan be found in [10, Theorem 4]. Theorem 2.7.
Let Ω ⊂ Z d . If ( F d ) Ω is a full spark matrix, then so is the submatrixof F d built from the rows indexed by(1) any translations of Ω , Ω + r = { r + i : i ∈ Ω } . (2) r Ω = { ri : i ∈ Ω } where r is coprime to d .(3) the complement of Ω in Z d . In general, it is challenging to give a characterization of finding deterministic fullspark matrices from rows of F d . In [12], the authors thought that the difficulty maycome from the existence of nontrivial subgroups of Z d . In the 1920s, Chebotar¨ e vgave the first characterization to the special case when d is a prime. We can find anintroduction and a proof of this result in the survey paper [31]. Theorem 2.8 (Chebotar¨ e v) . Let d be prime. Then every square submatrix of F d isinvertible. Later [10] and [24] generalized the techniques developed by Chebotar¨ e v and gavea characterization to the special case when d is a power of prime. We list the resultsin [10] here. To understand their results, we need the definition in [10] here. Definition 2.9.
We say a subset Ω ⊂ Z d is uniformly distributed over the divisorsof d if, for every divisor m of d , the m cosets of h m i partition Ω into subsets, eachof size ⌊ | Ω | m ⌋ or ⌈ | Ω | m ⌉ . For example, when d is prime, every subset of Z d is uniformly distributed over thedivisors of d . Then { , , · · · , L − } is uniformly distributed over the divisors of d for any L ≤ d . The following characterization can be found in [10, Theorem 9]. Theorem 2.10.
Let d be a prime power. We select rows indexed by Ω ⊂ Z d from F d to build the submatrix ( F d ) Ω . Then ( F d ) Ω is full spark if and only if Ω is uniformlydistributed over the divisors of d . As a consequence of Theorem 2.10 and Theorem 2.8, we can state an immediatecorollary.
Corollary 2.11.
Let Ω ⊂ Z d with | Ω | = L and Y = { e i , Ae i , · · · , A N A − e i : i ∈ Ω } . (1) If d is prime, then, for any A ∈ A L , Y is a frame for ℓ ( Z d ) .(2) Assume d is some power of prime. Then, for any A ∈ A L , Y is a frame for ℓ ( Z d ) if and only if Ω is uniformly distributed over the divisors of d . The proof of Theorem 2.10 in [10] also shows the necessity of having row indicesuniformly distributed over the divisors of d in general. However, it remains opento prove that it is a sufficient condition for arbitrary d . So far, we only know theselections of consecutive rows and their variants given in Theorem 2.7 produce fullspark matrices for any F d .In the spatiotemporal sampling problem of the heat diffusion process, a commonapproach is to place the sensors indexed by Ω in a periodic nonuniform way ([21, 23,26, 27] and reference therein). Let m be a positive divisor of d such that d = mJ ,and we also investigate when a union of periodic sets Ω = { m Z d + r, r ∈ W ⊂ Z m } is an admissible set for an operator A defined by a convolution kernel a ∈ ℓ ( Z d ).We ask similar questions in this case: what is the minimal cardinality of W suchthat Ω is an admissible set for A and how to find such W . It turns out that theanswers are also related to the geometric multiplicity of eigenvalues of A . We define a k ∈ ℓ ( Z m ) by(8) a k = [ ˆa ( k ) , ˆa ( k + J ) , · · · , ˆa ( k + ( m − J )] T , and D k ∈ C m × m by(9) D k = diag ( a k )for k = 0 , · · · , J − . Let B L = { A ∈ C d × d : Af = a ∗ f for some a ∈ ℓ ( Z d ), max k =0 , ··· ,J − M D k = L } . We are going to show that for A ∈ B L , the minimal cardinality of W is L and wealso find universal admissible unions of periodic sets Ω with | W | = L for all A ∈ B L . Theorem 2.12.
Assume d is a positive integer and m is a positive divisor of d suchthat d = mJ ( m > . Assume that W ⊂ Z m consists of L < m elements. Let
Ω = { m Z d + r : r ∈ W } ⊂ Z d and (10) Y = { e i , Ae i , · · · , A m − e i : i ∈ Ω } . (1) If Y is a frame of ℓ ( Z d ) for an operator A ∈ B L , then | W | ≥ L . Moreover,if Ω is admissible for an operator A ∈ B L , then | W | ≥ L .(2) Y is a frame for ℓ ( Z d ) for any A ∈ B L if and only if the submatix ( F m ) W is a | W | × m full spark matrix.Proof. (1) Assume A ∈ B L and its convolution kernel is a . Let f ∈ ℓ ( Z d ) andassume that it is orthogonal to Y .For a fixed r ∈ W , recall the definition of subsampling operator definedin Subsection 1.3, we let y s,r = S m Z d + r (( A s ) ∗ f ) for s = 0 , , · · · , m −
1. Ifwe take the discrete Fourier transform on y s,r , and use Poisson SummationFormula and the convolution theorem, then we can obtain identities(11) ˆy s,r ( k ) = 1 m m − X l =0 e πir ( k + Jl ) d ˆa ( k + J l ) s ˆf ( k + J l ) , k = 0 , · · · , J − s = 0 , · · · , m −
1. For k = 0 , , · · · , J −
1, we define A m,k = ··· ˆa ( k ) ˆa ( k + J ) ··· ˆa ( k +( m − J ) ... ... ... ˆa m − ( k ) ˆa m − ( k + J ) ··· ˆa m − ( k +( m − J ) and h r,k = e πirkd e πir ( k + J ) d ... e πir ( k +( m − J ) d . We also define y ( r ) k = ˆy ,r ( k ) ˆy ,r ( k ) ... ˆy m − ,r ( k ) and f k = ˆf ( k ) ˆf ( k + J ) ... ˆf ( k +( m − J ) . For a fixed k and r , we put identities (11) with s = 0 , , · · · , m − m y ( r ) k = A m,k diag ( h r,k ) f k . By the assumption that f is orthogonal to Y , for each s , y s,r = and hence ˆy s,r = . So y ( r ) k = , we conclude that f k ∈ ker( A m,k diag ( h r,k )). Let r take values in all elements of W and k = 0 , , · · · , J −
1, we see that f = ifand only if(13) \ r ∈ W ker( A m,k diag ( h r,k )) = { } for k = 0 , · · · J −
1. Let ker( A m,k ) ⊥ denote the orthogonal complement ofker( A m,k ) in ℓ ( Z m ), using the basic linear algebra, we know that showing(13) is equivalent to showing(14) X r ∈ W (ker( A m,k diag ( h r,k ))) ⊥ = ℓ ( Z m ) . for k = 0 , · · · J − a k for the kernel a in (8), we let { Λ j,k : j =1 , · · · , n k } be the level sets of a k . Note that its complex conjugate a k hasthe same level sets with a k . Let { P Λ j,k : j = 1 , · · · , n k } be the orthogonalprojections determined by { Λ j,k : j = 1 , · · · , n k } , i.e, P Λ j,k is the orthogonalprojection onto the subspace of ℓ ( Z m ) spanned by { e i ∈ ℓ ( Z m ) : i ∈ Λ j,k } .Let v = [1 , , · · · , T ∈ ℓ ( Z m ), observing that A m,k is a Vandermondematrix, it is not difficult to see that { P Λ j,k v : j = 1 , · · · , n k } is an orthogonalbasis for ker( A m,k ) ⊥ . Next, using the relationker( A m,k diag ( h r,k )) ⊥ = diag ∗ ( h r,k ) ker( A m,k ) ⊥ , we let b r = F m e r for r ∈ W and then we can see that { P Λ j,k b r : j =1 , · · · , n k } is an orthogonal basis for ker( A m,k diag ( h r,k )) ⊥ . Hence, for each k , showing (14) is equivalent to showing(15) { P Λ j,k b r : r ∈ W, j = 1 , · · · , n k } is complete on ℓ ( Z m ). Note that { P Λ j,k : j = 1 , · · · , n k } is a set of pairwiseorthogonal projections, (15) is true if and only if { P Λ j,k b r : r ∈ W } is complete on the range space E j,k of P Λ j,k for j = 1 , · · · , n k . Now let ussummarize what we have proved: assume that f ∈ ℓ ( Z d ) and it is orthogonalto Y , then f = if and only if for each k = 0 , , · · · , J − { P Λ j,k b r : r ∈ W } is complete on the range space E j,k of P Λ j,k for j = 1 , · · · , n k . By the defini-tion of B L , max j,k dim ( E j,k ) = L . Given the condition that Y is a frame for ℓ ( Z d ), we conclude that | W |≥ L .We point out that the condition l i = m − { m Z d + r, r ∈ W } is an admissible set for an operator A ∈ B L ,then | W |≥ L . Since if Y defined in (10) is not a frame for ℓ ( Z d ), then nomatter how large we increase each l i for i ∈ Ω, the new obtained Y will neverbe a frame. This fact follows from the special structure of Vandermondematrix.(2) Using the characterization summarized in (16) and the same argument as inthe proof of Theorem 2.4, Y defined in (10) is a frame for any A ∈ B L if andonly if ( F m ) W is full spark. (cid:3) Remark.
Proposition 3.1 in [6] says that W = { } will be an admissible set for all A ∈ B , which can be viewed as this theorem’s special case. In fact, this theoremshows that any W ⊂ Z m with | W | = 1 is an admissible set for B .As an immediate corollary, we get Corollary 2.13.
Suppose we have the same settings with Theorem 2.12.(1) If W = { , , · · · , L − } , then for any A ∈ B L , Y is a frame for ℓ ( Z d ) .(2) If m is prime, then for any A ∈ B L , Y is a frame for ℓ ( Z d ) .(3) If m is a power of prime and W is uniformly distributed over the divisor of m , then for any A ∈ B L , Y is a frame for ℓ ( Z d ) . Since m is a divisor of d , we can choose m to be prime or some power of a prime.If this is the case, we immediately know how to construct all possible W to give anadmissible union of periodic set Ω for B L .3. Two Variable Case I = Z d × Z d In this section, we consider the case I = Z d × Z d , which is the product group of twoidentical groups Z d . Suppose A is a circular convolution operator defined by a con-volution kernel a ∈ ℓ ( Z d × Z d ). One natural way to consider Problem 1.2 in the twovariable setting is viewing it as a single variable setting since we can map Z d × Z d to Z d by sending ( i, j ) to di + j . Under this identification, the operator A correspondsto a linear operator ˜ A acting on ℓ ( Z d ) and ˜ A = ( F d ⊗ F d ) ∗ diag ( ˜ˆa )( F d ⊗ F d ), where ˜ˆa is the image of ˆa in ℓ ( Z d ) under the above identification. Using the labelling method described in Subsection 1.3, we state a similar version of Proposition 2 . Proposition 3.1.
Let A be a circular convolution operator with a kernel a ∈ ℓ ( Z d × Z d ) and { Λ k : k = 1 , · · · , N A } be the level sets of ˆa . Suppose Ω ⊂ Z d × Z d . Let Y = { e j ,j , Ae j ,j , · · · , A l j ,j e j ,j : ( j , j ) ∈ Ω } .(1) If Y is a frame for ℓ ( Z d × Z d ) , then for each k , the submatrix of F d × F d builtfrom rows indexed by Λ k and columns indexed by Ω has rank | Λ k | . Hence itis necessary to have | Ω | ≥ max k | Λ k | .(2) For each ( j , j ) ∈ Ω , we define by r j ,j the degree of the A -annihilator of e j ,j . If for each ( j , j ) ∈ Ω and each k , l j ,j ≥ r j ,j − and the submatrixof F d × F d built from rows indexed by Λ k and columns indexed by Ω has rank | Λ k | , then Y is a frame for ℓ ( Z d × Z d ) .(3) If for each ( j , j ) ∈ Ω and each k , A l j ,j +1 e j ,j is in the space spanned byY and the submatrix of F d × F d built from rows indexed by Λ k and columnsindexed by Ω has rank | Λ k | , then Y is a frame for ℓ ( Z d × Z d ) . Similar to the one variable case, the problem of finding a common minimal ad-missible set Ω for 2D convolution operators with eigenvalues subject to the samelargest geometric multiplicity is equivalent to finding full spark matrices from rowsof F d ⊗ F d indexed by Ω. Unlike F d , F d ⊗ F d is not a Vandermonde matrix. In thecase of F d , the submatrix built from any consecutive L ≤ d rows of F d is full spark.This fact follows from the Vandermonde structure of F d . But the following lemmashows that this is not true for F d ⊗ F d . In fact, we prove that it requires at least d + 1 rows of F d ⊗ F d to build a nontrivial full spark matrix. Lemma 3.2.
For a positive integer < L ≤ d , given any L rows of F d ⊗ F d , thereexist L columns of F d ⊗ F d such that the resulting L × L submatrix is not invertible.Proof. Let { ( k j , l j ) : j = 1 , · · · L } be the row indices of F d ⊗ F d . If 1 < L ≤ d ,we claim that there exist column indices { ( s j , p j ) : j = 1 , · · · , L } such that theresulting L × L submatrix has two identical rows. Let G be the additive subgroupof Z d × Z d generated by ( k − k , l − l ) ∈ Z d × Z d , then | G | ≤ d. It is known fromPontryagin duality theory (see [30]) that there exists a corresponding annihilatorsubgroup H ⊂ Z d × Z d of size d | G | , such that for any ( s, p ) ∈ H ,(17) ω ( k − k ) s +( l − l ) pd = 1 . Since | H | = d | G | ≥ d , we can choose any subset of H consisting of L elements ascolumn indices. By (17), ω k s + l pd = ω k s + l pd , which means that the first two rows of the built submatrix are identical. (cid:3) We can also prove that for some 2 D convolution operators, it may happen that thelower bound for the cardinality of admissible sets is more than the largest geometric multiplicity of their eigenvalues. While we know a construction of full spark matrixbuilt from rows of F d with any given spark between 1 and d , Lemma 3.2 shows that itis impossible to find a full spark matrix with spark less than ( d + 1) from submatricesbuilt from rows of F d ⊗ F d . To our best knowledge, in the case of F d ⊗ F d , there isno deterministic formula that gives a way to construct full spark matrices for any d .However, we can draw a similar conclusion as Theorem 2.7. Proposition 3.3.
If the submatrix of F d ⊗ F d built from rows indexed by Ω ⊂ Z d × Z d is full spark, so is the submatrix of F d ⊗ F d build from rows indexed by(1) Ω + ( s, p ) for any ( s, p ) ∈ Z d × Z d ,(2) ( c , c )Ω = { ( c i, c j ) : ( i, j ) ∈ Ω } for any ( c , c ) ∈ Z d × Z d such that both c , c are coprime to d ,(3) Ω c = Z d × Z d − Ω . In modeling physical or biological phenomena, the convolution kernel a usuallypossesses certain symmetries in the frequency domain. We introduce several typesof symmetric convolution kernels and consider the problem of finding minimal ad-missible Ω for these symmetric convolution kernels. For the convenience of state-ment, we assume d is odd and we identify the level sets of ˆa with their moduloimages in {− d − , · · · , d − } × {− d − , · · · , d − } . In the rest of paper, we denote I = {− d − , · · · , d − } × {− d − , · · · , d − } . The following definitions can be found in[28]. Definition 3.4.
Let a be a D array defined on Z d × Z d . Recall that ˆa denote itsunnormalized discrete Fourier transform.(1) Given any level set Λ of ˆa , if all elements in Λ have the same ℓ ∞ norm, a issaid to possess ℓ ∞ -symmtery in frequency response.(2) If the level sets of ˆa consist of the sets in the form of { ( s, p ) , ( s, − p ) , ( − s, p ) , ( − s, − p ) } for ( s, p ) ∈ I , a is said to possess quadrantal symmetry in frequency response.(3) If the level sets of ˆa consist of the sets in the form of { ( p, s ) , ( s, p ) , ( − p, − s ) , ( − s, − p ) } for ( s, p ) ∈ I , a is said to possess diagonal symmetry in frequency response.(4) If the level sets of ˆa consist of the sets in the form of { ( s, p ) , ( p, s ) , ( − p, s ) , ( − s, p ) , ( − s, − p ) , ( − p, − s ) , ( p, − s ) , ( s, − p ) } for ( s, p ) ∈ I , a is said to possess octagonal symmetry in frequency response. For convolution kernels a with the same symmetry in frequency response, theirdiscrete Fourier transform have the same level sets. By Proposition 3.1, it is possiblefor us to construct minimal admissible sets and then determine l i,j for 2D convolutionkernels that are subject to the same symmetry conditions in frequency response.The specific constructions we will show are inspired by the ideas stemmed from themultivariable interpolation theory. Theorem 3.5.
Let A be a circular convolution operator defined by a convolu-tion kernel a ∈ ℓ ( Z d × Z d ) . Let Ω be a proper subset of Z d × Z d and Y = { e i,j , Ae i,j , · · · , A l i,j e i,j : ( i, j ) ∈ Ω } . (1) Assume that a possesses ℓ ∞ symmetry in frequency response. If we choose Ω = { , } × Z d ∪ Z d × { , } and for each ( i, j ) ∈ Ω , l i,j = d − , then Y is aframe for ℓ ( Z d × Z d ) .(2) Assume that a possesses quadrantal symmetry in frequency response. Supposethe elements i , i , j and j of Z d satisfy the conditions gcd ( | i − i | , d ) = 1 and gcd ( | j − j | , d ) = 1 . If we choose Ω = { ( i , j ) , ( i , j ) , ( i , j ) , ( i , j ) } ,and for each ( i, j ) ∈ Ω , l i,j = ( d +1) − , then Y is a frame for ℓ ( Z d × Z d ) .(3) Assume that a possesses diagonal symmetry in frequency response. Sup-pose the elements i , i , j and j of Z d satisfy the conditions gcd ( i , d ) = 1 and gcd ( j , d ) = 1 . If we choose Ω = { ( i , , ( i + i , , ( i + 2 i , , ( i +3 i , } or Ω = { (0 , j ) , (0 , j + j ) , (0 , j + 2 j ) , (0 , j + 3 j ) } , and for each ( i, j ) ∈ Ω , l i,j = ( d +1) − , then Y is a frame for ℓ ( Z d × Z d ) .(4) Assume that a possesses octagonal symmetry in frequency response. Supposethe elements i , i , j and j of Z d satisfy the conditions gcd ( | i − i | , d ) = 1 and gcd ( j , d ) = 1 . If we choose Ω = { i , i } × { j , j + j , j + 2 j , j + 3 j } ,and for each ( i, j ) ∈ Ω , l i,j = ( d +1)( d +3)8 − , then Y is a frame for ℓ ( Z d × Z d ) .Alternatively, suppose the elements i , i , j and j of Z d satisfy the conditions gcd ( | j − j | , d ) = 1 and gcd ( i , d ) = 1 . If we choose Ω = { i , i + i , i +2 i , i + 3 i } × { j , j } , and for each ( i, j ) ∈ Ω , l i,j = ( d +1)( d +3)8 − , then Y is a frame for ℓ ( Z d × Z d ) .Proof. (1) Since a has ℓ ∞ symmetry in frequency response, ˆa has d +12 levelsets which are given by Λ l = { ( s, p ) : ( s, p ) ∈ I , max( | s | , | p | ) = l } for l = 0 , · · · , d − . Then by computation, we get N A = d +12 and M A = 4 d − . Let Ω = { , } × Z d ∪ Z d × { , } . We first show that the (4 d − × (4 d − S built from rows indexed by Λ d − and columns indexed by Ω of F d ⊗ F d is invertible. Assume that there exists a vector c = ( c ( k, l )) ( k,l ) ∈ Ω such that Sc = . For each ( s, p ) ∈ Λ d − , we have(18) X ( k,l ) ∈ Ω c ( k, l ) ω ksd ω lpd = 0 . Reordering the terms in (18) by collecting the coefficients together for thepower of ω sd , we get(19) d − X l =0 c (0 , l ) ω lpd + ( d − X l =0 c (1 , l ) ω lpd ) ω sd + d − X k =2 ( c ( k,
0) + c ( k, ω pd )( ω sd ) k = 0 . Denote Λ ′ d − = {− d − , · · · , d − } × {− d − , d − } . Then obviously Λ ′ d − ⊂ Λ d − . Since (19) holds for ( s, p ) ∈ Λ ′ d − , we get(20) ω − d − d · · · ( ω − d − d ) d − ω − d − d · · · ( ω − d − d ) d − ... ... . . . ...1 ω d − d · · · ( ω d − d ) d − d − P l =0 c (0 ,l ) ω lpdd − P l =0 c (1 ,l ) ω lpd ... c ( d − , c ( d − , ω pd = for p = − d − and d − . Note that the matrix on the left of (20) is an invertibleVandermonde matrix, therefore the coefficient vector on the right of (20) is . Then we obtain(21) d − X l =0 c ( i, l )( ω ld ) p = 0 for i = 0 , p = ± d − , and ω − d − d ω d − d (cid:20) c ( k, c ( k, (cid:21) = for 2 ≤ k ≤ d −
1. Solving the above linear equations, we get c ( k,
0) = c ( k,
1) = 0 for 2 ≤ k ≤ d − . We can also reorder the terms in (18) bycollecting the coefficients together for the power of ω pd and get(22) d − X k =0 c ( k, ω skd + ( d − X k =0 c ( k, ω skd ) ω pd + d − X l =2 ( c (0 , l ) + c (1 , l ) ω sd )( ω pd ) l = 0 . Since equation (22) holds for ( s, p ) ∈ Λ ′ d − , similar to the case of reorderingwith the power of ω sd , we get c (0 , l ) = c (1 , l ) = 0 for 2 ≤ l ≤ d − . Nowsubstituting the entries of c we just solved into (21), and we can get c (0 ,
0) = c (0 ,
1) = c (1 ,
0) = c (1 ,
1) = 0 . Thus c = , which implies that S is invertible.The same arguments can show that the 8 l × l submatrix of F d ⊗ F d builtby rows indexed by Λ l and { , } × { , · · · , l } ∪ { , · · · , l } × { , } ⊂ Ω isinvertible for l = 1 , · · · , d − . Therefore, we have proved that the submatrixof F d ⊗ F d formed by rows indexed by Λ l and columns indexed by Ω is offull row rank for l = 0 , · · · , N A − . By (2) of Proposition 3.1, the conclusionfollows.(2) Assume that a has quadrantal symmetry in frequency response, it can beeasily computed that M A = 4 and N A = ( d +1) . LetΛ s,p = { ( s, p ) , ( s, − p ) , ( − s, − p ) , ( − s, p ) } be a level set of ˆa . Given Ω = {{ i , i }×{ j , j } : gcd ( | i − i | , d ) = 1 , gcd ( | j − j | , d ) = 1 } , the submatrix built from rows indexed by Λ s,p and columnsindexed by Ω for the case s = 0 and p = 0 is ω si pj d ω si pj d ω si pj d ω si pj d ω si − pj d ω si − pj d ω si − pj d ω si − pj d ω − si − pj d ω − si − pj d ω − si − pj d ω − si − pj d ω − si pj d ω − si pj d ω − si pj d ω − si pj d , which is (cid:20) ω si d ω si d ω − si d ω − si d (cid:21) ⊗ (cid:20) ω pj d ω pj d ω − pj d ω − pj d (cid:21) . Since the two matrices listed above are always invertible by given constraintson i , i , j and j , the Kronecker product of them is also invertible. For thecases when one of s, p is zero, Λ s,p has only two pairs. It is easy to checkthat the 2 × s,p and Ω has rank 2. In the event that s = p = 0, it is obvious that the submatrix formed by Λ s,p and Ω has rank1. Therefore, all conditions stated in (2) of Proposition 3 . { ( i , , ( i + i , , ( i + 2 i , , ( i + 3 i ,
0) : gcd ( i , d ) = 1 } , the other one follows in a similar way. We just need toprove the case i = 0 since any translation of Ω is also an admissible set byProposition 3.1. Given a has diagonal symmetry in frequency response, welet Λ s,p be a level set of ˆ a consisting of pairs { ( s, p ) , ( − s, − p ) , ( − p, − s ) , ( p, s ) } . For the cases when s, p = 0 and s = p , the submatrix built from rows indexedby Λ s,p and columns indexed by Ω is a Vandermonde matrix with 4 distinctbases:(23) ω si d ω si d ω si d ω − si d ω − si d ω − si d ω pi d ω pi d ω pi d ω − pi d ω − pi d ω − pi d . So it is invertible. For the cases s = 0 or p = 0 or s = p , it is easy to verifythe corresponding submatrix has full row rank. The conclusion follows.(4) We only prove the case Ω = { i , i : gcd ( | i − i | , d ) = 1 } × { j , j + j , j +2 j , j + 3 j : gcd ( j , d ) = 1 } , the other one follows similarly. Given a hasoctagonal symmetry in frequency response, we let Λ s,p be a level set of ˆ a consisting of pairs { ( s, p ) , ( p, s ) , ( − p, s ) , ( − s, p ) , ( − s, − p ) , ( − p, − s ) , ( p, − s ) , ( s, − p ) } . For the cases when ( s, p ) satisfies that s = ± p and s, p = 0, we denote by S s,p the 8 × F d ⊗ F d built from rows indexed by Λ s,p andcolumns indexed by Ω . We will use a similar argument to the proof of the ℓ symmetry case. Assume c = ( c ( k, l )) ( k,l ) ∈ Ω and S s,p c = . Then for every ( s , p ) ∈ Λ s,p , we have(24) X i =0 ( c ( i , j + ij ) ω p ( j + ij ) d ) ω s i d + ( X i =0 c ( i , j + ij ) ω p ( j + ij ) d ) ω s i d = 0 . Plugging ( s , p ) = ( s, p ) , ( − s, p ) into (24), and note that the matrix h ω si ω si ω − si ω − si i is invertible, we get P i =0 c ( i n , j + ij ) ω p ( j + ij ) = 0 for n = 1 ,
2. Plug-ging ( s , p ) = ( s, − p ) , ( − s, − p ), ( s , p ) = ( p, s ) , ( − p, s ) and ( s , p ) =( p, − s ) , ( − p, − s ) into (24), we can get(25) X i =0 c ( i n , j + ij ) ω − p ( j + ij ) d = 0 , X i =0 c ( i n , j + ij ) ω s ( j + ij ) d = 0 , X i =0 c ( i n , j + ij ) ω − s ( j + ij ) d = 0 . for n = 1 ,
2. Now we can write the above linear equations as a linear systemand solve c . Since the 4 × ω pj d ω pj + pj d ω pj +2 pj d ω pj +3 pj d ω − pj d ω − pj − pj d ω − pj − pj d ω − pj − pj d ω sj d ω sj + sj d ω sj +2 sj d ω sj +3 sj d ω − sj d ω − sj − sj d ω − sj − sj d ω − sj − sj d is a nonzero constant times an invertible Vandermonde matrix , we concludethat c = . Hence S s,p is invertible and has full row rank. For other casesof Λ s,p , it can be reduced to the case of quadrantal symmetry or diagonalsymmetry and by previous results, we know the corresponding submatrix hasfull row rank. Finally, It is easy to compute that M A = 8 , N A = ( d +1)( d +3)8 . By Proposition 3.1, the conclusion follows. (cid:3)
Similarly, we can also provide various constructions of minimal admissible Ω con-sist of unions of periodic sets. We summarize them as follows.
Theorem 3.6.
Let A be a circular convolution operator defined by a convolutionkernel a ∈ ℓ ( Z d × Z d ) . Suppose d is odd and d = mJ ( m > . Let Ω = { m Z d + r : r ∈ W ⊂ Z m } and Y = { e i,j , Ae i,j , · · · , A m − e i,j : ( i, j ) ∈ Ω } .(1) Assume that a possesses ℓ ∞ symmetry in frequency response. If W = { , }× Z m ∪ Z m × { , } , then Y is a frame for ℓ ( Z d × Z d ) . (2) Assume that a possesses quadrantal symmetry in frequency response. Supposethe elements i , i , j and j of Z m satisfy the conditions gcd ( | i − i | , m ) = 1 and gcd ( | j − j | , m ) = 1 . If W = { ( i , j ) , ( i , j ) , ( i , j ) , ( i , j ) } , then Y is a frame for ℓ ( Z d × Z d ) .(3) Assume that a possesses diagonal symmetry in frequency responses. Supposethe elements i , i , j and j of Z m satisfy the conditions gcd ( i , m ) = 1 and gcd ( j , m ) = 1 . If W = { ( i , , ( i + i , , ( i + 2 i , , ( i + 3 i , } or W = { (0 , j ) , (0 , j + j ) , (0 , j +2 j ) , (0 , j +3 j ) } , then Y is a frame for ℓ ( Z d × Z d ) .(4) Assume that a possesses octagonal symmetry in frequency response. Supposethe elements i , i , j and j of Z m satisfy the conditions gcd ( | i − i | , m ) = 1 and gcd ( j , m ) = 1 . If W = { i , i } × { j , j + j , j + 2 j , j + 3 j } , then Y is a frame for ℓ ( Z d × Z d ) . Alternatively, suppose the elements i , i , j and j of Z m satisfy the conditions gcd ( | j − j | , m ) = 1 and gcd ( i , m ) = 1 . If W = { i , i + i , i + 2 i , i + 3 i }× { j , j } , then Y is a frame for ℓ ( Z d × Z d ) .Proof. From the proof of Theorem 2.12, we see that it suffices to deal with certainsubmatrices of F m ⊗ F m . We can then follow a similar argument as the proof ofTheorem 3.5. (cid:3) General case
In this section, we will briefly discuss Problem 1.2 for a finite abelian group I andgeneralize the algebraic characterizations given in Proposition 2.1 and Proposition3.1. We will illuminate the relations between the problem under consideration, rep-resentations of I that commute with the evolution operator A , and the charactertable of I . We will recall several classical results of representation theory of finiteabelian groups.We assume I = { i , i , · · · , i d − } , where d is a positive integer in this section. Acharacter of a finite abelian group I is a group homomorphism χ : I → S . Thecharacters of I form a finite abelian group with respect to the pointwise product.This group is called the character group of I, and denoted by ˆ I . In fact, we canprove that I is isomorphic to ˆ I . Thus, I can serve as index set for ˆ I and we assumeˆ I = { χ i , χ i , · · · , χ i d − } . We choose an enumerate of ˆ I such that χ i is the trivialcharacter of I . Let X = (cid:0) χ i s ( i t ) (cid:1) be the matrix whose rows are indexed by ˆ I andcolumns are indexed by I . Then X is a square matrix of dimension d and it is calledthe character table of group I . We can further show that the matrix √ d X is unitary.Let us consider the signal space ℓ ( I ). The distinguishing feature of functionson I is that the group acts on itself by left translation, thereby moving around thefunctions on it. More specifically, for i ∈ I , the translate T i f of a function f by i isthe function on I defined by( T i f )( j ) = f ( i + j ) , for function f ∈ ℓ ( I ) , and i, j ∈ I. It is easy to check that the map ρ : i → T i from I to the C − linear automorphismsof ℓ ( I ) is a group homomorphism. The pair ( ρ, ℓ ( I )) is called the left regularrepresentation of I . As we can see, a character χ i s of I is indeed a square summablefunction defined on I . Using the fact that χ s is a group homomorphism, it followsthat χ s is an eigenvector for all translation operators { T i : i ∈ I } . Note that wehave d group characters, so the translation operators { T i : i ∈ I } are simultaneouslydiagonalizable by characters of I .By an abuse of notation, suppose A is an evolution operator on ℓ ( I ) such that f is evolving under the iterated action of A . This discrete evolution process is called spatially invariant if the operator A commutes with any spatial translations on ℓ ( I ).It turns out that A is indeed a circular convolution operator. The following theoremcan be found in [22, Theorem 5.1.3]: Theorem 4.1.
Let A be a linear operator on ℓ ( I ) . Then A commutes with spatialtranslations { T i : i ∈ I } if and only if A is a circular convolution operator withsome convolution kernel a ∈ ℓ ( I ) . Now let a ∈ ℓ ( I ), we definite ˆa ∈ ℓ ( ˆ I ) by ˆa ( χ s ) = h a , χ s i , s ∈ I. Suppose A is a circular convolution operator with kernel a . It follows from Theorem4 . I are also eigenvectors of A . By computation, we canobtain(26) A = ( 1 √ d X ) T diag ( ˆa ) 1 √ d X , where X denotes the complex conjugate of the matrix X . In the case of I = Z d andˆ I = { χ , χ , · · · , χ d − } , where χ s (1) = e πisd for s ∈ Z d , we can show that √ d X = F d and the equation (26) is exactly the same with the equation (3). Now we are readyto extend Proposition 2.1 and Proposition 3.1 to the general case. Proposition 4.2.
Let A be a circular convolution operator with a kernel a ∈ ℓ ( I ) and { Λ k : k = 1 , · · · , N A } be the level sets of ˆa . Suppose that Ω ⊂ I and Let Y = { e i , Ae i , · · · , A l i e i : i ∈ Ω } .(1) If Y is a frame for ℓ ( I ) , then for each k , the submatrix of X built from rowsindexed by Λ k and columns indexed by Ω has rank | Λ k | .(2) For each i ∈ Ω , we define by r i the degree of the A -annihilator of e i . If foreach i ∈ Ω and each k , l i ≥ r i − and the submatrix of X built from rowsindexed by Λ k and columns indexed by Ω has rank | Λ k | , then Y is a framefor ℓ ( I ) .(3) If for each i ∈ Ω and each k , A l i +1 e i is in the space spanned by Y and thesubmatrix of X built from rows indexed by Λ k and columns indexed by Ω hasrank | Λ k | , then Y is a frame for ℓ ( I ) . An immediate corollary we can get is
Corollary 4.3.
Let A be a circular convolution operator with a kernel a ∈ ℓ ( I ) .Suppose Ω ⊂ I . If the set Ω is an admissible set for A , then so is any translation of Ω . Concluding Remarks
In this paper, we have characterized universal spatiotemporal sampling sets fordiscrete spatially invariant evolution systems. In the one variable case I = Z d , wehave shown that the lower bound of sensors numbers we derived is achievable andhow to construct minimal universal irregular and unions of periodic sensor locationsfor convolution operators with eigenvalues subject to the same largest geometricmultiplicity. In the two variable case I = Z d × Z d , we have shown that the lowerbound of sensor numbers derived may not be achievable and the problem of findinguniversal spatiotemporal sampling sets is less favorable. We restricted ourselves tothe evolution systems in which the convolution operators have certain symmetriesin the Fourier domain and gave various constructions of minimal universal irregularand unions of periodic admissible sets. The ideas can be easily generalized to thethree or higher variable case I = Z d × · · · × Z d n . As we have already seen, theproblem of finding a deterministic admissible set Ω with | Ω | as few as possible for a2D convolution operator A is not easy in general; however, if we relax the conditionof finding full spark matrices to finding matrices with restricted isometry property (the definition can be found in [11] ) from discrete Fourier matrices, our algebraicperspective allows us to obtain random constructions of universal (but not minimal)admissible sets with high probability, which are parallel to those classical results inthe literature of compressed sensing, e.g., [12]. The results in the unions of periodiccase allow us to obtain near-optimal deterministic constructions of admissible setsΩ by choosing the period m to be prime and making use of Weil sum; see relatedresults in [32]. Finally, we have also studied the general finite abelian group case andestablished a connection between the problem under consideration with the charactertable of I . 6. Acknowledgements
The author would like to sincerely thank Akram Aldroubi for his careful readingof the manuscript and insightful comments. The author is indebted to anonymousreviewers for their valuable advice to improve the paper substantially. The authorwould also like to thank Xuemei Chen, Keaton Hamm and James Murphy for theiruseful suggestions to revise the presentation of the paper.
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