A Generalized Walsh System for Arbitrary Matrices Received October 14, 2018; Accepted December 18, 2018

Demonstr. Math. 2019; 52:40–55

Research Article Open Access Steven N. Harding and Gabriel Picioroaga* A generalized Walsh system for arbitrary matrices https://doi.org/10.1515/dema-2019-0006 Received October 14, 2018; accepted December 18, 2018

Abstract: In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. 2 Appl., 2014, 409(2),1128-1139] by means of representations of the Cuntz algebra ON on L [0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to flter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compres- sion, similar to the discrete wavelet transform.

Keywords: Cuntz algebras, Walsh basis, Hadamard matrix

1 Introduction { } It is a well-known fact that the collection of exponential functions e2πinx forms an orthonormal basis n∈Z (ONB) for the Hilbert space L2[0, 1]. However, motivated by applications in signal processing, we are inter- ested in working with discrete analogs of the exponential ONB. The collection of Rademacher functions, al- 2 though orthonormal and piecewise constant with range {−1, 1}, do not form a complete set in L [0, 1]. In [2] the Rademacher functions were used to construct an ONB that we call in this paper the classic Walsh system. Many generalizations can be found in the literature, e.g., identifying the Walsh functions as characters over the dyadic group in [3], identifying the Rademacher functions with N-adic exponentials in [4], etc. In this paper we study in more detail the generalized Walsh ONB system found in [1]. We will observe that many interesting results about periodicity, invertibility, transform continuity of these type of functions can actually be obtained in a slightly more general setting, where the system is generated by an arbitrary N × M matrix. We begin by presenting a few of the results of [1] regarding how a Cuntz algebra representation generates 2 × the construction of the generalized Walsh ONB on L [0, 1]. Let A ∈ CN N , the space of N × N complex-valued √ matrices, be unitary with frst row elements 1/ N. We adopt the convention that all indexing starts at zero. Letting be the characteristic function supported on , we defne the flter functions N−1 on 2[0, 1] as χA A {mi}i=0 L follows:

√ N−1 ( ) = ∑ ( ). mi x N Ai,j χ[j/N,(j+1)/N) x j=0

Steven N. Harding: Iowa State University, Department of Mathematics, 411 Morrill Road, Ames, IA 50011, U.S.A.; E-mail:[email protected] *Corresponding Author: Gabriel Picioroaga: University of South Dakota, Department of Mathematical Sciences, 414 E. Clark Street, Vermillion, SD 57069, U.S.A.; E-mail: [email protected]

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Note that 1. Letting ( ) = mod 1, we defne the operators N−1 on 2[0, 1] as follows: m0 ≡ r x Nx {Si}i=0 L

Si f (x) = mi(x)(f ∘ r)(x). (1.1)

Note that S01 = 1.

N−1 O 2[0, 1] Theorem 1.1. [1] The operators {Si}i=0 form a representation of the Cuntz algebra N on L , i.e., * = , Si Sj δi,j IL2[0,1] N−1 ∑ * = . Si Si IL2[0,1] i=0

Theorem 1.2. [1] The family WA := 1 Z , , , 0, , − 1 {Sw1 ...Swn | n ∈ + w1 ... wn ∈ { ... N }} 2 A is an orthonormal basis for L [0, 1], discarding repetitions generated by the fact that S01 = 1. We refer to W as the generalized Walsh basis corresponding to the matrix A.

One description of the elements of WA is as follows: Let n ∈ Z+, and consider its usual decomposition in base N,

k n = i0 + i1N + ... + ik N , (1.2) where i0, i1,..., ik−1 ∈ {0, 1,..., N − 1}. The general Walsh function of index n is then given by ( ) = ( ) ( )... ( k ), (1.3) Wn,A x mi0 x mi1 rx mik r x where rk = r ∘ r ∘ ... ∘ r for k functions. As the name suggests, the general Walsh ONB transcends the classic Walsh ONB. To be precise, the classic Walsh ONB coincides with the general Walsh ONB corresponding to the matrix

1 (1 1 ) A = √ . 2 1 −1 See [5] for applications of the general Walsh ONB to signal processing.

× A natural way to extend (1.3) and (1.1) for arbitrary matrices A ∈ CN M is to mimic the above construction without emphasis on the size of . Specifcally we redefne ( ) = mod 1 as well as the flters N−1 as A r x Mx {mi}i=0 follows:

√ M−1 ( ) = ∑ ( ). (1.4) mi x M Ai,j χ[j/M,(j+1)/M) x j=0 2 2 Further defne Wn,A : [0, 1] → C and Si : L [0, 1] → L [0, 1] as in (1.3) and (1.1), respectively, with flters as in (1.4). We will continue to refer to the collection ∞ for CN×M as general Walsh; however, we {Wn,A}n=0 A ∈ note that the general Walsh set does not form an ONB when N ≠ M. In this paper, we examine properties and applications of these rectangular matrix constructions.

2 Periodicity and invertibility of the general Walsh set

Many benefts arise from having various modes of periodicity, e.g., reduction of computational complexity. The periodic nature of the classic Walsh functions, in terms of the dyadic intervals of the unit interval, are

Brought to you by | Iowa State University Authenticated Download Date | 3/22/19 5:17 PM 42 Ë Steven N. Harding and Gabriel Picioroaga apparent from the Rademacher function construction. In fact, every classic Walsh function has some mode of periodicity, yet this is not the case for general Walsh functions. In this section, we characterize the periodic structure of the general Walsh set, as well as mention some of their algebraic properties. The construction of the general Walsh function in (1.3) suggests an intrinsic periodic nature which is closely related to the dimension of the associated matrix. In the fgure below, we provide the plots of a few general Walsh functions associated with the following matrix to illustrate this observation: √ √ √ ⎛ 1/ 3 1/ 3 1/ 3 ⎞ ⎜ ⎟ ⎜ √ √ √ ⎟ = ⎜1/ 14 2/ 14 −3/ 14⎟ . (2.1) A ⎜ ⎟ ⎜ ⎟ ⎝ √ √ √ ⎠ 5/ 42 −4/ 42 −1/ 42 Notice that plots (D), (E), and (F) exhibit repeating units, whose units of repetition may be described in terms of plots (A), (B), and (C), respectively. Furthermore, the number of repeating units can be ascertained by the ratio of their indices. For example, the order of the general Walsh function depicted in (E) is 32 times the order of its corresponding function depicted in (B); consequently, plot (E) may be described by repeating units attained by compressing plot (B) into the interval [0, 1/32) and then extending periodically. See [6] for results pertinent to these observations. Here we present an alternative approach to characterizing periodicity. First, we observe the following lemma.

(a) W1,A(x) (b) W4,A(x) (c) W17,A(x)

(d) W27,A(x) (e) W36,A(x) (f) W51,A(x)

Figure 1: The plots of six general Walsh functions associated corresponding to (2.1).

N×M Lemma 2.1. Let A ∈ C be with no row consisting entirely of zeros. Let n ∈ N0, and consider its base-N decomposition (1.2). Then ( ) = ( ) ( ) Wn,A x mi0 ,A x W(n−i0)/N,A rx

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0, 1,..., − 1 [ j , j+1 ) ( ) ≠ 0 and, for every j ∈ { M }, there exists an interval Ij ⊂ M M , such that W(n−i0)/N,A rx for x ∈ Ij.

Proof. It is straightforward to show the identity, so we omit that part of the proof. Instead, we prove the latter statement. We begin by fxing j ∈ {0, 1,..., M − 1}. Since every row of A contains a nonzero element, let 0, 1,..., − 1 , such that ≠ 0 for every 1 ≤ ≤ . Now consider the chain: jt ∈ { M } Ait ,jt t k

[ ) k−1 k−2 j j + 1 jM + j1M + ... + j −1 I := k , k + k ⊂ ... j Mk+1 Mk+1 Mk [ ) [ ) [ ) j2 j2 + 1 jM + j1 j1 j1 + 1 j j j + 1 ... ⊂ , + ⊂ , + ⊂ , . M3 M3 M2 M2 M2 M M M

Then, for x ∈ Ij, we have

( ) = ( ) ( 2 )... ( k ) W(n−i0)/N,A rx mi1 rx mi2 r x mik r x = ... Ai1 ,j1 Ai2 ,j2 Aik ,jk ≠ 0 which completes the proof of the lemma.

By the nature of the function , we observe that the function exhibits periodicity. In partic- r W(n−i0)/N,A ∘ r ular, ( ( + / )) = ( ( )). W(n−i0)/N,A r x j M W(n−i0)/N,A r x We will regard this property as periodic on the unit interval with the usual notion of periodicity. Then, from Lemma 2.1, we may characterize when a general Walsh function is periodic in terms of the flter . mi0

CN×M Theorem 2.2. Let A ∈ be with no row consisting entirely of zeros. Wn,A is periodic if and only if mi0 is periodic.

If is periodic, then there exists a positive integer < , such that ( ) = ( + / ) for Proof. mi0 s M mi0 x mi0 x s M 0 ≤ x < 1 − s/M. From Lemma 2.1, we fnd that Wn,A is periodic because

( + / ) = ( + / ) ( ( + / )) Wn,A x s M mi0 ,A x s M W(n−i0)/N,A r x s M = ( ) ( ) mi0 ,A x W(n−i0)/N,A rx

= Wn,A(x).

Conversely, if Wn,A is periodic, then there exists a positive integer t < M, such that t/M is the period of Wn,A. Then, from the identity in Lemma 2.1, we have

( + / ) ( ) = ( + / ) ( ( + / )) mi0 x t M W(n−i0)/N,A rx mi0 x t M W(n−i0)/N,A r x t M

= Wn,A(x + t/M)

= Wn,A(x) = ( ) ( ), mi0 x W(n−i0)/N,A rx for 0 ≤ < 1 − / . Again by Lemma 2.1, for every 0 ≤ ≤ − − 1, there exists an interval [ j , j+1 ), x t M j M t Ij ⊂ M M such that ( ) ≠ 0 for . Hence, W(n−i0)/N,A rx x ∈ Ij ( + / ) = ( ), mi0 x t M mi0 x for . Since is constant on -adic intervals, it follows that is periodic. x ∈ Ij mi0 M mi0

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Now we will discuss some algebraic properties of the general Walsh set. Let × be the operation on L2[0, 1] given by (f × g)(x) = f (x)g(x). Although × is neither binary nor commutative on the general Walsh set, we will investigate right and left invertibility in ( ∞ , ×). We will impose the conditions: {Wi,A}i=0 (i) No row other than the frst row can be a scalar multiple of the all ones vector.

(ii) The Schur product of rows ri and rj, i ≠ j, is not a scalar multiple of the all ones vector. 2 (iii) The ℓ -norm of any row vector is 1. Examples of matrices satisfying all three conditions are N × M submatrices of an M × M Hadamard or Fourier matrix. However, there is a plethora of examples which are not submatrices of the Fourier or Hadamard that satisfy the requirements above, for example: √ √ √ (1 3 1 3 1 3) A = 0 0 1

To characterize invertibility under these conditions, we begin with the following lemma.

CN×M ( ∞ , ×) Lemma 2.3. Let A ∈ satisfy conditions (i) and (ii). Then Wn,A is left or right invertible in {Wi,A}i=0 if and only if |Wn,A(x)| = 1.

Proof. The proof is similar for both left and right invertibility. Hence, suppose that Wn,A is right invertible in ( ∞ , ×). Then there exists N , such that × = 1. Consider the base- decomposition of {Wi,A}i=0 m ∈ 0 Wn,A Wm,A N n and m,

k n = i0 + i1N + ... + ik N l m = j0 + j1N + ... + jl N .

We may assume without loss of generality that k ≥ l since Wn,A × Wm,A = Wm,A × Wn,A = 1. However, assume for a contradiction that k > l. Note that ik ≠ 0. Upon regrouping, we have ( )( ) ( ) ( ) ( ) ( ) ( ) ... ( l ) ( l ) ( l+1 )..., ( k ) = 1. mi0 x mj0 x mi1 rx mj1 rx mil r x mjl r x mil+1 r x mik r x

By the proof of Lemma 2.1, we may choose an interval I small enough, such that for all x ∈ I all terms in the product above except ( k ) are constant. By i) this would not be possible unless = 0, i.e., the frst row mik r x ik in matrix A. Hence k = l. The same argument can be made to show that il = jl, ..., i0 = j0 by condition (ii). The converse is straightforward, and this concludes the proof.

N×M Theorem 2.4. Let A ∈ C satisfy conditions (i), (ii) and (iii). Let n ∈ N0, and consider its base-N decom- (1.2) ( ∞ , ×) position . Then Wn,A is left or right invertible in {Wi,A}i=0 if and only if 2 = 2 = ... = 2 = 1/ for 0 ≤ ≤ . |Aij ,0| |Aij ,1| |Aij ,M−1| M j k

Suppose that is invertible in ( ∞ , ×). Consider the product of flter functions Proof. Wn,A {Wi,A}i=0 ( ) = ( ) ( )... ( k−1 ) ( k ). Wn,A x mi0 x mi1 rx mik−1 r x mik r x − For x ∈ [0, 1/Mk), we have rl(x) ∈ [0, 1/Mk l) when l ≤ k. In particular, when l < k, the inequality rl(x) < 1/M implies that ( l ) is constant. Thus, on the interval [0, 1/ k), the product ( ) ( )... ( k−1 ) is mil r x M mi0 x mi1 rx mik−1 r x independent of x. Let b be this constant, so we have

k k ( ) = k ( ). Wn,A|[0,1/M ) x b mik |[0,1/M ) r x From the defnition of the flter function, we have

√ M−1 k ∑ m |[0,1/ k)(r x) = M A , χ +1 +1 (x). ik M ik l [l/Mk ,(l+1)/Mk ) l=0

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2 = 2 ( 2 + 2 + ... + 2) = . |b| M |b| M |Aik ,0| |Aik ,1| |Aik ,M−1| M Therefore, = 1 and consequently 2 = 2 = ... = 2 = 1/ . Moreover, this implies |b| |Aik ,0| |Aik ,1| |Aik ,M−1| M ( ) = 1, and we have |mik x | ( ) = ( ) ( )... ( k−1 ) = 1. |Wn,A x | |mi0 x mi1 rx mik−1 r x |

We repeat this argument until we have exhausted all of the ij. The converse follows directly from Lemma 2.3.

( ∞ , ×) Corollary 2.5. Let A be a Hadamard matrix. Then every general Walsh function is invertible in {Wi,A}i=0 .

3 Continuity and error estimates

In this section, we discuss results pertinent to error approximation analysis much like that of the following theorem from [7].

× √ Theorem 3.1. [7] Let A ∈ CN N be a unitary matrix with 1/ N for frst row entries. If f is constant on the N-adic [ / q , ( + 1)/ q) Nq−1 intervals { k N k N }k=0 , then

Nq−1 ∑ f(x) = ⟨f , Wn,A⟩Wn,A(x), n=0 for every x ∈ [0, 1).

A function which is constant on N-adic intervals may be treated as an encoding of data and, hence, fnds applications in signal processing, e.g., estimating the error in reconstructing a signal f from its frequencies N×M {⟨f , Wn,A⟩} using an approximated matrix B. We will make this explicit next. Let A, B ∈ C , and let q ∈ Z+. Let PCq be the collection of piecewise constant functions f : [0, 1] → C of the form

Mq−1 = ∑ , f αj χ[j/Mq ,(j+1)/Mq) j=0

q Mq−1 Mq ( )M −1 where { } ⊂ C. We defne the analysis operator : → C by = ⟨ , , ⟩ , and the αj j=0 ΘA PCq ΘA f f Wn A n=0 mixed frame operator is then defned by * , that is ΘB ΘA

Mq−1 * = ∑ , . (3.1) ΘB ΘA f ⟨f Wn,A⟩Wn,B n=0 √ When A = B is an N × N unitary matrix with frst row elements 1/ N, Theorem 3.1 guarantees the recovery * = . In the context of general and , we will observe this reconstruction identity, as well ΘB ΘA f f A B as upper estimates on the error operator − * on . To that end, let : CMq be given by I ΘB ΘA PCq θ PCq → θf(j) = f ((j + 1/2)/Mq). We will perform our analysis of the error operator in the sequence space through this map.

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× × We begin by recalling the tensor product. Let A ∈ CN1 M1 and B ∈ CN2 M2 . Recall the tensor product of A with B, denoted A ⊗ B, is defned as follows: ⎛ ⎞ A0,0BA0,1B...A0,M1−1B ⎜ A1,0BA1,1B...A1, −1B ⎟ ⎜ M1 ⎟ A ⊗ B = ⎜ . . . . ⎟ . ⎜ . . .. . ⎟ ⎝ ⎠ AN1−1,0BAN1−1,1B...AN1−1,M1−1B The following lemma gives a method by which we may compute the general Walsh function. This will be needed in characterizing recovery.

× Lemma 3.2. Let A ∈ CN M. Let n ∈ {0, 1,..., Nq − 1} and m ∈ {0, 1,...Mq − 1}, and consider their respective base decompositions:

q−1 n = k0 + k1N + ... + kq−1N q−1 m = i0 + i1M + ... + iq−1M .

Then, in the context of sequences, we have √ ( )( ) = q ... . θWn,A m M Ak0 ,iq−1 Ak1 ,iq−2 Akq−1 ,i0

We evaluate each ( )( s ) for 0 ≤ ≤ − 1. Let be the vector of length q−s, Proof. θmks r m s q As M √ ( ) = , , , , , , , , , , As M Aks ,0 ... Aks ,0 Aks ,1 ... Aks ,1 ... Aks ,M−1 ... Aks ,M−1 with each string, e.g., ,..., , consisting of q−s−1 elements. Note that s is the vector ( )⊕Ms , Aks ,0 Aks ,0 M θ∘mks ∘r As s q−s q−s q−1 that is the direct sum As ⊕ ... ⊕ As for M -many vectors. Since M divides iq−s M + ... + iq−1M , we have s q−s−1 s that (θm )(r m) is the element of As whose index is i0 + i1M + ... + i − −1M . Thus, we fnd (θm )(r m) = √ ks q s ks . MAks ,iq−s−1

∞ To estimate the pointwise error of recovery from (3.1), we will impose the L -norm on PCq. To emphasize that we are restricting ourselves to the subspace ∞[0, 1], we will denote the norm by · . It is clear PCq ⊂ L ‖ ‖PCq that = ∞ q . If we let be an operator on and be the corresponding matrix representation ‖f ‖PCq ‖θf ‖ℓ (CM ) T PCq S of T, i.e., θ(Tf ) = S(θf), then

= sup = sup ( ) ∞ q = ∞ q . (3.2) ‖T‖PCq ‖Tf ‖PCq ‖S θf ‖ℓ (CM ) ‖S‖ℓ (CM ) ‖f‖ =1 ‖θf‖ q =1 PCq ℓ∞(CM )

The next theorem characterizes perfect recovery. Furthermore, its proof may be used to estimate an upper bound on the norm of the error operator on PCq in terms of the matrices A and B that were implemented in its construction.

, CN×M − * = 0 * = Theorem 3.3. Let A B ∈ . We have I ΘB ΘA if and only if A B IM.

We determine the matrix representation of the mixed frame operator. Let Mq−1 be the canonical Proof. S {em}m=0 Mq ′ q ONB of C , and let δm ∈ PCq where em = θδm. Let m, m ∈ {0, 1,..., M − 1}, and consider their base-M decomposition:

q−1 m = i0 + i1M + ... + iq−1M ′ q−1 m = j0 + j1M + ... + jq−1M

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From Lemma 3.2, we fnd ( )( ′) = [ * ]( ′) S em m θ ΘB ΘA δm m Mq−1 = ∑ , ( )( ′) ⟨δm Wn,A⟩L2[0,1] θWn,B m n=0 Mq−1 , ∑ ⟨em θWn,A⟩ℓ2(CMq ) ′ = (θW , )(m ) Mq n B n=0 Mq−1 1 ∑ ′ = (θW , )(m)(θW , )(m ) Mq n A n B n=0 N−1 = ∑ ...... Ak0 ,iq−1 Ak1 ,iq−2 Akq−1 ,i0 Bk0 ,jq−1 Bk1 ,jq−2 Bkq−1 ,j0

k0 ,...,kq−1=0 = ( * ) ( * ) ...( * ) . A B iq−1 ,jq−1 A B iq−2 ,jq−2 A B i0 ,j0 ( )⊗q Thus the matrix representation of * is ( * )T , and the conclusion follows directly from this ΘB ΘA A B observation.

Note that we may attain perfect recovery precisely when A has full column rank, and a uniform upper bound on the pointwise error in reconstructing signals in PCq from (3.1) may be derived by Theorem 3.3. The M×M following corollary makes use of the well-known fact that, given D, E ∈ C , ‖D ⊗ E‖ℓ∞ = ‖D‖ℓ∞ ‖E‖ℓ∞ , as induced matrix norms.

× × Corollary 3.4. Let A ∈ CN M be with full column rank. Let ϵ > 0 and q ∈ Z+. Then, for B ∈ CN M satisfying

* √q ‖B A − IM‖ℓ∞ < ϵ + 1 − 1, − * < we have ‖I ΘB ΘA‖PCq ϵ.

Proof. ⃦ ⊗q⃦ * ⃦ ( * T) ⃦ − = q − ( ) ‖I ΘB ΘA‖PCq ⃦IM A B ⃦ ⃦ ⃦ℓ∞ ⃦ ⊗q⃦ ⃦ ( * T) ⃦ = ⃦IMq − IM + (A B − IM) ⃦ ⃦ ⃦ℓ∞ q ( ) k ∑ q ⃦ * T⃦ ≤ ⃦(A B − IM) ⃦ k ⃦ ⃦ℓ∞ k=1 q (⃦ * T⃦ ) = ⃦(A B − IM) ⃦ + 1 − 1 ⃦ ⃦ℓ∞ (⃦ * ⃦ )q = ⃦B A − IM⃦ + 1 − 1 ⃦ ⃦ℓ∞ < ϵ.

We note that this upper bound is a signifcant improvement of the one presented in [6].

4 Operators on L2[0, 1] satisfying Cuntz-like relations

An important topic in signal analysis is the method by which we may decompose and then recover data. Only half of the relations in the Cuntz algebra setting sufce to perform this task, whereas the remaining Cuntz

Brought to you by | Iowa State University Authenticated Download Date | 3/22/19 5:17 PM 48 Ë Steven N. Harding and Gabriel Picioroaga relations establish a splitting of the signal into orthogonal components. In this section we describe a proce- dure for constructing a fnite collection of linear operators with the fexibility of partially or fully satisfying the Cuntz-like relations. Let , CN×M. Defne the collection of flters N−1 corresponding to by A B ∈ {mj,A}j=0 A

M−1 ∑ mj,A(x) = Aj,k fk(x), k=0 √ where ( ) are either ( ) or 2πikx. We refer to the collection of flters as when fk x Mχ[k/M,(k+1)/M) x e piecewise-type fk are the characteristic functions and as exponential-type when fk are the complex exponential functions. Depending on the context, there are advantages to using one type of flter over the other. For example, it is not too complicated to see that, in expressing mj,C in terms of mj,A and mj,B, it is substantially simpler to use flters of piecewise-type for C = A ⊕ B and flters of exponential-type for C = A ⊗ B.

× × Proposition 4.1. Let A ∈ CN1 M1 and B ∈ CN2 M2 . For piecewise-type flter functions, ⎧ ( + ) ⎪ M1 M2 0 ≤ ≤ − 1; ⎨⎪mj,A x if j N1 ( ) = M1 mj,A⊕B x ( + ) ⎪ M1 M2 − M1 ≤ ≤ + − 1. ⎪mj−N1 ,B x if N1 j N1 N2 ⎩ M2 M2

For exponential-type flter functions,

( ) = ( ) ( ). mj,A⊗B x m⌊j/N2⌋,A M2x m(j mod N2),B x

Proof. The frst identity is fairly clear to see, so we omit the proof. Instead, we show the second identity:

M1 M2−1 ∑ 2πikx mj,A⊗B(x) = (A ⊗ B)j,k e k=0

M1−1 (l+1)M2−1 ∑ ∑ 2πikx = (A ⊗ B)j,k e l=0 k=lM2

M1−1 (l+1)M2−1 = ∑ ∑ 2πikx A⌊j/N2⌋,l B(j mod N2),k−lM2 e l=0 k=lM2

M1−1 M2−1 = ∑ 2πilM2 x ∑ 2πikx A⌊j/N2⌋,l e B(j mod N2),k e l=0 k=0 as desired.

Now let us defne the operators N−1 corresponding to on 2[0, 1] by {Sj,A}j=0 A L

Sj,A f(x) = mj,A(x)f (r(x)), where r(ω) = Mω mod 1. A simple calculation provides the adjoint operator

* 1 ∑ S , f(x) = m , (ω)f (ω). j A M j A ω : r(ω)=x The result that follows characterizes when these operators intermingle to satisfying some Cuntz-like relations.

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× Theorem 4.2. Let A, B ∈ CN M. The following results hold when the flter functions are of piecewise-type or of exponential-type. The following conditions are equivalent:

( ) * = , 0, 1,..., − 1 i Sj,A Sk,B δj,k I for all j k ∈ { N } * (ii) BA = IN

Moreover, the conditions are also equivalent:

N−1 ( ) ∑ * = iii Sj,B Sj,A I j=0 * (iv) A B = IM

Proof. We will prove the theorem for the case of piecewise-type flter functions and defer the case of exponential-type to the next section, as it is more technical.

We ascertain the equivalence of conditions (i) and (ii) through the following calculation:

* 1 ∑ S , S , f(x) = m , (ω)m , (ω)f (r(ω)) j A k B M j A k B ω : r(ω)=x M−1 M−1 = ( ) ∑ ∑ ( ) ∑ ( ) f x Aj,t χ[t/M,(t+1)/M) ω Bk,s χ[s/M,(s+1)/M) ω ω : r(ω)=x t=0 s=0 M−1 = ( ) ∑ ∑ ( ) f x Aj,t Bk,t χ[t/M,(t+1)/M) ω ω : r(ω)=x t=0 * = f (x)(BA )k,j .

Now, suppose condition (iv) is satisfed.

N−1 N−1 ∑ * ∑ 1 ∑ S , S , f (x) = m , (x) m , (ω)f (ω) j B j A j B M j A j=0 j=0 ω : r(ω)=r(x) N−1 M−1 M−1 = ∑ ∑ ( ) ∑ ∑ ( ) ( ) Bj,s χ[s/M,(s+1)/M) x Aj,t χ[t/M,(t+1)/M) ω f ω j=0 s=0 ω : r(ω)=r(x) t=0 M−1 M−1 N−1 = ∑ ∑ ∑ ∑ ( ) ( ) ( ) Bj,s Aj,t χ[s/M,(s+1)/M) x χ[t/M,(t+1)/M) ω f ω s=0 t=0 j=0 ω : r(ω)=r(x) M−1 M−1 = ∑ ∑( * ) ∑ ( ) ( ) ( ) A B t,s χ[s/M,(s+1)/M) x χ[t/M,(t+1)/M) ω f ω s=0 t=0 ω : r(ω)=r(x) M−1 = ∑ ∑ ( ) ( ) ( ) χ[s/M,(s+1)/M) x χ[s/M,(s+1)/M) ω f ω s=0 ω : r(ω)=r(x) M−1 = ( ) ∑ ( ) f x χ[s/M,(s+1)/M) x s=0 = f(x).

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This concludes that condition (iii) is valid. Conversely, suppose condition (iii) is satisfed. Fix an integer 0 ≤ k ≤ M − 1, and consider the characteristic function supported on [k/M, (k + 1)/M),

N−1 ( ) = ∑ * ( ) χ[k/M,(k+1)/M) x Sj,B Sj,A χ[k/M,(k+1)/M) x j=0 N−1 ∑ 1 ∑ = m , (x) m , (ω)χ (ω). j B M j A [k/M,(k+1)/M) j=0 ω : r(ω)=r(x) Then, for x ∈ [r/M, (r + 1)/M), we have

N−1 ∑ * δk,r = Bj,r Aj,k = (A B)k,r , j=0 where δk,r is the Kronecker delta. This establishes that condition (iv) is valid.

CN×N N−1 O Corollary 4.3. Let A ∈ . The collection of operators {Sj,A}j=0 generates the Cuntz algebra N if and only if A is unitary.

5 Operators on ℓ2(Z) satisfying Cuntz-like relations

In this section, inspired by Appendix C: A tale of two Hilbert spaces in [8], we describe a collection of operators 2 2 on ℓ (Z) which are closely connected to the operators Sj,A on L [0, 1] of the previous section. We then take 2 advantage of the appealing feature that describing signals in subspaces of ℓ (Z) is much more manageable for discrete data. Finally, we show an example of an image signal decomposed in a manner similar to discrete wavelet transform, where the operators Sj,A correspond to a matrix A without a constant frst row. In this case the Walsh system is not an ONB; however, the frst half of the Cuntz relations allows decomposition and × reconstruction of a signal. Throughout this section, we assume that A, B ∈ CN M and that the flter functions mj,A are of exponential-type, i.e.,

M−1 ∑ 2πikx mj,A(x) = Aj,k e . k=0 Let us defne the operators ˜ N−1 corresponding to on 2(Z) by {Sj,A}j=0 A ℓ ˜ ∑ Sj,A v(n) = hj,A(n − Mk)v(k), k∈Z 2 where hj,A ∈ ℓ (Z) is given by { Aj,k if k ∈ {0, 1,.., M − 1}; h , (k) = j A 0 otherwise

A simple calculation provides the adjoint operator

˜* ( ) = ∑ ( − ) ( ). Sj,A v n hj,A k Mn v k k∈Z ˜ The diagram below illustrates an intertwining feature of the operators Sj,A and Sj,A. More precisely, the ˜ 2 2 diagram commutes, that is Sj,A∘θ = θ∘Sj,A, where θ : L [0, 1] → ℓ (Z) is the canonical isometric isomorphism given by

2 θf(n) = ⟨f , e πinx⟩. (5.1)

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Thus θ is unitary, and its adjoint is * ∑ 2 θ v = v(k)e πikx . k∈Z 2 2 Then, to convert between the operators on L [0, 1] and the operators on ℓ (Z), we have the conjugation *˜ ˜ * tranformations: Sj,A = θ Sj,A θ and Sj,A = θSj,A θ .

Sj,A L2[0, 1] L2[0, 1]

θ θ ˜ S , 2 j A 2 ℓ (Z) ℓ (Z)

˜ Figure 2: Diagram of the Intertwining of Sj,A and Sj,A.

N×M ˜ ˜ Theorem 5.1. Let A ∈ C . The operator (5.1) intertwines the operators Sj,A and Sj,A, that is Sj,A ∘θ = θ∘Sj,A, for all j ∈ {0, 1,..., N − 1}.

˜ 2πinx 2πinx Proof. It is sufcient to show Sj,A θ(e ) = θSj,A(e ) for all n ∈ Z. We frst evaluate the left hand-side at the index m ∈ Z:

˜ 2πinx Sj,A θ(e )(m) = hj,A(m − Mn) { A , − if m − Mn ∈ {0, 1,.., M − 1}; = j m Mn 0 otherwise

2πinx We next evaluate the right-hand side at the index m ∈ Z. We begin by expanding Sj,A(e ),

M−1 2πinx ∑ 2πikx 2πin r(x) Sj,A(e ) = Aj,k e e k=0 M−1 ∑ 2πi(k+nM)x = Aj,k e . k=0 Then we have

2πinx 2πinx 2πimx θSj(e )(m) = ⟨Sj(e ), e ⟩ M−1 ∑ 2πi(k+Mn)x 2πimx = Aj,k⟨e , e ⟩ k=0 { A , − if m − Mn ∈ {0, 1,.., M − 1} = j m Nn 0 else as desired. 2 We now proceed towards reproducing Theorem 4.2 for the operators {Sj,A} on L [0, 1] generated by the ˜ exponential-type flter functions. This is done by working with the associated operators {Sj,A} in the intertwining of Theorem 5.1 through the matrix representation of the operators

˜* ˜ for , 0, 1,..., − 1 , (5.2) Sj,A Sk,B j k ∈ { N } N−1 ∑ ˜ ˜* . (5.3) Sk,B Sk,A k=0

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2 In what follows, we denote the canonical ONB of ℓ (Z) by the collection {en}n∈Z, i.e., en(m) = δm,n. Recall 2 2 that the matrix representation of an operator T : ℓ (Z) → ℓ (Z) is the bi-infnite matrix whose (n, m)-entry is 2 T(em)(n). For convenience, we denote the matrix representation of the identity operator on ℓ (Z) by I∞. The matrix representations of the operators above are fairly simple to derive, so we state the following proposition without proof.

× Proposition 5.2. Let A, B ∈ CN M. The matrix representations of (5.2) and (5.3) are the block diagonal ma- * * T trices (BA )k,j I∞ and I∞ ⊗ (A B) , respectively.

The following corollary is an immediate consequence of Theorem 5.1 and Proposition 5.2.

, CN×M * = ( *) Corollary 5.3. Let A B ∈ . Then Sj,A Sk,B BA k,j I.

Before proceeding, we frst take a short digression to make the action of the matrix representation * T 2 I∞ ⊗ (A B) on v ∈ ℓ (Z) explicit. Letting vn = (v(Mn), v(Mn + 1),..., v(M(n + 1) − 1)), we observe that * T * T * T (I∞ ⊗ (A B) )v = ⊕n∈Z(A B) vn. Then, by considering I∞ ⊗ (A B) as acting on orthogonal subspaces, we have the following result.

CM×M = Lemma 5.4. Let C ∈ . Then ‖I∞ ⊗ C‖ℓ2(Z) ‖C‖ℓ2(CM ).

Now we may determine exactly how close the operator (5.3) is to the identity operator with respect to the operator norm.

× Proposition 5.5. Let A, B ∈ CN M. Then

⃦ −1 ⃦ ⃦ N ⃦ ⃦ ⃦ − ∑ ˜ ˜* = − * . ⃦I∞ Sk,B Sk,A⃦ ⃦IM A B⃦ ⃦ ⃦ ⃦ ⃦ℓ2(CM ) ⃦ k=0 ⃦ℓ2(Z)

Proof. From Proposition 5.2 and Lemma 5.4, we have

⃦ −1 ⃦ ⃦ N ⃦ ⃦ ⃦ − ∑ ˜ ˜* = − ( * )T ⃦I∞ Sk,B Sk,A⃦ ⃦I∞ I∞ ⊗ A B ⃦ ⃦ ⃦ ⃦ ⃦ℓ2(Z) ⃦ k=0 ⃦ℓ2(Z) ⃦ ( * T)⃦ = ⃦I∞ ⊗ IM − (A B) ⃦ ⃦ ⃦ℓ2(Z) ⃦ * T⃦ = ⃦IM − (A B) ⃦ ⃦ ⃦ℓ2(CM ) ⃦ * ⃦ = ⃦IM − A B⃦ . ⃦ ⃦ℓ2(CM )

× Corollary 5.6. Let A, B ∈ CN M. Then

⃦ −1 ⃦ ⃦ N ⃦ ⃦ ⃦ − ∑ * = − * . ⃦I Sk,B Sk,A⃦ ⃦IM A B⃦ ⃦ ⃦ ⃦ ⃦ℓ2(CM ) ⃦ k=0 ⃦L2[0,1]

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Proof. To condense the notation, let Ψ and ψ be the operators

N−1 = − ∑ * , Ψ I Sk,B Sk,A k=0 N−1 = − ∑ ˜ ˜* . ψ I∞ Sk,B Sk,A k=0

By Theorem 5.1, we have Ψ = θ*ψθ. Since θ is unitary, Proposition 5.5 implies that

= sup ‖Ψ‖L2[0,1] ‖Ψf ‖L2[0,1] 2 =1 ‖f ‖L = sup ‖ψθf ‖ℓ2(Z) =1 ‖θf‖ℓ2 = ‖ψ‖ℓ2(Z) = − * . ‖IM A B‖ℓ2(CM )

Altogether, Corollary 5.3 and Corollary 5.6 complete the proof of Theorem 4.2 for the operators Sj,A and Sj,B defned by the exponential-type flters. This was accomplished by showing (Theorem 5.2 and Proposition 5.5) ˜ ˜ that the associated operators Sj,A and Sj,B intermingle to mimic the Cuntz-like relations, so we have those additional results. In this section we develop an application of the operators ˜ N−1 to discrete data. For each Z , we {Sj}j=0 q ∈ + ˜ 2 ˜ { ⃒ q } denote by Hq the subspace of ℓ (Z) given by Hq = span ek ⃒ k = 0, 1,..., M − 1 . The matrix representation ˜ q+1 q of the linear operator S | ˜ is the M × M matrix j Hq ⎛ ⎞ Aj,0 0 ... 0 ⎜ . . . . ⎟ ⎜ . . .. . ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜Aj,M−1 ... ⎟ ⎜ ⎟ ⎜ 0 A ,0 ... 0 ⎟ ⎜ j ⎟ ⎜ . . .. . ⎟ ⎜ . . . . ⎟ ⎜ ⎟ . ⎜ 0 0 ⎟ ⎜ Aj,M−1 ... ⎟ ⎜ . . . ⎟ ⎜ . . .. . ⎟ ⎜ . . . . ⎟ ⎜ ⎟ ⎜ 0 0 ...A ,0 ⎟ ⎜ j ⎟ ⎜ . . . . ⎟ ⎜ . . .. . ⎟ ⎝ ⎠ 0 0 ...Aj,M−1 ˜ ˜ Since the range of S | ˜ is embedded in H +1, we may view its matrix representation as infating the di- j Hq q * (˜ ) ˜* mension of data by a factor of M. The matrix representation of the adjoint operator S | ˜ = S | ˜ is j Hq j Hq+1 ˜* ˜ the adjoint of the matrix above. Here, since the range of S | ˜ is embedded in Hq, we may view its matrix j Hq+1 representation as defating the dimension of the data by a factor of M.

Example 5.7. In the classic case N = 2 there are two flters, lowpass and highpass, however for N ≥ 3 this is √ not necessarily maintained anymore. If A is a 3 × 3 unitary matrix having a constant 1/ 3 row, then the flter corresponding to it can be interpreted as lowpass but there are intermediate ones whose action on the image may give negative values. We are still able to ‘see‘ the action of the transform and the multiresolutions through a normalization of the values of the transformed signal. The pictures in the fgure above represent frst and second iterates on columns followed by rows of the image signal f according to the cascade algorithm, as seen here:

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(a) Transform A, First Iterate (b) Transform A, Second Iterate

Figure 3: Generalized discrete wavelet transform for matrices A and B.

˜* ˜* S0, S0, f u: A A uu uu uu uu ˜* / ˜* ˜* S? 0,A f I S1,A S0,A f ~ II ~~ II ~~ II ~~ II ~~ $ / ˜* ˜* ˜* f @ S1, f S2, S0, f @@ A A A @@ @@ @ ˜* S2,A f

The wavelet-like transform was implemented using the following matrices: √ √ √ √ ⎛ 1/3 1/2 (1/6) 23 ⎞ ⎛ 1/ 3 1/ 3 1/ 3⎞ √ √ √ A = ⎜(−1/9) 23 3 0 (2/9) 3 ⎟ , B = ⎜ −.2 −.5855 .7855⎟ . ⎝ √ √ √ √ ⎠ ⎝ ⎠ (1/9) 3 (−1/2) 3 (1/18) 23 3 −.7916 .5690 .2226

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Notice both matrices are unitary (B’s entries were truncated) but matrix A does not satisfy the constant row condition.

References

[1] Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128–1139 [2] Walsh J. L., A closed set of normal orthogonal functions, Amer. J. Math, 1923, 45(1), 5–24 [3] Vilenkin N., On a class of complete orthonormal systems, Bull. Acad. Sci. Math. [Izv. Akad. Nauk SSSR Ser. Mat.], 1947, 11(4), 363–400 [4] Chrestenson H. E., A class of generalized Walsh functions, Pacic. J. Math.,1955, 5(1), 17–31 [5] Dutkay D. E., Picioroaga G., Silvestrov S., On generalized Walsh bases, Acta Appl. Math., 2018,1–18, https://doi.org/10. 1007/s10440-018-0214-x [6] Harding S. N., Generalized Walsh transforms, Cuntz algebras representations and applications in signal processing, Uni- versity of South Dakota, Master Thesis, 2015, Copyright - Database copyright ProQuest LLC [7] Dutkay D. E., Picioroaga G., Generalized Walsh bases and applications, Acta Appl. Math., 2014, 133(1), 1–18, https://doi. org/10.1007/s10440-013-9856-x [8] Jorgensen P. E. T., Analysis and Probability: Wavelets, Signals, Fractals, Springer-Verlag New York, 2006

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