A generalized Walsh system for arbitrary matrices

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. Appl., 2014, 409(2),1128-1139] by means of representations of the Cuntz algebra ON on L2[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 filter variation. We consider such transforms in the casewhen the orthogonality conditions in Cuntz relations are removed.We show that these transformswhich still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.


Introduction
It is a well-known fact that the collection of exponential functions 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 the construction of the generalized Walsh ONB on L 2 [0, 1]. Let A ∈ C N×N , the space of N × N complex-valued matrices, be unitary with first row elements 1/ √ N. We adopt the convention that all indexing starts at zero. Letting χ A be the characteristic function supported on A, we define the filter functions {m i } N−1 i=0 on L 2 [0, 1] as follows: Note that m 0 ≡ 1. Letting r(x) = Nx mod 1, we define the operators {S i } N−1 i=0 on L 2 [0, 1] as follows: (1.1) Note that S 0 1 = 1. is an orthonormal basis for L 2 [0, 1], discarding repetitions generated by the fact that S 0 1 = 1. We refer to W A as the generalized Walsh basis corresponding to the matrix A.
One description of the elements of W A is as follows: Let n ∈ Z+, and consider its usual decomposition in base N, where i 0 , i 1 , ..., i k−1 ∈ {0, 1, ..., N − 1}. The general Walsh function of index n is then given by where r k = 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 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 ∈ C N×M is to mimic the above construction without emphasis on the size of A. Specifically we redefine r(x) = Mx mod 1 as well as the filters {m i } N−1 i=0 as follows: for A ∈ C N×M as general Walsh; however, we 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. 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 figure below, we provide the plots of a few general Walsh functions associated with the following matrix to illustrate this observation: 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 3 2 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, 1/3 2 ) 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. 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 fixing j ∈ {0, 1, ..., M − 1}. Since every row of A contains a nonzero element, let j t ∈ {0, 1, ..., M − 1}, such that A it ,jt ≠ 0 for every 1 ≤ t ≤ k. Now consider the chain: Then, for x ∈ I j , we have which completes the proof of the lemma.
By the nature of the function r, we observe that the function W (n−i0)/N,A ∘ r exhibits periodicity. In particular, 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 filter m i0 . Conversely, if W n,A is periodic, then there exists a positive integer t < M, such that t/M is the period of W n,A . Then, from the identity in Lemma 2.1, we have Since m i0 is constant on M-adic intervals, it follows that m i0 is periodic. Now we will discuss some algebraic properties of the general Walsh set. Let × be the operation on L 2 [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 ({W i,A } ∞ i=0 , ×). We will impose the conditions: (i) No row other than the first row can be a scalar multiple of the all ones vector. (ii) The Schur product of rows r i and r j , i ≠ j, is not a scalar multiple of the all ones vector. (iii) The ℓ 2 -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: To characterize invertibility under these conditions, we begin with the following lemma.

Lemma 2.3. Let A ∈ C N×M satisfy conditions (i) and (ii). Then W n,A is left or right invertible in ({W
Proof. The proof is similar for both left and right invertibility. Hence, suppose that W n,A is right invertible in . Then there exists m ∈ N 0 , such that W n,A × W m,A = 1. Consider the base-N decomposition of n and m, We may assume without loss of generality that k ≥ l since W n,A × W m,A = W m,A × W n,A = 1. However, assume for a contradiction that k > l. Note that i k ≠ 0. Upon regrouping, we have 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 m i k (r k x) are constant. By i) this would not be possible unless i k = 0, i.e., the first row in matrix A. Hence k = l. The same argument can be made to show that i l = j l , ..., i 0 = j 0 by condition (ii). The converse is straightforward, and this concludes the proof.
Theorem 2.4. Let A ∈ C N×M satisfy conditions (i), (ii) and (iii). Let n ∈ N 0 , and consider its base-N decom- . Consider the product of filter functions .
From the definition of the filter function, we have It follows from this and Lemma 2.3 that By summing the square of these terms and observing condition (iii), we find Therefore, |b| = 1 and consequently We repeat this argument until we have exhausted all of the i j . The converse follows directly from Lemma 2.3.

Continuity and error estimates
In this section, we discuss results pertinent to error approximation analysis much like that of the following theorem from [7].
A function which is constant on N-adic intervals may be treated as an encoding of data and, hence, finds applications in signal processing, e.g., estimating the error in reconstructing a signal f from its frequencies {⟨f , W n,A ⟩} using an approximated matrix B. We will make this explicit next. Let A, B ∈ C N×M , and let q ∈ Z+. Let PCq be the collection of piecewise constant functions f : [0, 1] → C of the form , and the mixed frame operator is then defined by Θ * B Θ A , that is To that end, let θ : PCq → C M q be given by θf (j) = f (︀ (j + 1/2)/M q )︀ . We will perform our analysis of the error operator in the sequence space through this map.
We begin by recalling the tensor product. Let A ∈ C N1×M1 and B ∈ C N2×M2 . Recall the tensor product of A with B, denoted A ⊗ B, is defined as follows: The following lemma gives a method by which we may compute the general Walsh function. This will be needed in characterizing recovery.
Then, in the context of sequences, we have Proof. We evaluate each (θm ks )(r s m) for 0 ≤ s ≤ q − 1. Let As be the vector of length M q−s , 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 PCq ⊂ L ∞ [0, 1], we will denote the norm by ‖·‖ PCq . It is clear that ‖f ‖ PCq = ‖θf ‖ ℓ ∞ (C M q ) . If we let T be an operator on PCq and S be the corresponding matrix representation of T, i.e., θ(Tf ) = S(θf ), then 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.
Thus the matrix representation of Θ * B Θ A is (︁ (A * B) T )︁ ⊗q , and the conclusion follows directly from this 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 following corollary makes use of the well-known fact that, given D, E ∈ C M×M , ‖D ⊗ E‖ ℓ ∞ = ‖D‖ ℓ ∞ ‖E‖ ℓ ∞ , as induced matrix norms.

Corollary 3.4.
Let A ∈ C N×M be with full column rank. Let ϵ > 0 and q ∈ Z+. Then, for B ∈ C N×M satisfying Proof.
We note that this upper bound is a significant improvement of the one presented in [6].

Operators on L 2 [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 suffice to perform this task, whereas the remaining Cuntz relations establish a splitting of the signal into orthogonal components. In this section we describe a procedure for constructing a finite collection of linear operators with the flexibility of partially or fully satisfying the Cuntz-like relations. Let A, B ∈ C N×M . Define the collection of filters {m j,A } N−1 j=0 corresponding to A by We refer to the collection of filters as piecewise-type when f k are the characteristic functions and as exponential-type when f k are the complex exponential functions.
Depending on the context, there are advantages to using one type of filter over the other. For example, it is not too complicated to see that, in expressing m j,C in terms of m j,A and m j,B , it is substantially simpler to use filters of piecewise-type for C = A ⊕ B and filters of exponential-type for C = A ⊗ B.
For exponential-type filter functions, Proof. The first identity is fairly clear to see, so we omit the proof. Instead, we show the second identity: where r(ω) = Mω mod 1. A simple calculation provides the adjoint operator The result that follows characterizes when these operators intermingle to satisfying some Cuntz-like relations.
Proof. We will prove the theorem for the case of piecewise-type filter 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: Now, suppose condition (iv) is satisfied.
This concludes that condition (iii) is valid. Conversely, suppose condition (iii) is satisfied. Fix an integer 0 ≤ k ≤ M − 1, and consider the characteristic function supported on [k/M, (k + 1)/M), Then, for x ∈ [r/M, (r + 1)/M), we have where δ k,r is the Kronecker delta. This establishes that condition (iv) is valid.

Operators on ℓ 2 (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 on ℓ 2 (Z) which are closely connected to the operators S j,A on L 2 [0, 1] of the previous section. We then take advantage of the appealing feature that describing signals in subspaces of ℓ 2 (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 S j,A correspond to a matrix A without a constant first row. In this case the Walsh system is not an ONB; however, the first half of the Cuntz relations allows decomposition and reconstruction of a signal. Throughout this section, we assume that A, B ∈ C N×M and that the filter functions m j,A are of exponential-type, i.e., Let us define the operators {S j,A } N−1 j=0 corresponding to A on ℓ 2 (Z) bỹ where h j,A ∈ ℓ 2 (Z) is given by A simple calculation provides the adjoint operator The diagram below illustrates an intertwining feature of the operators S j,A andS j,A . More precisely, the diagram commutes, that isS j,A ∘θ = θ∘S j,A , where θ : L 2 [0, 1] → ℓ 2 (Z) is the canonical isometric isomorphism given by θf (n) = ⟨f , e 2πinx ⟩. (5.1) Thus θ is unitary, and its adjoint is Then, to convert between the operators on L 2 [0, 1] and the operators on ℓ 2 (Z), we have the conjugation tranformations: S j,A = θ *S j,A θ andS j,A = θS j,A θ * . In what follows, we denote the canonical ONB of ℓ 2 (Z) by the collection {en} n∈Z , i.e., en(m) = δm,n. Recall that the matrix representation of an operator T : ℓ 2 (Z) → ℓ 2 (Z) is the bi-infinite matrix whose (n, m)-entry is T(em)(n). For convenience, we denote the matrix representation of the identity operator on ℓ 2 (Z) by I∞. The matrix representations of the operators above are fairly simple to derive, so we state the following proposition without proof. .
Proof. From Proposition 5.2 and Lemma 5.4, we have . Corollary 5.6. Let A, B ∈ C N×M . Then .
Proof. To condense the notation, let Ψ and ψ be the operators Since the range ofS j |H q is embedded inH q+1 , we may view its matrix representation as inflating the dimension of data by a factor of M. The matrix representation of the adjoint operator (︁S j |H q )︁ * =S * j |H q+1 is the adjoint of the matrix above. Here, since the range ofS * j |H q+1 is embedded inHq, we may view its matrix representation as deflating the dimension of the data by a factor of M.
Example 5.7. In the classic case N = 2 there are two filters, 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 filter 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 figure above represent first and second iterates on columns followed by rows of the image signal f according to the cascade algorithm, as seen here: