The density theorem of a class of dilation-and-modulation systems on the half real line

In the practice, time variable cannot be negative. The space $L^2(\Bbb R_+)$ of square integrable functions defined on the right half real line $\Bbb R_+$ models causal signal space. This paper focuses on a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$. The density theorem for Gabor systems in $L^2(\Bbb R)$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $L^2(\Bbb R)$ in terms of the index set alone-independently of window functions. The space $L^2(\Bbb R_+)$ admits no nontrivial Gabor system since $\Bbb R_+$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$ and the notion of $\Theta$-transform matrix. Using $\Theta$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $L^2(\Bbb R_+)$ under the condition that $\log_ba$ is a positive rational number, where $a$ and $b$ are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $L^2(\Bbb R_+)$ is that $\log_ba \leq 1$. Simultaneously, we obtain a $\Theta$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $L^2(\Bbb R_+)$.


Introduction
Gabor analysis in the three cases has some similarity although there exist many differences in some aspects.
The idea of considering frame theory on locally compact abelian groups has appeared in several publications including [3,4,6,7,9,19,21,22]. Write R + = (0, ∞). Given a, b > 0, a measurable function h defined on R + is said to be b-dilation periodic if h(b·) = h(·) on R + . Define the sequence {Λ m } m∈Z of b-dilation periodic functions by Our focus in this paper will be on the dilation-and-modulation systems (MD-systems) in L 2 (R + ) of the form: with ψ ∈ L 2 (R + ) under the following General setup: General setup: (i) a and b are two constants greater than 1.
(ii) log b a = p q , p and q are two coprime positive integers.
The space L 2 (R + ) can be considered as a closed subspace of L 2 (R) consisting of all functions in L 2 (R) which vanish outside R + . And it models causal signal space. In the practice, time variable cannot be negative. Mathematically, we are inspired by the following observations to study MD-systems of the form (1.4): • The space L 2 (R + ) admits no nontrivial shift invariant system of the form { T na ψ(·) : n ∈ Z } since ψ = 0 is a unique L 2 (R + )-solution to T na ψ(·) = 0 on (−∞, 0) for n ∈ Z.
This implies that it admits no nontrivial wavelet or Gabor system.
• An MD-system in L 2 (R + ) cannot be derived from a wavelet system in the Hardy space H 2 (R) via Fourier transform since the Fourier transform version of {D a j T m ψ : m, j ∈ Z} is {M a j m D a jψ : m, j ∈ Z}.
• The space L 2 (R + ) is not closed under the Fourier transform since the Fourier transform of a compactly supported nonzero function in L 2 (R + ) lies outside this space.
• R, Z and C L are locally compact abelian groups according to the usual addition and topology, while R + is not since the difference between two numbers in R + may be negative. So the analysis in L 2 (R + ) differs from that in L 2 (R). Also R + is a locally compact abelian group according to the usual multiplication and topology.
These observations inspire us to study multiplication-based frames for L 2 (R + ) of the form (1.4). In (1.4), the dilation periodicity and expression on [1, b) of Λ m are to match the dilation operation on ψ and to apply the Fourier series theory.
The density theorem essentially states that necessary and sufficient conditions for a Gabor system to be complete, a frame, a Riesz basis, or a Riesz sequence in L 2 (R) or general L 2 (R d ) can be formulated in terms of the index set alone-independently of the window function. It has a long and very involved history from the one-dimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor systems, and most recently beyond the setting of Gabor systems to abstract localized systems. For details, see [1,2,12,13,17,20,23,25,26] and references therein.
Let a and b be as in the general setup. We always write In this paper, we study the density theorem for the systems of the form (1.4) in L 2 (R + ). Section 2 is an auxiliary one. In this section, we introduce the notions of Θ β -transform and Θ β -transform matrix, and study their properties. In Section 3, using Θ β -transform matrix method we characterize complete MD-systems and MD-frames, obtain a parametrized expression of all complete systems and frames of the form (1.4) in L 2 (R + ), and finally we derive the density theorem. It turns out that " log b a ≤ 1" is necessary and Before proceeding, we introduce some notations and notions. Throughout this paper, the relation of quality, inclusion or inequality between two measurable sets is understood up to a set of measure zero, and the relation of quality or inequality between two measurable functions is understood in almost everywhere sense. We denote by I t the t × t identity matrix, and by N t the set for t ∈ N.
. So we say that S 1 and S 2 are αZ-congruent (T 1 and T 2 are α Z -dilation congruent) in this case. Also observe that Z is the superscript of α in the dilation congruence, and that only finitely many S 1,k among {S 1,k : k ∈ Z} are nonempty if both S 1 and S 2 are bounded in addition. Similarly, only finitely many Given d, L ∈ N and a measurable set E ⊂ R d , we denote by L 2 (E, C L ) the vector-valued Hilbert space where f l and g l denote the l-th components of f and g respectively. We denote by

Θ β -transform and Θ β -transform matrix
Let a and b be as in the general setup, and β be defined as in (1.5). This section is devoted to Θ β -transform matrix and related properties, which is an auxiliary one to following sections.
By a standard argument, we have the following two lemmas which partially appeared in [24, Lemma 2.1].
Lemma 2.2. (i) Θ β has the quasi-periodicity: (iii) The mappings Θ β and Γ are unitary operators from Then f is well-defined and the unique function satisfying for a.e. (x, ξ) ∈ S × [0, 1). This is in turn equivalent to by the definition of f .
Proof. By Remark 1.2, the conclusion of this lemma is equivalent to the fact that if (r 1 , s 1 , k) = (r 2 , s 2 , 0) for (r 1 , s 1 ), (r 2 , s 2 ) ∈ N q × N p and k ∈ Z, and that the set is Z-congruent to [0, 1).
This implies that p (s 2 − s 1 ) and q (r 1 − r 2 ) since p and q are coprime. It follows that (r 1 , s 1 ) = (r 2 , s 2 ) due to r 1 , r 2 ∈ N q and s 1 , s 2 ∈ N p . The proof is completed.
Under the hypothesis of Lemma 2.4, suppose that (r ′ , s ′ ) ∈ N q × N p is such that pr ′ + qs ′ = pq + 1.
Define L q (ξ) = 0 I q−r ′ e 2πiξ I r ′ 0 and R p (ξ) = 0 I s ′ e −2πiξ I p−s ′ 0 . (2.12) Then they are uniquely determined by p and q.
Lemma 2.5. Let a and b be as in the general setup, and ψ ∈ L 2 (R + ). We have for (l, m) ∈ Z × N q and a.e. (x, ξ) ∈ R + × R, where U m (ξ) = 0 I q−m e 2πiξ I m 0 .
Since pr ′ + qs ′ = pq + 1, we have It follows that and thus for (r, s) ∈ N q × N p and a.e. (x, ξ) ∈ R + × R by Lemma 2.2 (i) and a simple computation. So for a.e. (x, ξ) ∈ R + × R. This implies (ii). The proof is completed. Proof. When q = 1, we have b = a 1 p , and thus (2.16) is exactly (2.17). Next we prove their equivalence for the case q > 1.
By Lemma 2.5 and a simple computation, we have is as in (2.12). Also observe that which is in turn equivalent to (2.17). The proof is completed.
Let a and b be as in the general setup. In this section, using Θ β -transform matrix method, we characterize complete MD-systems and MD-frames, present a parameterized expression of them, and derive the density theorem for MD-systems of the form MD(ψ, a, b) in L 2 (R + ).
Lemma 3.1. Let a and b be as in the general setup, and ψ ∈ L 2 (R + ). Then for f ∈ L 2 (R + ).
Proof. By Lemma 2.2 (ii) and (iii), we have by Lemma 2.1 (ii). The proof is completed.
The following lemma is borrowed from [15], and it is a variation of [15, Corollary 2.4].
Lemma 3.2. An arbitrary µ × ν matrix-valued measurable function A on a measurable set E in R d must have the form where U (·) and V (·) are µ × µ and ν × ν unitary matrix-valued measurable functions on E respectively, and Λ(·) is a min(µ, ν)×min(µ, ν) diagonal matrix-valued measurable function on E.
By an easy application of the spectrum theorem for self-adjoint matrices (see also [8, p.978]), we have  Let P ker(Ψ(x, ξ)) be the orthogonal projection from C p onto the kernel space ker(Ψ(x, ξ)) of Ψ(x, ξ), and {e 1 , e 2 , · · · , e p } be the canonical orthonormal basis for C p , i.e., every e l with 1 ≤ l ≤ p is the vector in C p with the l-th component being 1 and others being zero. Then span{P ker(Ψ(x, ξ)) e l : 1 ≤ l ≤ p} = ker(Ψ(x, ξ)).
is necessary for the existence of complete MD-systems in L 2 (R + ). if and only if By a standard argument, (3.8) is equivalent to This is equivalent to by Lemma 2.6.
Collecting Theorems 3.1-3.3 and Remark 3.1, we obtain the following density theorem: Theorem 3.4. Let a and b be as in the general setup. Then the following are equivalent: (i) log b a ≤ 1.
Finally, we conclude this paper by the following conjecture.
Conjecture. In Theorem 3.4, log b a is required to be a rational number. This is a technical condition in all our arguments. We conjecture that, for general a, b > 1, (i), (ii) and (iii) are equivalent to each other.