1 Introduction
Before proceeding, we recall some notions and notations. An at most countable sequence \(\{e_{i}\}_{i\in\mathcal{I} }\) in a separable Hilbert space \(\mathcal{H}\) is called a frame for \(\mathcal{H}\) if there exist \(0< C_{1}\le C_{2}<\infty\) such that where \(C_{1}\), \(C_{2}\) are called frame bounds; it is called a Bessel sequence in \(\mathcal{H}\) if the right-hand side inequality in (1.1) holds, where \(C_{2}\) is called a Bessel bound. In particular, \(\{e_{i}\}_{i\in\mathcal{I} }\) is called a Parseval frame if \(C_{1}=C_{2}=1\) in (1.1). Given a frame \(\{e_{i}\}_{i\in\mathcal{I} }\) for \(\mathcal{H}\), a sequence \(\{\tilde{e}_{i}\}_{i\in\mathcal{I} }\) is called a dual of \(\{e_{i}\}_{i\in\mathcal{I} }\) if it is a frame such that It is easy to check that \(\{e_{i}\}_{i\in\mathcal{I} }\) is also a dual of \(\{\tilde{e}_{i}\}_{i\in\mathcal{I} }\) if \(\{\tilde{e}_{i}\}_{i\in \mathcal{I} }\) is a dual of \(\{e_{i}\}_{i\in\mathcal{I} }\). So, in this case, we say \(\{ e_{i}\}_{i\in\mathcal{I} }\) and \(\{\tilde{e}_{i}\}_{i\in\mathcal{I} }\) form a pair of dual frames for \(\mathcal{H}\). It is well known that \(\{e_{i}\}_{i\in\mathcal{I} }\) and \(\{\tilde{e}_{i}\}_{i\in\mathcal{I} }\) form a pair of dual frames for \(\mathcal{H}\) if they are Bessel sequences and satisfy (1.2). The fundamentals of frames can be found in [1‐3]. The Fourier transform of \(c\in l^{2}(\mathbb {Z})\) is defined by \(\hat{c}(\cdot)=\sum_{m\in \mathbb {Z}}c(m)e^{-2\pi im\cdot}\). For two sequences c and d on \(\mathbb {Z}\), the convolution
\(c*d\) is defined by if it is well defined. The Kronecker delta is defined by \(\delta_{n, m} = \left \{ \scriptsize{\begin{array}{@{}l@{\quad}l} 1 &\mbox{if }n=m; \\ 0& \mbox{if }n\neq m. \end{array}} \right .\)
\(l_{0}(\mathbb {Z})\) denotes the set of finitely supported sequences on \(\mathbb {Z}\). We denote by I the identity operator on \(l^{2}(\mathbb {Z})\), and by \(\chi_{E}\) its characteristic function for a set E. Write \(\mathbb {R}_{+}=(0, \infty)\). For a positive number \(a>1\), a function h defined on \(\mathbb {R}_{+}\) is said to be a-dilation periodic if \(h(a\cdot )=h(\cdot)\) on \(\mathbb {R}_{+}\). For a function f defined on \([1, a)\), we define the function f̃ on \(\mathbb {R}_{+}\) by which is called the a-dilation periodization of f. Obviously, it is a-dilation periodic.
$$ C_{1}\|f\|^{2}\le\sum _{i \in\mathcal{I}} \bigl\vert \langle f, e_{i}\rangle \bigr\vert ^{2}\le C_{2}\|f\|^{2} \quad\mbox{for }f\in \mathcal{H}, $$
(1.1)
$$ f= \sum_{i \in\mathcal{I}}\langle f, \tilde{e}_{i}\rangle e_{i}\quad\mbox{for } f\in\mathcal{H}. $$
(1.2)
$$c*d(k)=\sum_{m\in \mathbb {Z}}c(k-m)d(m)\quad\mbox{for }k\in \mathbb {Z} $$
$$\tilde{f}(\cdot)=f\bigl(a^{-l}\cdot\bigr)\quad\mbox{on } \bigl[a^{l}, a^{l+1}\bigr)\mbox{ for }l\in \mathbb {Z}, $$
The translation operator \(T_{x_{0}}\), the modulation operator \(M_{x_{0}}\) with \(x_{0}\in \mathbb {R}\), and the dilation \(D_{c}\) with \(c>0\) are, respectively, defined by and for \(f\in L^{2}(\mathbb {R})\). They are the basis of wavelet analysis. Affine systems of the form \(\{D_{a^{j}}T_{bk}\psi: j, k\in \mathbb {Z}\}\) with \(\psi\in L^{2}(\mathbb {R})\) and a, \(b>0\), and Gabor systems of the form \(\{E_{mb}T_{na}g: m, n\in \mathbb {Z}\}\) with \(g\in L^{2}(\mathbb {R})\) and a, \(b>0\) have been extensively studied. However, dilation-and-modulation systems of the form have not been extensively studied. This paper focuses on the following systems that are like (1.3): where We will investigate the theory of \(L^{2}(\mathbb {R}_{+})\)-frames of the form (1.4). It is obvious that \(L^{2}(\mathbb {R}_{+})\) can be considered as the Fourier transform of the Hardy space \(H^{2}(\mathbb {R})\) defined by where the Fourier transform is defined by and extended to \(L^{2}(\mathbb {R})\) by the Plancherel theorem. Wavelet frames in \(H^{2}(\mathbb {R})\) of the form \(\{D_{a^{j}}T_{m}\varphi: j, m\in \mathbb {Z}\}\) were studied in [4, 5], and some variations can be found in [6‐11]. By the Plancherel theorem, an \(H^{2}(\mathbb {R})\)-frame \(\{D_{2^{j}}T_{m}\varphi: j, m\in \mathbb {Z}\}\) leads to a \(L^{2}(\mathbb {R}_{+})\)-frame In (1.6), \(e^{-2\pi i2^{-j}m\cdot}\) is \(2^{j}\mathbb {Z}\)-periodic with respect to additive operation, and the period depends on the dilation factor \(2^{j}\). However, \(\widetilde{\psi_{m}}\) in (1.4) is a-dilation periodic and unrelated to j. Therefore, frames of the form (1.4) are different from ones of the form (1.6) for \(L^{2}(\mathbb {R}_{+})\) and of independent interest. They are related to a kind of function-valued frames in [12]. In [13], numerical experiments were made to establish that the nonnegative integer shifts of the Gaussian function formed a Riesz sequence in \(L^{2}(\mathbb {R}_{+})\). In [14], a sufficient condition was obtained to determine whether the nonnegative translates form a Riesz sequence on \(L^{2}(\mathbb {R}_{+})\).
$$\begin{aligned}& T_{x_{0}}f(\cdot)=f(\cdot-x_{0}),\\& M_{x_{0}}f(\cdot)=e^{2\pi ix_{0}\cdot}f(\cdot), \end{aligned}$$
$$D_{c}f(\cdot)=\sqrt{c}f(c\cdot) $$
$$ \{M_{mb}D_{a^{j}}\psi: m, j\in \mathbb {Z} \} \quad\mbox{with }a, b>0 $$
(1.3)
$$ \{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z} \}\quad\mbox{with }a>0, $$
(1.4)
$$ \psi_{m}(\cdot)=\frac{1}{\sqrt{a-1}}e^{2\pi i\frac{m\cdot }{a-1}}\quad\mbox{on }[1, a)\mbox{ for }m\in \mathbb {Z}. $$
(1.5)
$$H^{2}(\mathbb {R})=\bigl\{ f\in L^{2}(\mathbb {R}):\hat{f}(\cdot)=0 \mbox{ a.e. on } (-\infty, 0)\bigr\} , $$
$$\hat{f}(\cdot)= \int_{\mathbb {R}}f(x)e^{-2\pi ix\cdot}\,dx\quad\mbox{for }f\in L^{1}(\mathbb {R})\cap L^{2}(\mathbb {R}) $$
$$ \bigl\{ e^{-2\pi i2^{-j}m\cdot}\hat{\varphi}\bigl(2^{-j}\cdot \bigr): j, m\in \mathbb {Z} \bigr\} . $$
(1.6)
Anzeige
The rest of this paper is organized as follows. Section 2 is devoted to characterizing frames and dual frames for \(L^{2}(\mathbb {R}_{+})\) with the structure of (1.4). Section 3 is devoted to Parseval frames and orthonormal bases for \(L^{2}(\mathbb {R}_{+})\) of the form (1.4). It turns out that Parseval frames, orthonormal bases, and orthonormal systems in \(L^{2}(\mathbb {R}_{+})\) of the form (1.4) are mutually equivalent to each other. It is worth noting that neither affine systems nor Gabor systems have such a property.
2 Frame and dual frame characterization
This section characterizes \(L^{2}(\mathbb {R}_{+})\)-frames and dual frames of the form (1.4). For this purpose, we need some notations and lemmas. For \(f\in L^{2}(\mathbb {R}_{+})\), we define and
$$ \mathcal{G}(f,\cdot)= \bigl(\overline{D_{a^{j+l}}f(\cdot)} \bigr)_{j, l\in \mathbb {Z}} $$
(2.1)
$$ F(\cdot)= \bigl(D_{a^{l}}f(\cdot) \bigr)_{l\in \mathbb {Z}}. $$
(2.2)
Lemma 2.1
For
m, \(j\in \mathbb {Z}\), define
\(\psi_{m}\)
as in (1.5), then
(i)
\(\{\psi_{m}: m\in \mathbb {Z}\}\)
is an orthonormal bases for
\(L^{2}([1, a))\);
(ii)
for
\(f\in L^{1}([1, a))\).
$$ \int_{[1, a)}\bigl\vert f(x)\bigr\vert ^{2}\,dx=\sum _{m\in \mathbb {Z}}\biggl\vert \int _{[1, a)}f(x)\overline{\psi_{m}(x)}\,dx\biggr\vert ^{2} $$
(2.3)
Proof
(i) By a simple computation, we see that \(T_{1}D_{(a-1)^{-1}}\) is a unitary operator from \(L^{2}([0, 1))\) onto \(L^{2}([1, a))\), and for \(m\in \mathbb {Z}\). Also observing that \(\{e^{2\pi im\cdot}: m\in \mathbb {Z}\}\) is an orthonormal basis for \(L^{2}([0, 1))\), we have (i).
$$\psi_{m}(\cdot)=e^{\frac{2\pi im}{a-1}}T_{1}D_{(a-1)^{-1}}e^{2\pi im\cdot} $$
Anzeige
(ii) By (i), (2.3) holds if \(f\in L^{2}([1, a))\). When \(f\in L^{1}([1, a))\setminus L^{2}([1, a))\), the left-hand side of (2.3) is infinity. Now we prove by contradiction that the right-hand side of (2.3) is also infinity. Suppose it is finite. Since \(\{ \psi_{m}: m\in \mathbb {Z}\}\) is an orthonormal basis for \(L^{2}([1, a))\), we see that belongs to \(L^{2}([1, a))\), and thus to \(L^{1}([1, a))\). It has the same Fourier coefficients as f. So \(f=g\) by the uniqueness of Fourier coefficients, and thus \(f\in L^{2}([1, a))\). This is a contradiction. The proof is completed. □
$$g=\sum_{m\in \mathbb {Z}} \biggl( \int_{[1, a)}f(x)\overline{\psi _{m}(x)}\,dx \biggr) \psi_{m} $$
Lemma 2.2
Let
\(c, d\in l^{2}(\mathbb {Z})\). Suppose
\(c\ast d\in l^{2}(\mathbb {Z})\), then
$$(c\ast d)^{\wedge}(\cdot)=\hat{c}(\cdot)\hat{d}(\cdot)\quad\textit{a.e. on } \mathbb {R}. $$
Proof
Since \(c\ast d\in l^{2}(\mathbb {Z})\), we have \((c\ast d)^{\wedge }\in L^{2}([0, 1))\), and thus \((c\ast d)^{\wedge}\in L^{1}([0, 1))\). Also observing that \(\hat{c}\cdot\hat{d}\in L^{1}([0, 1))\), to finish the proof we only need to prove that Define \(c_{n}\) by for \(n\in \mathbb {N}\). Then \(c_{n}\in l^{1}(\mathbb {Z})\), and Since \(c_{n}\in l^{1}(\mathbb {Z})\) and \(d\in l^{2}(\mathbb {Z})\), we have \((c_{n}\ast d)^{\wedge}=\hat{c}_{n}\hat{d}\in L^{2}([0, 1))\), and thus Arbitrarily fix \(k\in \mathbb {Z}\). Observe that and as \(n\rightarrow\infty\) by (2.5). So, letting \(n\rightarrow \infty\) in (2.6) we have (2.4). The proof is completed. □
$$ \int_{[0, 1)}\hat{c}(\xi)\hat{d}(\xi)e^{2\pi ik\xi}\,d\xi=c\ast d(k)\quad\mbox{for }k\in \mathbb {Z}. $$
(2.4)
$$c_{n}(k) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} c(k) & \mbox{if }\vert k\vert \leq n; \\ 0 & \mbox{if }\vert k\vert > n \end{array}\displaystyle \right . $$
$$ \|c_{n}-c\|_{l^{2}(\mathbb {Z})}=\|\hat{c}_{n}- \hat{c}\|_{L^{2}([0, 1))}\rightarrow0\quad\mbox{as } n\rightarrow\infty . $$
(2.5)
$$ \int_{[0, 1)}\hat{c}_{n}(\xi)\hat{d}( \xi)e^{2\pi ik\xi}\,d\xi =\sum_{l\in \mathbb {Z}}c_{n}(k-l)d(l) \quad\mbox{for }k\in \mathbb {Z}. $$
(2.6)
$$\biggl\vert \sum_{l\in \mathbb {Z}}c_{n}(k-l)d(l)- \sum_{l\in \mathbb {Z}}c(k-l)d(l)\biggr\vert \leq \|c_{n}-c\|_{l^{2}(\mathbb {Z})}\|d\|_{l^{2}(\mathbb {Z})}\rightarrow0 $$
$$\biggl\vert \int_{[0, 1)}\hat{c}_{n}(\xi)\hat{d}( \xi)e^{2\pi ik\xi }\,d\xi- \int_{[0, 1)}\hat{c}(\xi)\hat{d}(\xi)e^{2\pi ik\xi}\,d\xi \biggr\vert \leq\|\hat{c}_{n}-\hat{c}\|_{L^{2}([0, 1))}\|\hat{d}\| _{L^{2}([0, 1))}\rightarrow0 $$
Lemma 2.3
For
\(f, \psi\in L^{2}(\mathbb {R}_{+})\), we have
$$ \sum_{j\in \mathbb {Z}}\sum _{m\in \mathbb {Z}}\bigl\vert \langle f, \widetilde{\psi_{m}}D_{a^{j}} \psi\rangle_{L^{2}(\mathbb {R}_{+})}\bigr\vert ^{2}= \int_{1}^{a}\bigl\| \mathcal{G}(\psi, x)F(x) \bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx. $$
(2.7)
Proof
Since \(f(\cdot)\overline{\widetilde{\psi_{m}}(\cdot )D_{a^{j}}\psi(\cdot)}\in L^{1}(\mathbb {R}_{+})\), we have It follows that by Lemma 2.1(ii), and thus (2.7) holds. The proof is completed. □
$$\begin{aligned} \langle f, \widetilde{\psi_{m}}D_{a^{j}}\psi\rangle _{L^{2}(\mathbb {R}_{+})} =& \int_{1}^{a}\sum_{l\in \mathbb {Z}}a^{l}f \bigl(a^{l}x\bigr)\overline {\widetilde{\psi_{m}} \bigl(a^{l}x\bigr)}\overline{(D_{a^{j}}\psi) \bigl(a^{l}x\bigr)}\,dx \\ =& \int_{1}^{a}\overline{\psi_{m}(x)}\sum _{l\in \mathbb {Z}}(D_{a^{l}}f) (x)\overline{D_{a^{j+l}} \psi(x)}\,dx. \end{aligned}$$
$$\sum_{j\in \mathbb {Z}}\sum_{m\in \mathbb {Z}} \bigl\vert \langle f, \widetilde{\psi_{m}}D_{a^{j}}\psi \rangle_{L^{2}(\mathbb {R}_{+})}\bigr\vert ^{2}=\sum _{j\in \mathbb {Z}} \int_{1}^{a} \biggl\vert \sum _{l\in \mathbb {Z}}(D_{a^{l}}f) (x)\overline {(D_{a^{j+l}}\psi) (x)}\biggr\vert ^{2}\,dx $$
Theorem 2.1
Let
\(\psi\in L^{2}(\mathbb {R}_{+})\). The following are equivalent:
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is a Bessel sequence in
\(L^{2}(\mathbb {R}_{+})\)
with Bessel bound
B.
(ii)
\(\mathcal{G}^{\ast}\mathcal{G}(\psi, \cdot)\leq BI\)
a.e. on
\([1, a)\).
(iii)
\(\vert \sum_{l\in \mathbb {Z}}a^{\frac {l}{2}}\psi(a^{l}x)e^{-2\pi il\xi} \vert \leq\sqrt{B}\)
for a.e. \((x, \xi)\in[1, a)\times[0, 1)\).
Proof
Since we have \(\sum_{j\in \mathbb {Z}}|D_{a^{j}}\psi(\cdot)|^{2}<\infty\) a.e. on \([1, a)\). So all rows and columns of \(\mathcal{G}(\psi, x)\) belong to \(l^{2}(\mathbb {Z})\) for a.e. \(x\in[1, a)\). It follows that \(\mathcal{G}^{\ast} \mathcal{G}(\psi, \cdot)\) is well defined a.e. on \([1, a)\) by the Cauchy-Schwartz inequality. Thus \(\langle\mathcal{G}^{\ast}\mathcal{G}(\psi , \cdot)c, c\rangle_{l^{2}(\mathbb {Z})}\) and \(\|\mathcal{G}(\psi, \cdot )c\|_{l^{2}(\mathbb {Z})}^{2}\) are well defined, and a.e. on \([1,a)\) for each \(c\in l_{0}(\mathbb {Z})\). First we show the equivalence between (i) and (ii). Suppose \(\mathcal{G}^{\ast}\mathcal{G}(\psi, \cdot)\leq BI\) a.e. on \([1, a)\). Arbitrarily fix \(f\in L^{2}(\mathbb {R}_{+})\). It is easy to check that So \(F(x)\in l^{2}(\mathbb {Z})\), and thus for a.e. \(x\in[1, a)\). It follows that Therefore, by Lemma 2.3, and thus \(\{\widetilde{\psi_{m}}D_{a^{j}}\psi : m, j\in \mathbb {Z}\}\) is a Bessel sequence in \(L^{2}(\mathbb {R}_{+})\) with Bessel bound B.
$$\int_{1}^{a}\sum_{j\in \mathbb {Z}} \bigl\vert D_{a^{j}}\psi(x)\bigr\vert ^{2}\,dx= \int_{0}^{\infty}\bigl|\psi(x)\bigr|^{2}\,dx< \infty, $$
$$ \bigl\langle \mathcal{G}^{\ast}\mathcal{G}(\psi, \cdot)c, c \bigr\rangle _{l^{2}(\mathbb {Z})}=\bigl\| \mathcal{G}(\psi, \cdot)c\bigr\| _{l^{2}(\mathbb {Z})}^{2} $$
(2.8)
$$\int_{1}^{a}\bigl\| F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx= \|f\|_{L^{2}(\mathbb {R}_{+})}^{2}< \infty. $$
$$\bigl\| \mathcal{G}(\psi, x)F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}=\bigl\langle \mathcal{G}^{\ast }\mathcal{G}(\psi, x)F(x), F(x)\bigr\rangle _{l^{2}(\mathbb {Z})} \leq B\bigl\| F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}, $$
$$\int_{1}^{a}\bigl\| \mathcal{G}(\psi, x)F(x) \bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx\leq B \int _{1}^{a}\bigl\| F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx=B \|f\|_{L^{2}(\mathbb {R}_{+})}^{2}. $$
$$\sum_{j\in \mathbb {Z}}\sum_{m\in \mathbb {Z}} \bigl\vert \langle f, \widetilde{\psi_{m}}D_{a^{j}}\psi \rangle_{L^{2}(\mathbb {R}_{+})}\bigr\vert ^{2}\leq B\|f\|_{L^{2}(\mathbb {R}_{+})}^{2}, $$
Now suppose (i) holds. We prove (ii) by contradiction. Suppose \(\mathcal{G}^{\ast}\mathcal{G}(\psi, \cdot)>BI\) on some \(E\subset[1, a)\) with \(|E|>0\). Take \(0\neq c\in l_{0}(\mathbb {Z})\) and define \(f\in L^{2}(\mathbb {R}_{+})\) by Then f is well defined since Let us check \(\int_{1}^{a}\|\mathcal{G}(\psi, x)F(x)\|_{l^{2}(\mathbb {Z})}^{2}\,dx\): So \(\sum_{j\in \mathbb {Z}}\sum_{m\in \mathbb {Z}}\vert \langle f, \widetilde{\psi_{m}}D_{a^{j}}\psi\rangle_{L^{2}(\mathbb {R}_{+})}\vert ^{2}>B\|f\|_{L^{2}(\mathbb {R}_{+})}^{2}\) by Lemma 2.3, contradicting the fact that \(\{\widetilde{\psi _{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\) is a Bessel sequence in \(L^{2}(\mathbb {R}_{+})\) with Bessel bound B.
$$F(\cdot)=\chi_{E}(\cdot)c\quad\mbox{on }[1, a). $$
$$\|f\|_{L^{2}(\mathbb {R}_{+})}^{2}= \int_{1}^{a}\bigl\| F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx=|E| \|c\|_{l^{2}(\mathbb {Z})}^{2}. $$
$$\begin{aligned} \int_{1}^{a}\bigl\| \mathcal{G}(\psi, x)F(x) \bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx =& \int_{1}^{a}\bigl\langle \mathcal{G}^{\ast} \mathcal{G}(\psi, x)F(x), F(x)\bigr\rangle _{l^{2}(\mathbb {Z})}\,dx \\ =& \int_{E}\bigl\langle \mathcal{G}^{\ast}\mathcal{G}( \psi, x)F(x), F(x)\bigr\rangle _{l^{2}(\mathbb {Z})}\,dx \\ >&B \int_{1}^{a}\bigl\| F(x)\bigr\| _{l^{2}(\mathbb {Z})}^{2}\,dx \\ =&B\|f\|_{L^{2}(\mathbb {R}_{+})}^{2}. \end{aligned}$$
Next we prove the equivalence between (ii) and (iii). By (2.8) and the density of \(l_{0}(\mathbb {Z})\) in \(l^{2}(\mathbb {Z})\), (ii) is equivalent to which is equivalent to For x satisfying (2.9), define \(\alpha_{x}\in l^{2}(\mathbb {Z})\) by Then (2.9) can be rewritten as equivalently, by Lemma 2.2, this is also equivalent to by the density of trigonometric polynomials in \(L^{2}([0, 1))\). It is obvious that (2.10) is equivalent to (iii). The proof is completed. □
$$\bigl\| \mathcal{G}(\psi, x)c\bigr\| ^{2}_{l^{2}(\mathbb {Z})}\leq B\|c \|^{2}_{l^{2}(\mathbb {Z})}\quad\mbox{for }c\in l_{0}(\mathbb {Z})\mbox{ and a.e. }x\in[1, a), $$
$$ \bigl\| \overline{\mathcal{G}(\psi, x)}c\bigr\| ^{2}_{l^{2}(\mathbb {Z})} \leq B\|c\| ^{2}_{l^{2}(\mathbb {Z})}\quad\mbox{for }c\in l_{0}(\mathbb {Z})\mbox{ and a.e. }x\in[1, a). $$
(2.9)
$$\alpha_{x}(l)=a^{\frac{l}{2}}\psi\bigl(a^{l}x\bigr). $$
$$\|\alpha_{x}\ast d\|^{2}_{l^{2}(\mathbb {Z})}\leq B\|d \|^{2}_{l^{2}(\mathbb {Z})}\quad\mbox{ for }\,d\in l_{0}(\mathbb {Z})\mbox{ and a.e. }x\in[1, a), $$
$$\int_{0}^{1}\bigl|\hat{\alpha}_{x}(\xi) \hat{d}(\xi)\bigr|^{2}\,d\xi\leq B \int_{0}^{1}\bigl|\hat{d}(\xi)\bigr|^{2}\,d\xi \quad\mbox{for }\,d\in l_{0}(\mathbb {Z})\mbox{ and a.e. }x\in[1, a) $$
$$ \int_{0}^{1}\bigl|\hat{\alpha}_{x}( \xi)\bigr|^{2}\bigl|\hat{d}(\xi)\bigr|^{2}\,d\xi \leq B \int_{0}^{1}\bigl|\hat{d}(\xi)\bigr|^{2}\,d\xi \quad\mbox{for }\,d\in l^{2}(\mathbb {Z})\mbox{ and a.e. }x\in[1, a), $$
(2.10)
By the same procedure as in the proof of Theorem 2.1, we have the following theorem.
Theorem 2.2
Let
\(\psi\in L^{2}(\mathbb {R}_{+})\). The following are equivalent:
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is a frame for
\(L^{2}(\mathbb {R}_{+})\)
with frames bounds
A
and
B;
(ii)
\(AI\leq\mathcal{G}^{\ast}\mathcal{G}(\psi, \cdot)\leq BI\)
a.e. on
\([1, a)\);
(iii)
\(\sqrt{A}\leq \vert \sum_{l\in \mathbb {Z}}a^{\frac {l}{2}}\psi(a^{l}x)e^{-2\pi il\xi} \vert \leq\sqrt{B}\)
for a.e. \((x, \xi)\in[1, a)\times[0, 1)\).
Theorem 2.3
Let
ψ, \(\varphi\in L^{2}(\mathbb {R}_{+})\). Suppose
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
and
\(\{ \widetilde{\psi_{m}}D_{a^{j}}\varphi: m, j\in \mathbb {Z}\}\)
are Bessel sequences in
\(L^{2}(\mathbb {R}_{+})\). The following are equivalent:
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
and
\(\{ \widetilde{\psi_{m}}D_{a^{j}}\varphi: m, j\in \mathbb {Z}\}\)
form a pair of dual frames for
\(L^{2}(\mathbb {R}_{+})\).
(ii)
\(\sum_{j\in \mathbb {Z}}a^{j}\varphi(a^{j}\cdot)\overline{\psi (a^{j+l}\cdot)}=\delta_{l,0}\)
a.e. on
\([1, a)\)
for
\(l\in \mathbb {Z}\).
(iii)
\((\sum_{l\in \mathbb {Z}}a^{\frac {l}{2}}\varphi(a^{l}x)e^{-2\pi il\xi} ) (\sum_{l\in \mathbb {Z}} a^{\frac{l}{2}}\overline{\psi(a^{l}x)}e^{2\pi il\xi } )=1\)
for a.e. \((x, \xi)\in[1, a)\times[0, 1)\).
Proof
Since both \(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\) and \(\{\widetilde{\psi_{m}}D_{a^{j}}\varphi: m, j\in \mathbb {Z}\}\) are Bessel sequences in \(L^{2}(\mathbb {R}_{+})\), is well defined for each f, \(g\in L^{2}(\mathbb {R}_{+})\). By Theorem 2.1, \(\mathcal{G}(\psi, x)\) and \(\mathcal{G}(\varphi, x)\) are bounded linear operators on \(l^{2}(\mathbb {Z})\) for a.e. \(x\in[1, a)\). Let us check the \((k,l)\)-entry of \(\mathcal{G}^{\ast}(\varphi, \cdot) \mathcal{G}(\psi, \cdot)\): So (ii) is equivalent to By Lemma 2.3 and the polarization identity, we have for \(f, g\in L^{2}(\mathbb {R}_{+})\). By a simply computation, we also have for f, \(g\in L^{2}(\mathbb {R}_{+})\). So (i) holds if and only if So, to show the equivalence between (i) and (ii), next we only need to prove the equivalence between (2.11) and (2.12). Obviously, (2.11) implies (2.12). Now we prove the converse implication. Suppose (2.12) holds. Let \(e_{n}\) be the vector in \(l^{2}(\mathbb {Z})\) with the nth component being 1 and the others being zero for each \(n\in \mathbb {Z}\). Since all entries of \(\mathcal{G}^{\ast }(\varphi, \cdot)\mathcal{G}(\psi, \cdot)\) are in \(L^{1}[1, a)\), there exists \(E_{0}\subset[1, a)\) with \(|E_{0}|=0\) such that every point of \((1, a)\setminus E_{0}\) is a Lebesgue point of all entries of \(\mathcal{G}^{\ast}(\varphi, \cdot)\mathcal{G}(\psi, \cdot)\). Arbitrarily fix \(x_{0}\in(1, a)\setminus E_{0}\) and \(n_{0}\), \(m_{0}\in \mathbb {Z}\). Take \(\varepsilon>0\) so small that \((x_{0}-\varepsilon, x_{0}+\varepsilon)\subset(1, a)\), and take f and g such that a.e. on \([1, a)\). Then from (2.12) we have This leads to by letting \(\varepsilon\rightarrow0\). By the arbitrariness of \(x_{0}\), \(n_{0}\), and \(m_{0}\), we have on \([1, a)\setminus E_{0}\) for n, \(m\in \mathbb {Z}\). This implies that Therefore, we obtain the equivalence between (i) and (ii).
$$\sum_{(m,j)\in \mathbb {Z}^{2}}\langle f, \widetilde{\psi _{m}}D_{a^{j}}\psi\rangle_{L^{2}(\mathbb {R}_{+})}\langle\widetilde{\psi _{m}}D_{a^{j}}\varphi, g\rangle_{L^{2}(\mathbb {R}_{+})} $$
$$\begin{aligned} \bigl[\mathcal{G}^{\ast}(\varphi, \cdot)\mathcal{G}(\psi, \cdot ) \bigr]_{k,l} =&\sum_{j\in \mathbb {Z}}D_{a^{j+k}} \varphi(\cdot)\overline {D_{a^{j+l}}\psi(\cdot)} =\sum_{j\in \mathbb {Z}}a^{\frac{j+k}{2}}\varphi \bigl(a^{j+k}\cdot \bigr)a^{\frac{j+l}{2}}\overline{\psi \bigl(a^{j+l}\cdot\bigr)} \\ =& a^{\frac{l-k}{2}}\sum_{j\in \mathbb {Z}}a^{j}\varphi \bigl(a^{j}\cdot \bigr)\overline{\psi\bigl(a^{j+l-k}\cdot\bigr)}. \end{aligned}$$
$$ \mathcal{G}^{\ast}(\varphi, \cdot)\mathcal{G}(\psi, \cdot)=I\quad\mbox{a.e. on }[1, a). $$
(2.11)
$$\sum_{(m,j)\in \mathbb {Z}^{2}}\langle f, \widetilde{\psi _{m}}D_{a^{j}}\psi\rangle_{L^{2}(\mathbb {R}_{+})}\langle\widetilde{\psi _{m}}D_{a^{j}}\varphi, g\rangle_{L^{2}(\mathbb {R}_{+})}= \int _{1}^{a}\bigl\langle \mathcal{G}^{\ast}( \varphi, x)\mathcal{G}(\psi, x)F(x), G(x) \bigr\rangle _{l^{2}(\mathbb {Z})}\,dx $$
$$\langle f, g\rangle_{L^{2}(\mathbb {R}_{+})}= \int_{1}^{a}\bigl\langle F(x), G(x)\bigr\rangle _{l^{2}(\mathbb {Z})}\,dx $$
$$\begin{aligned} &\int_{1}^{a}\bigl\langle \mathcal{G}^{\ast}( \varphi, x)\mathcal{G}(\psi, x)F(x), G(x)\bigr\rangle _{l^{2}(\mathbb {Z})}\,dx \\ &\quad= \int_{1}^{a}\bigl\langle F(x), G(x)\bigr\rangle _{l^{2}(\mathbb {Z})}\,dx\quad\mbox{for }f, g\in L^{2}(\mathbb {R}_{+}). \end{aligned}$$
(2.12)
$$F(\cdot)=\chi_{(x_{0}-\varepsilon, x_{0}+\varepsilon)}(\cdot ) e_{n_{0}}\quad\mbox{and}\quad G( \cdot)=\chi_{(x_{0}-\varepsilon, x_{0}+\varepsilon)}(\cdot)e_{m_{0}} $$
$$\frac{1}{2\varepsilon} \int_{x_{0}-\varepsilon}^{x_{0}+\varepsilon }\bigl\langle \mathcal{G}^{\ast}( \varphi, x)\mathcal{G}(\psi, x)e_{n_{0}}, e_{m_{0}} \bigr\rangle _{l^{2}(\mathbb {Z})}\,dx=\delta_{n_{0},m_{0}}. $$
$$\bigl\langle \mathcal{G}^{\ast}(\varphi, x_{0})\mathcal{G}( \psi, x_{0})e_{n_{0}}, e_{m_{0}} \bigr\rangle _{l^{2}(\mathbb {Z})}=\delta_{n_{0},m_{0}}, $$
$$\bigl\langle \mathcal{G}^{\ast}(\varphi, \cdot)\mathcal{G}(\psi, \cdot )e_{n}, e_{m} \bigr\rangle _{l^{2}(\mathbb {Z})}= \delta_{n,m} $$
$$\mathcal{G}^{\ast}(\varphi, \cdot)\mathcal{G}(\psi, \cdot)=I\quad \mbox{on }(1, a)\setminus E_{0}. $$
Now we prove the equivalence between (ii) and (iii). Obviously, (ii) is equivalent to that is, For x satisfying (2.13), define \(\alpha_{x}, \beta_{x}\in l^{2}(\mathbb {Z})\) by Then (2.13) can be rewritten as This is equivalent to by Lemma 2.2, and thus equivalent to (iii) by its \(\mathbb {Z}\)-periodicity with respect to ξ. The proof is completed. □
$$\sum_{j\in \mathbb {Z}}a^{\frac{j}{2}}\varphi \bigl(a^{j}\cdot\bigr)a^{\frac {j+l}{2}}\overline{\psi \bigl(a^{j+l}\cdot\bigr)}=\delta_{l,0}\quad\mbox{ a.e. on }[1, a) \mbox{ for }l\in \mathbb {Z}, $$
$$ \sum_{j\in \mathbb {Z}}a^{-\frac{j}{2}}\varphi \bigl(a^{-j}\cdot \bigr)a^{\frac{l-j}{2}}\overline{\psi \bigl(a^{l-j}\cdot\bigr)}=\delta_{l,0} \quad\mbox{a.e. on }[1, a)\mbox{ for }l\in \mathbb {Z}. $$
(2.13)
$$\alpha_{x}(l)=a^{\frac{l}{2}}\overline{\psi\bigl(a^{l}x \bigr)}\quad \mbox{and}\quad\beta_{x}(l)=a^{-\frac{l}{2}}\varphi \bigl(a^{-l}x\bigr). $$
$$\alpha_{x}\ast\beta_{x}=\delta_{l,0}\quad \mbox{for a.e. }x\in[1, a)\mbox{ and }l\in \mathbb {Z}. $$
$$\biggl(\sum_{l\in \mathbb {Z}}a^{\frac{l}{2}}\overline{\psi \bigl(a^{l}x\bigr)}e^{-2\pi il\xi} \biggr) \biggl(\sum _{l\in \mathbb {Z}}a^{\frac{l}{2}}\varphi\bigl(a^{l}x \bigr)e^{2\pi il\xi} \biggr)=1 \quad\mbox{a.e. } (x, \xi)\in[1, a)\times[0, 1) $$
3 Parseval frames and orthonormal bases
The following theorem characterizes Parseval frames of the form (1.4). It is worth noting that the theorem shows all such Parseval frames must be orthonormal bases. This is not the case for Parseval wavelet (Gabor) frames.
Theorem 3.1
Let
\(\psi\in L^{2}(\mathbb {R}_{+})\). The following are equivalent:
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is a Parseval frames for
\(L^{2}(\mathbb {R}_{+})\).
(ii)
\(\sum_{j\in \mathbb {Z}}a^{j}\psi(a^{j}\cdot)\overline{\psi (a^{j+l}\cdot)}=\delta_{l,0}\)
a.e. on
\([1, a)\)
for
\(l\in \mathbb {Z}\).
(iii)
\(\vert \sum_{l\in \mathbb {Z}}a^{\frac {l}{2}}\psi(a^{l}x)e^{2\pi il\xi} \vert =1\)
for a.e. \((x, \xi)\in[1, a)\times[0, 1)\).
(iv)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is an orthonormal basis for
\(L^{2}(\mathbb {R}_{+})\).
Proof
Since (ii) and (iii) are equivalent, and (iii) implies that \(\{ \widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\) is a Bessel sequence by Theorem 2.1, we can obtain the equivalence among (i), (ii), and (iii) by taking \(\varphi=\psi\) in Theorem 2.3. It is obvious that (iv) implies (i). Now we prove the converse implication to finish the proof. Suppose (i) holds. Then (iv) holds if and only if equivalently, \(\|\psi\|_{L^{2}(\mathbb {R}_{+})}^{2}=a-1\). By the equivalence between (i) and (ii), we have Integrating this equation over \([1, a)\) leads to So (iv) holds. The proof is completed. □
$$\|\widetilde{\psi_{m}}D_{a^{j}}\psi\|_{L^{2}(\mathbb {R}_{+})}=1 \quad \mbox{for } m, j\in \mathbb {Z}, $$
$$\sum_{j\in \mathbb {Z}}a^{j}\bigl|\psi\bigl(a^{j} \cdot\bigr)\bigr|^{2}=1 \quad\mbox{a.e. on } [1, a). $$
$$\|\psi\|_{L^{2}(\mathbb {R}_{+})}^{2}=a-1. $$
Theorem 3.2
Let
\(0\ne\psi\in L^{2}(\mathbb {R}_{+})\). Then we have
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is an orthogonal system in
\(L^{2}(\mathbb {R}_{+})\)
if and only if
$$ \sum_{j\in \mathbb {Z}}a^{j}\psi \bigl(a^{j}\cdot\bigr)\overline{\psi \bigl(a^{j+l}\cdot\bigr)}= \frac{\|\psi\|^{2}_{L^{2}(\mathbb {R}_{+})}}{a-1}\delta _{l,0} \quad\textit{a.e. on }[1, a)\textit{ for }l\in \mathbb {Z}; $$
(3.1)
(ii)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is an orthonormal system for
\(L^{2}(\mathbb {R}_{+})\)
if and only if
$$ \sum_{j\in \mathbb {Z}}a^{j}\psi \bigl(a^{j}\cdot\bigr)\overline{\psi \bigl(a^{j+l}\cdot\bigr)}= \delta_{l,0}\quad\textit{a.e. on }[1, a)\textit{ for }l\in \mathbb {Z}. $$
(3.2)
Proof
(i) By a simple computation, we have for \((m_{1}, j_{1}), (m_{2}, j_{2})\in \mathbb {Z}^{2}\). It follows that \(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\) is an orthogonal system in \(L^{2}(\mathbb {R}_{+})\) if and only if Also observing that we have (i) by Lemma 2.1(i) and the uniqueness theorem of Fourier coefficients.
$$\langle\widetilde{\psi_{m_{1}}}D_{a^{j_{1}}}\psi, \widetilde{\psi _{m_{2}}}D_{a^{j_{2}}}\psi\rangle_{L^{2}(\mathbb {R}_{+})}= \frac{1}{\sqrt {a-1}}a^{\frac{j_{2}-j_{1}}{2}} \int_{1}^{a}\psi_{m_{1}-m_{2}}(x)\sum _{j\in \mathbb {Z}}a^{j}\psi\bigl(a^{j}x\bigr)\overline{ \psi\bigl(a^{j+j_{2}-j_{1}}x\bigr)}\,dx $$
$$ \int_{1}^{a}\psi_{m}(x)\sum _{j\in \mathbb {Z}}a^{j}\psi \bigl(a^{j}x\bigr) \overline{\psi\bigl(a^{j+l}x\bigr)}\,dx=0\quad\mbox{for }(0, 0)\neq(m, l) \in \mathbb {Z}^{2}. $$
(3.3)
$$ \|\psi\|^{2}_{L^{2}(\mathbb {R}_{+})}= \int_{1}^{a}\sum_{j\in \mathbb {Z}}a^{j}\bigl| \psi\bigl(a^{j}x\bigr)\bigr|^{2}\,dx, $$
(3.4)
(ii) \(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\) is an orthonormal system for \(L^{2}(\mathbb {R}_{+})\) if and only if it is an orthogonal system and \(\|\psi\|^{2}_{L^{2}(\mathbb {R}_{+})}=a-1\). So we have (ii) by (i) and (3.4). The proof is completed. □
Corollary 3.1
Let
\(0\ne\psi\in L^{2}(\mathbb {R}_{+})\). Then the following are equivalent:
(i)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is a Parseval frame for
\(L^{2}(\mathbb {R}_{+})\).
(ii)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is an orthonormal basis for
\(L^{2}(\mathbb {R}_{+})\).
(iii)
\(\{\widetilde{\psi_{m}}D_{a^{j}}\psi: m, j\in \mathbb {Z}\}\)
is an orthonormal system in
\(L^{2}(\mathbb {R}_{+})\).
(iv)
\(\sum_{j\in \mathbb {Z}}a^{j}\psi(a^{j}\cdot)\overline{\psi (a^{j+l}\cdot)}=\delta_{l,0}\)
a.e. on
\([1, a)\)
for
\(l\in \mathbb {Z}\).
(v)
\(\vert \sum_{l\in \mathbb {Z}}a^{\frac {l}{2}}\psi(a^{l}x)e^{2\pi il\xi} \vert =1\)
for a.e. \((x, \xi)\in[1, a)\times[0, 1)\).
Acknowledgements
Supported by the National Natural Science Foundation of China (Grant No. 11271037).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.