\documentclass[12pt]{report} \usepackage{amssymb} \usepackage{amsmath} \renewcommand{\baselinestretch}{1} %\setlength{\textheight}{8.2in} %\pagestyle{empty} \begin {document} \vspace{.2in}\parindent=0mm \begin{center} %Page No.1 {\bf{Affine and quasi-affine frames on positive half line}} \vspace{.1in}\parindent=0mm By \parindent=0mm \vspace{.1in} Abdullah \parindent=0mm \vspace{.1in} Department of Mathematics, Zakir Husain Delhi College, University of Delhi, JLN Marg, New Delhi-110 002, India. \parindent=0mm \vspace{.1in} E-mail: abd.zhc.du@gmail.com \end {center} \parindent=0mm \vspace{.2in} {\bf{Abstract.}} The notion of quasi-affine frame was recently introduced by Ron and Shen in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we have extended the notion of affine and quasi-affine frame on Euclidean space $\mathbb R^n$ to positive half-line. Furthermore, we establish the relation between affine and quasi-affine frames on positive half line and the preservation of frame bounds when changing an affine frame to a quasi-affine frame is also shown. \parindent=0mm \vspace{.2in}{\bf{Keywords:}} Wavelets, affine frame, quasi-affine frame, Wash-Fourier transform. \vspace{.2in}\parindent=0mm {\bf{Mathematics Subject Classification (2000):}}~42C15, 40A30. \parindent=0mm \vspace{.2in} {\bf{1. Introduction}} \parindent=8mm \vspace{.2in} A frame differs from a Riesz basis in that it may be linearly dependent. In signal and image processing, this freedom allows the possibility of redundancy. The concept of frames, first introduced by Duffin and Schaeffer [7] in the context of non-harmonic Fourier series. outside signal processing, frames did not seem to generate much interest until Daubechies, Grossmann, and Meyer [8] brought attention to it. They showed that Duffin and Schaeffer’s definition is an abstraction of a concept given by Gabor [11] for doing signal analysis. \parindent=8mm \vspace{.2in} Discrete affine systems are obtained by applying dilations to a given shift-invariant system. Affine system is invariant under the dilaton operator, but is not shift invariant. The notion of quasi-affine frame was recently introduced by Ron and Shen [18] in order to achieve shift-invariance of the discrete wavelet transform. The main theorem of Ron and Shen in this direction is called "fundamental theorem of affine frames". It states that, an affine system ia an affine frame if and only if the quasi-affine system derived from it is also a frame. An operator on $L^2(\mathbb R^s) \times L^2(\mathbb R^s)$ corresponding to two affine systems was itroduced by Chui et al. [6] and they established a criterion in which this operator agrees with its analogue for two quasi-affine systems. \parindent=8mm \vspace{.2in} Although there are many results for affine frames on the real-line $\mathbb R$ , the counterparts on positive half-line $\mathbb{R}^+$ are not yet reported. So this paper is devoted to affine systems also known as wavelet systems and quasi-affine systems on positive half-line. Farkov [9] has given general construction of compactly supported orthogonal p-wavelets in $L^2(\mathbb{R}^+)$. Farkov et al. [10] gave an algorithm for biorthogonal wavelets related to Walsh functions on positive half line. Shah and Debnath [19], studied Dyadic wavelet frames on a half-line using the Walsh-Fourier transform. Recently, Meenakshi et al. [16] studied NUMRA on positive half line. \parindent=8mm \vspace{.2in} The objective of this paper is establish the relation between affine and quasi-affine frames on positive half line and the two frames have the same frame bounds and also establish a criterion in which the operator on $L^2(\mathbb R^+) \times L^2(\mathbb R^+)$ corresponding to affine systems agrees with its analogue for quasi-affine systems. \parindent=0mm \vspace{.2in} {\bf{2. Notations and preliminaries}} \parindent=0mm \vspace{.2in} Throughout we shall denote $|A|$ and $\chi_{A}(\xi)$, respectively Lebesgue measure and characteristic function of $A$. If $N \geq 1$ is an integer, we define $$\Gamma_{N}^+= \{m N +j : m \in \mathbb{Z}^+,~j=0, 1, ..., N-1 \}$$ \parindent=0mm and if $r$ is any odd integer with $1 \leq r \leq 2 N-1$ such that $r$ and $N$ are relatively prime, the set $\Lambda_{r, N}^+= \left\{0, \frac{r}{N} \right\}+ \mathbb{Z}^+.$ \parindent=0mm \vspace{.2in} {\bf{2.1. Walsh-Fourier Analysis}} \parindent=0mm \vspace{.2in} Let $p$ be a fixed natural number greater than 1. As usual, let $\mathbb{R}^+ = [0, \infty)$ and $\mathbb{Z}^+ = \{0, 1, ... \}$. Denote by $[x]$ the integer part of $x$. For $x \in \mathbb{R}^+$ and for any positive integer $j$, we set $$x_j=[p^j x](\text{mod}\,p),~~x_{-j}=[p^{1-j} x](\text{mod}\,p), \eqno(2.1)$$ \parindent=0mm where $x_j,~x_{-j} \in \{0, 1, ...,p-1 \}$. \parindent=0mm \vspace{.2in} Consider the addition defined on $\mathbb{R}^+$ as follows: $$x \oplus y = \sum_{j < 0} \xi_j p^{-j-1} + \sum_{j>0} \xi_j p^{-j} \eqno(2.2)$$ \parindent=0mm with $$\xi_j= x_j + y_j (\text{mod}\,p),~~j \in \mathbb{Z} \backslash \{0 \}, \eqno(2.3)$$ \parindent=0mm where $\xi_j \in \{0, 1, 2, ..., p-1 \}$ and $x_j,~y_j$ are calculated by (2.1). Moreover, we write $z = x \ominus y$ if $z \oplus y = x$, where $\ominus$ denotes subtraction modulo $p$ in $\mathbb{R}^+$. \parindent=8mm \vspace{.2in} For $x \in [0,1)$, let $r_0(x)$ be given by $$r_0(x)= \left\{ \begin{array}{lll} \displaystyle 1,~~~~~~x \in \left[0, \frac{1}{p} \right), \\\\ \displaystyle \varepsilon_p^j, ~~~~~~ x \in \left[ j p^{-1}, (j+1)p^{-1} \right),~j=1, 2, ..., p-1, \end{array} \right. \eqno(2.4)$$ \parindent=0mm where $\varepsilon_p = \text{exp} \left( \frac{2 \pi i}{p} \right)$. The extension of the function $r_0$ to $\mathbb{R}^+$ is defined by the equality $r_0(x+1) = r_0 (x),~x \in \mathbb{R}^+$. Then the generalized Walsh functions $\{\omega_m(x) \}_{m \in \mathbb{Z}^+}$ are defined by $$\omega_0(x)=1,~~\omega_m(x)=\prod_{j=0}^{P} \left( r_0 \left( p^j x \right) \right)^{\mu_j}, $$ \parindent=0mm where $ m = \sum_{j=0}^{P} \mu_j p^j,~\mu_j \in \{0, 1, 2, ..., p-1 \},~\mu_P \not= 0.$ \parindent=8mm \vspace{.2in} For $x,~\omega \in \mathbb{R}^+$, let $$\chi(x, \omega)=\text{exp} \left( \frac{2 \pi i}{p} \sum_{j=1}^{\infty}(x_j \omega_{-j} +x_{-j} \omega_j) \right), \eqno(2.5)$$ \parindent=0mm where $x_j$ and $\omega_j$ are calculated by (2.1). \parindent=8mm \vspace{.2in} We observe that $$\chi\left(x,\,\frac{m}{p^{n-1}} \right) = \chi\left(\frac{x}{p^{n-1}},\,m \right) = \omega_m\left(\frac{x}{p^{n-1}} \right) ~~~\forall\,x \in [0,\,p^{n-1}),~m \in \mathbb{Z}^+. $$ \parindent=0mm The Walsh-Fourier transform of a function $f \in L^1(\mathbb{R}^+)$ is defined by $$\tilde{f}(\omega) = \int_{\mathbb{R}^+}f(x) \overline{\chi(x, \omega)} dx, \eqno(2.6)$$ \parindent=0mm where $\chi(x, \omega)$ is given by (2.5). \parindent=8mm \vspace{.2in} If $f \in L^2(\mathbb{R}^+)$ and $$J_af(\omega)= \int_{0}^{a}f(x) \overline{\chi(x,\,\omega)}dx~~~(a<0), \eqno(2.7)$$ \parindent=0mm then $\tilde{f}$ is defined as limit of $J_a f$ in $L^2(\mathbb{R}^+)$ as $a \rightarrow \infty$. \parindent=8mm \vspace{.2in} The properties of Walsh-Fourier transform are quite similar to the classical Fourier transform. It is known that systems $\{\chi(\alpha, .) \}_{\alpha = 0}^{\infty}$ and $\{\chi(., \alpha) \}_{\alpha = 0}^{\infty}$ are orthonormal bases in $L^2(0, 1)$. Let us denote by $\{ \omega \}$ the fractional part of $\omega$. For $l \in \mathbb{Z}^+$, we have $\chi(l, \omega)= \chi(l,\{\omega \})$. \parindent=0mm \vspace{.2in} If $x, y, \omega \in \mathbb{R}^+$ and $x \oplus y$ is $p$-adic irrational, then $$\chi(x \oplus y, \omega) =\chi(x, \omega) \chi(y, \omega),~~\chi(x \oplus y, \omega) =\chi(x, \omega) \overline{\chi(y, \omega)}, \eqno(2.8)$$ \parindent=0mm \vspace{.2in} {\bf{3. Main results}} \parindent=0mm \vspace{.2in} {\bf{Definition 3.1.}} Let $\Psi=\{\psi_1, \psi_2, ..., \psi_L \}$ be a finite family of functions in $L^2(\mathbb R^+)$. The affine system generated by $\Psi$ is the collection $$X(\Psi)= \{ \psi_{l,j,k}: 1 \leq l \leq L, j \in \mathbb Z, k \in \mathbb Z^+ \}$$ \parindent=0mm where $\psi_{l,j,k}(x)= p^{j/2} \psi_l(p^j x \ominus k)$. The quasi-affine system generated by $\Psi$ is $${X}^q(\Psi)= \{ \tilde{\psi}_{l,j,k}: 1 \leq l \leq L, j \in \mathbb Z, k \in \mathbb Z^+ \},$$ \parindent=0mm where $$\psi^q_{l,j,k}(x)= \left\{ \begin{array}{l} p^{j/2} \psi_l(p^j x \ominus k), ~~~~~~j\geq 0,~~ k \in \mathbb Z^+ \\ p^{j} \psi_l(p^j (x \ominus k)), ~~~~~~ j< 0,~~ k \in \mathbb Z^+. \end{array}\right. \eqno(3.1)$$ \parindent=8mm We say that $\Psi$ is a set of basic wavelets of $L^2(\mathbb R^+)$ if the affine system $X(\Psi)$ forms an orthonormal basis for $L^2(\mathbb R^+)$. \parindent=0mm \vspace{.2in} {\bf{Definition 3.2.}} $X \subset L^2(\mathbb R^+)$ is a Bessel family if there exists $B>0$ so that $$\sum_{\eta \in X} ||^2 \leq B \| f \|^2 ~~~~~~~~~{\text{for}} ~ f \in L^2(\mathbb R^+). \eqno(3.2)$$ \parindent=0mm If, in addition, there exists a constant $A>0$, $A \leq B$ such that $$A \| f \|^2 \leq \sum_{\eta \in X} ||^2 \leq B \| f \|^2 ~~~~~~~~~{\text{for all}} ~ f \in L^2(\mathbb R^+), \eqno(3.3)$$ \parindent=0mm then $X$ is called a frame. The frame is tight if we can choose $A$ and $B$ such that $A=B$. The (quasi) affine system $X(\Psi)$ (resp. $X^q(\Psi)$) is a (quasi) affine frame if (3.3) holds for $X=X(\Psi)$ ($X=X^q(\Psi)$). \parindent=8mm \vspace{.2in} Let $\Psi, \tilde{\Psi} \subset L^2(\mathbb R^+)$ be two finite sets in $L^2(\mathbb R^+)$ with the same cardinality. The operator $$P_{\Psi, \tilde{\Psi}} (f,g)= \sum_{l=1}^{L} \sum_{j \in \mathbb Z} \sum_{k \in \mathbb Z^+} \langle f, \psi_{l,j,k} \rangle \langle \tilde{\psi}_{l,j,k},g \rangle,~~~f,g \in L^2(\mathbb R^+), \eqno(3.4)$$ \parindent=0mm plays a very important role in our investigation. It is bounded linear operator on $L^2(\mathbb R^+) \times L^2(\mathbb R^+)$, if both $X(\Psi)$ and $X(\tilde{\Psi})$ are affine Bessel families. Our frame analysis is based on the relation to the operator $$P^q_{\Psi, \tilde{\Psi}} (f,g)= \sum_{l=1}^{L} \sum_{j \in \mathbb Z} \sum_{k \in \mathbb Z^+} \langle f, \psi^q_{l,j,k} \rangle \langle \tilde{\psi}^q_{l,j,k},g \rangle,~~~f,g \in L^2(\mathbb R^+), \eqno(3.5)$$ \parindent=0mm which is associated with the quasi-affine systems. The following terminology will be used. \parindent=0mm \vspace{.2in} {\bf{Definition 3.3.}} Let $\Psi, \tilde{\Psi} \subset L^2(\mathbb R^+)$ be two finite sets in $L^2(\mathbb R^+)$ with the same cardinality that generate two Bessel families $X(\Psi)$ and $X(\tilde{\Psi})$. Then $\tilde{\Psi}$ is called an affine dual of $\Psi$, if $$P_{\Psi, \tilde{\Psi}} (f,g) = \langle f, g \rangle, ~~~~f,g \in L^2(\mathbb R^+), $$ \parindent=0mm and a quasi-affine dual if $$P^q_{\Psi, \tilde{\Psi}} (f,g) = \langle f, g \rangle, ~~~~f,g \in L^2(\mathbb R^+). $$ \parindent=8mm Affine system $X(\Psi)$ are dilation-invariant in the sense that $$X(\Psi) = D(X(\Psi))$$ \parindent=0mm This is the reason that $P_{\Psi, \tilde{\Psi}}$ in (3.4) has the property $$P_{\Psi, \tilde{\Psi}} (Df, Dg) = P_{\Psi, \tilde{\Psi}} (f,g),~~~f,g \in L^2(\mathbb R^+). \eqno(3.6)$$ \parindent=0mm On the other hand, it can be seen easily from the definition of quasi-affine systems, that $$P^q_{\Psi, \tilde{\Psi}} (T_k f, T_k g) = P^q_{\Psi, \tilde{\Psi}} (f,g),~~~f,g \in L^2(\mathbb R^+),~k \in \mathbb Z^+. $$ \parindent=0mm Hence, $P^q_{\Psi, \tilde{\Psi}}$ is invariant with respect to multi-integer shifts of both arguments. \parindent=8mm \vspace{.2in} For $J \geq 0$, we let $Q_J$ denote a complete set of representatives of $\mathbb Z^+ / (p^J \mathbb Z^+)$, so that $$\# Q_J = p^J.$$ \parindent=0mm The special choice of representatives is $$ Q_J = \mathbb Z^+ \cap p^J ([1,2)).\eqno(3.7)$$ \parindent=0mm We use the notations $$P_{j} (f,g)= \sum_{l=1}^{L} \sum_{k \in \mathbb Z^+} \langle f, \psi_{l,j,k} \rangle \langle \tilde{\psi}_{l,j,k},g \rangle, \eqno(3.8)$$ \parindent=0mm and $$P^q_{j} (f,g)= \sum_{l=1}^{L} \sum_{k \in \mathbb Z^+} \langle f, \psi^q_{l,j,k} \rangle \langle \tilde{\psi}^q_{l,j,k},g \rangle, \eqno(3.9)$$ \parindent=0mm for any $j \in \mathbb Z$. In particular, $P_j = P_j^q$ holds for all $j \geq 0$. \parindent=0mm \vspace{.2in} {\bf{Lemma 3.4.}} Let $\Psi$ be a finite subset of $L^2(\mathbb R^+)$ and $\tilde{\Psi}$ be the dual of $\Psi$. Let $J>0$ be an integer. Then for all $j \geq -J$ and $f,g \in L^2(\mathbb R^+)$, $$P_j^q (f,g)= p^{-J} \sum_{v \in Q_J} P_j(T_v f, T_v g).$$ \parindent=0mm \vspace{.1in} {\bf{Proof.}} For any $j \geq 0$ the operator $P_j$ is invariant with respect to multi-integer shifts, so that $$P^q_j(f,g) = P_j(f,g) = P_j(T_v f, T_v g),~~~~v \in Q_J.$$ \parindent=0mm For any $-J \leq j < 0$, we first observe that $P_j$ is invariant with respect to shifts $k \in p^{-j} \mathbb Z^+$. Furthermore, $H= \mathbb Z^+ / p^{-j} \mathbb Z^+$ is a normal subgroup of $G= \mathbb Z^+ / p^{J} \mathbb Z^+ $ and there is a group isomorphism of $G/H$ and $ p^{-j} \mathbb Z^+ / p^{J} \mathbb Z^+ $. Hence, $|G/H|=p^{J+j}$, and $$p^{-J} \sum_{v \in Q_J} P_j (T_v f, T_v g) = p^{-J} \sum_{\lambda \in Q_{-j}} \sum_{ {\underset{ v-\lambda \in p^{-j} \mathbb Z^+ } {v \in Q_J} }} P_j (T_v f, T_v g) $$ $$= p^{j} \sum_{\lambda \in Q_{-j}} P_j (T_\lambda f, T_\lambda g) .$$ \parindent=0mm By inserting the definition (3.1) of $\psi^q_{l,j,k}$ we see that the last expression is equal to $P^q_j(f,g)$. \parindent=0mm \vspace{.2in} {\bf{Lemma 3.5.}} Let $\Psi = \{ \psi_1, \psi_2, ..., \psi_L \} \subset L^2(\mathbb R^+)$ and let $f \in L^2(\mathbb R^+)$ be a function with compact support. Then $$\lim_{N \rightarrow \infty} \sum_{j<0} P^q_j(D^N f, D^N f) = 0 \eqno(3.10)$$ \parindent=0mm and $$\lim_{N \rightarrow \infty} p^{-N} \sum_{j<-N} \sum_{v \in Q_N} P_j(T_v f, T_v f) = 0, \eqno(3.11)$$ \parindent=0mm where the special choice of representatives (3.7) is used in (3.11) for any $N \in \mathbb N$. \parindent=0mm \vspace{.2in} {\bf{Proof.}} Let $\Omega$ denote the support of $f$ and let us first choose $N_0 > 0$ so large that $D^{-N} \Omega$ is contained in a ball of radius $1/2$ around the origin for all $N \geq N_0$. In order to prove (3.10), we consider $$ \sum_{j<0} P^q_j(D^N f, D^N f) = \sum_{j<0} p^j \sum_{l=1}^{L} \sum_{k \in \mathbb Z^+} \left| \langle f, D^{-N} T_k D^j \psi_l \rangle \right|^2 $$ $$ = \sum_{j<0} p^j \sum_{l=1}^{L} \sum_{k \in \mathbb Z^+} \left| \langle T_{p^N k}f, D^{j-N} \psi_l \rangle \right|^2 $$ $$ \leq \sum_{j<0} p^j \sum_{l=1}^{L} \sum_{k \in \mathbb Z^+} \|f\|^2 \int_{\Omega + p^N k} \left| D^{j-N} \psi_l(x) \right|^2 dx $$ $$ = \|f\|^2 \sum_{j<0} p^j \sum_{k \in \mathbb Z^+} \int_{p^{j-N}\Omega + p^j k} \sum_{l=1}^{L} \left| \psi_l(x) \right|^2 dx. \eqno(3.12) $$ \parindent=0mm By our previous choice of $N_0$, we obtain that $$ \sum_{j<0} P^q_j(D^N f, D^N f) \leq \|f\|^2 \int_{\mathbb R^+} w_N(x) \sum_{l=1}^{L} \left| \psi_l(x) \right|^2 dx $$ \parindent=0mm holds for all $N \geq N_0$, where $$w_N(x) = \sum_{j<0} p^j \chi_{p^j(\mathbb Z^+ + p^{-N} \Omega)} (x),~~~x \in \mathbb R^+ . \eqno(3.13)$$ \parindent=0mm Since $$w_N(x) \leq \sum_{j<0} p^j = \frac{p}{p-1},~~~N \geq N_0, $$ \parindent=0mm and since $\sum_{l=1}^{L} |\psi_l|^2 \in L^1(\mathbb R^+)$, the dominated convergence theorem can be applied to the above integral. It thus suffices to show that $$ \lim_{N \rightarrow \infty} w_N(x)=0 ~~~ \text{for all}~x \in U= \mathbb R^+ \backslash {\underset{j<0 } {\cup} } p^j \mathbb Z^+ . $$ \parindent=0mm This last assertion can be shown as follows. By the compactness of $\Omega$ the sequence of numbers $$ r_j = \sup \{\| p^j y\|: y \in \Omega \} $$ \parindent=0mm tends to zero as $j \rightarrow -\infty$. If we fix $x \in U$, then all terms in (3.13) which satisfy $$ r_{j-N} < \text{dist} (x, p^j \mathbb Z^+) = d_j(x),~~~j<0,$$ \parindent=0mm vanish. In other words, $$w_N(x) \leq \sum_{ {\underset{ r_{j-N} \geq d_j(x)} {j<0} }} p^j \rightarrow 0 ~~\text {as} ~~N \rightarrow \infty. $$ \parindent=8mm In the second relation (3.11), we let $N \geq N_0$ and use similar transformations as in (3.12) in order to obtain $$p^{-N} \sum_{v \in Q_N} \sum_{j<-N} P_j (T_v f, T_v f) $$ $$ \leq \| f \|^2 p^{-N} \sum_{v \in Q_N} \sum_{j<-N} \sum_{k \in \mathbb Z^+} \int_{p^j(\Omega + v)+k} \sum_{l=1}^{L} |\psi_l(x)|^2 dx. $$ \parindent=0mm We now define $$v_N(x) = p^{-N} \sum_{v \in Q_N} \sum_{j<-N} \sum_{k \in \mathbb Z^+} \chi_{p^j (\Omega + v)+k } (x), ~~~x \in \mathbb R^+ . \eqno(3.14)$$ \parindent=0mm Our remaining task is to show that $v_N$ is uniformly bounded in $N$, as $N$ tends to infinity, and converges to zero pointwise a.e. We will even show that $v_N$ tends to zero uniformly. Recall that $Q_N$ is assumed to be of the form (3.7) and hence $$p^{-N} (\Omega + v) \subset S= (1/2, 3/2)~~~\text{for all}~N \geq N_0,~v \in Q_N.$$ \parindent=0mm In order to resolve the various summations in (3.14), we fix $j$ and $k$ and first observe that $$ \sum_{v \in Q_N} \chi_{p^j(\Omega +v)+k} \leq c_1 \chi_{p^{j+N}S+k}$$ \parindent=0mm holds for all $N \geq N_0,~j < -N$ and $k \in \mathbb Z^+$ and the constant $c_1 > 0$ only depends on the compact set $\Omega$. Next the properties of $p$ and the set $S$ where the special choice of $Q_N$ enters, gives the estimate $$ \sum_{j< -N} \chi_{p^{j+N}S+k} \leq c_2 \chi_{B_r(0)+k}$$ \parindent=0mm for all $N \geq N_0,~ k \in \mathbb Z^+$, where the constant $c_2, r> 0$ only depends on $p$. Finally, summation over $k \in \mathbb Z^+$ gives another contant $c_3 >0$ which only depends on $r$ (hence on $p$) such that $$\sum_{k \in \mathbb Z^+} \chi_{B_r(0)+k} \leq c_3.$$ \parindent=0mm Combining all these relations yields $$ v_N(x) \leq c_1c_2c_3 p^{-N},~~~ x \in \mathbb R^+.$$ \parindent=0mm and hence the uniform convergence of $v_N$ to zero. This completes the proof of the lemma. \parindent=0mm \vspace{.2in} {\bf{Theorem 3.6.}} Let $\Psi$ be a finite subset of $L^2(\mathbb R^+)$. Then \parindent=0mm \vspace{.1in} (a) $X(\Psi)$ is a Bessel family if and only if $X^q(\Psi)$ is a Bessel family. Furthermore, their exact upper bounds are equal. \parindent=0mm \vspace{.1in} (b) $X(\Psi)$ is an affine frame if and only if $X^q(\Psi)$ is a quasi-affine frame. Furthermore, their lower and upper exact bounds are equal. \parindent=0mm \vspace{.2in} {\bf{Proof.}} (a) Let $\Psi = \tilde{\Psi}$. All summands of $P_{\Psi, \Psi}$ and $P^q_{\Psi, \Psi}$ are nonnegative. If $\Psi$ generates an affine Bessel family with upper bound $B \geq 0$, then $$P^q_{\Psi, \Psi}(f,f) = \lim_{J \rightarrow \infty} \sum_{j \geq -J} P_j^q(f,f)$$ $$= \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} \sum_{j \geq -J} P_j(T_v f, T_v f) $$ $$ \leq \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} P_{\Psi, \Psi}(T_v f, T_v f) $$ $$ \leq \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} B \|T_v f \|^2 = B \| f \|^2 $$ \parindent=0mm holds for all $f \in L^2(\mathbb R^+)$. Here, we used the translation invariance of the norm. We have thus shown that the quasi-affine frame $X^q(\Psi)$ is also a Bessel family with the same upper bound $B$. \parindent=8mm Conversely, let us assume that $X^q(\Psi)$ is a Bessel family with upper bound $B_q \geq 0$. Let us further assume that there exists an $f \in L^2(\mathbb R^+)$, such that $$ \| f \| = 1 ~~\text{and}~~ P_{\Psi, \Psi}(f,f) > B_q.$$ \parindent=0mm Then by the dilation invariance of $X(\Psi)$, we can find $N \in \mathbb N$ such that $$ \sum_{j = -N}^{\infty} P_j(f,f) = \sum_{j=0}^{\infty} P_j (D^N f, D^N f) > B_q. $$ But this contradicts the definition of $B_q$, since $$P_{\Psi, \Psi}^q (D^N f, D^N f) \geq \sum_{j=0}^{\infty} P^q_j (D^N f, D^N f) = \sum_{j=0}^{\infty} P_j (D^N f, D^N f), $$ \parindent=0mm and the dilation $D$ is an isometry. Thus, we can conclude that $X(\Psi)$ must be a Bessel family with upper bound $B_q$. \parindent=0mm \vspace{.2in} (b) It only remains to consider the lower frame bounds $A$ and $A_q$. The proof follows the same argument as the proof of (a). The only differences are the use of lemma 3.5 at certain places e.g., the relation $A_q \geq A$ follows by using (3.11) in the third line of $$P^q_{\Psi, \Psi}(f,f) = \lim_{J \rightarrow \infty} \sum_{j \geq -J} P_j^q(f,f)$$ $$= \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} \sum_{j \geq -J} P_j(T_v f, T_v f) $$ $$ = \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} \sum_{j \in \mathbb Z} P_{j}(T_v f, T_v f) $$ $$ = \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} P_{\Psi, \Psi}(T_v f, T_v f) $$ $$ \geq \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} A \|T_v f \|^2 = A \| f \|^2 $$ \parindent=0mm for all $f \in L^2(\mathbb R^+)$ which have compact support. Since this is the dense subset of $L^2(\mathbb R^+)$, the relation $P^q_{\Psi, \Psi}(f, f) \geq A \|f\|^2$ holds for all $f \in L^2(\mathbb R^+)$. The opposite relation $A_q \leq A$ is shown by assuming the contrary, so that $$P_{\Psi, \Psi}(f, f) \leq A_q-\varepsilon ~~~\text{for some } f \in L^2(\mathbb R^+),~~\| f \| = 1,$$ \parindent=0mm and some $\varepsilon > 0$. Without loss of generality, we can assume that $f$ has compact support. The dilation invariance of the operator $P_{\Psi, \Psi}(f, f)$ gives $$P_{\Psi, \Psi}(D^N f, D^N f) \leq A_q-\varepsilon ~~~\text{for all } N \in \mathbb N.$$ \parindent=0mm By (3.10), there exists $N \in \mathbb N$ such that $$P^q_{\Psi, \Psi}(D^N f, D^N f) < \sum_{j=0}^{\infty} P^q_{j}(D^N f, D^N f) + \frac{\varepsilon}{2} = \sum_{j=0}^{\infty} P_{j}(D^N f, D^N f) + \frac{\varepsilon}{2}$$ $$ \leq P_{\Psi, \Psi}(D^N f, D^N f)+ \frac{\varepsilon}{2} \leq A_q - \frac{\varepsilon}{2}$$ \parindent=0mm which contradicts with the definition of the lower frame bound $A_q$ of $X^q(\Psi)$. \parindent=0mm \vspace{.2in} {\bf{Theorem 3.7.}} Let $\Psi$ be a finite subset of $L^2(\mathbb R^+)$ and $\tilde{\Psi}$ be the dual of $\Psi$. Assume that $\Psi$ and $\tilde{\Psi}$ generate two affine Bessel families. Then $P_{\Psi, \tilde{\Psi}}$ translation-invariant if and only if $$P_{\Psi, \tilde{\Psi}} = P^q_{\Psi, \tilde{\Psi}}. \eqno(3.15)$$ \parindent=0mm \vspace{.2in} {\bf{Proof.}} First we assume that $P_{\Psi, \tilde{\Psi}}$ is translation-invariant. Then, as in the previous proof, we have $$P^q_{\Psi, \tilde{\Psi}}(f,g) = \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} P_{\Psi, \tilde{\Psi}} (T_v f, T_v g)$$ \parindent=0mm for all $f, g \in L^2(\mathbb R^+)$ with compact support. Since $P_{\Psi, \tilde{\Psi}}$ is assumed to be translation-invariant, the right-hand side equals $$ \lim_{J \rightarrow \infty} p^{-J} \sum_{v \in Q_J} P_{\Psi, \tilde{\Psi}} ( f, g) = P_{\Psi, \tilde{\Psi}}(f,g) $$ \parindent=0mm The equality extends to all functions in $L^2(\mathbb R^+)$ by density and boundedness of both operator. On the other hand, as a consequence of (3.15), the operator $P_{\Psi, \tilde{\Psi}}$ is invariant with respect to shifts $T_k,~k \in \mathbb Z^+$. 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