\(^{1}\) Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar-190 006, Jammu and Kashmir, India, e-mail: siawoahmad@gmail.com.

\(^{2}\)Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar-190 006, Jammu and Kashmir, India, e-mail: neyaznit@yahoo.co.in.

\(^{3}\)Department of Mathematics, Faculty of Science, Jazan University, Jazan-45142, Saudi Arabia, e-mail: msyed@jazanu.edu.sa.

1 Introduction

Duffin and Schaeffer [ 14 ] introduced the concept of frame in separable Hilbert space while dealing with some deep problems in non-harmonic Fourier series. Frames are basis-like systems that span a vector space but allow for linear dependency, which can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases.

During the last two decades, there is a substantial body of work that has been concerned with the construction of wavelets on local fields. Even though the structures and metrics of local fields of zero and positive characteristics are similar, their wavelet and MRA (multiresolution analysis) theory are quite different. For example, R. L. Benedetto and J. J. Benedetto [ 12 ] developed a wavelet theory for local fields and related groups. They did not develop the multiresolution analysis (MRA) approach, their method is based on the theory of wavelet sets and only allows the construction of wavelet functions whose Fourier transforms are characteristic functions of some sets. Khrennikov, Shelkovich and Skopina [ 21 ] constructed a number of scaling functions generating an MRA of \(L^2(\mathbb Q_p\)). But later on in [ 10 ] , Albeverio, Evdokimov and Skopina proved that all these scaling functions lead to the same Haar MRA and that there exist no other orthogonal test scaling functions generating an MRA except those described in [ 21 ] . Some wavelet bases for \(L^2(\mathbb Q_p\)) different from the Haar system were constructed in [ . These wavelet bases were obtained by relaxing the basis condition in the definition of an MRA and form Riesz bases without any dual wavelet systems. For some related works on wavelets and frames on \(\mathbb Q_p\), we refer to [ . On the other hand, Lang [ constructed several examples of compactly supported wavelets for the Cantor dyadic group. Farkov [ has constructed many examples of wavelets for the Vilenkin \(p\)-groups. Jiang et al. [ 18 ] pointed out a method for constructing orthogonal wavelets on local field \(\mathbb K\) with a constant generating sequence and derived necessary and sufficient conditions for a solution of the refinement equation to generate a multiresolution analysis of \(L^2(\mathbb K)\). In the series of papers [ , we have obtained various results related to wavelet and Gabor frames on local fields.

The study of periodic bi-frames was carried by Li and Jia [ 35 ] but the parallel development on local fields is not reported yet. In this paper, we introduce the notion of periodic wavelet bi-frames on local field of positive characteristic and establish the theory for the construction of periodic wavelet bi-frames on local fields.

The rest of the article is structured as follows. In section 2, we discuss the preliminaries of local fields and some basic definitions which plays vital role in the rest of the paper. In section 3, we establish some results related to periodic Bessel sequences on local fields of positive characteristic. section 4 is devoted to the construction of periodic wavelet bi-frames on local fields.

2 Preliminaries on Local Fields

Let \(K\) be a field and a topological space. Then \(K\) is called a local field if both \(K^+\) and \(K^*\) are locally compact Abelian groups, where \(K^+\) and \(K^*\) denote the additive and multiplicative groups of \(K\), respectively. If \(K\) is any field and is endowed with the discrete topology, then \(K\) is a local field. Further, if \(K\) is connected, then \(K\) is either \(\mathbb R\) or \(\mathbb C\). If \(K\) is not connected, then it is totally disconnected. Hence by a local field, we mean a field \(K\) which is locally compact, non-discrete and totally disconnected. The \(p\)-adic fields are examples of local fields. In the rest of this paper, we use the symbols \(\mathbb N\), \(\mathbb N_0\) and \(\mathbb Z\) to denote the sets of natural, non-negative integers and integers, respectively.

Let \(K\) be a local field. Let \(dx\) be the Haar measure on the locally compact Abelian group \(K^{+}\). If \(\alpha \in K\) and \(\alpha \ne 0\), then \(d(\alpha x)\) is also a Haar measure. Let \(d(\alpha x)=|\alpha |dx\). We call \(|\alpha |\) the absolute value of \(\alpha \). Moreover, the map \(x\to |x|\) has the following properties: (a) \(|x|=0\) if and only if \(x = 0;\) (b) \(|xy|=|x|\cdot |y|\) for all \(x, y \in K\); and (c) \(|x+y|\le \max \left\{ |x|, |y|\right\} \) for all \(x, y\in K\). Property (c) is called the ultrametric inequality. The set \({\mathfrak D}= \left\{ x \in K: |x| \le 1\right\} \) is called the ring of integers in \(K.\) Define \({\mathfrak B}= \left\{ x \in K: |x| {\lt} 1\right\} \). The set \({\mathfrak B}\) is called the prime ideal in \(K\). The prime ideal in \(K\) is the unique maximal ideal in \({\mathfrak D}\) and hence as result \({\mathfrak B}\) is both principal and prime. Since the local field \(K\) is totally disconnected, so there exist an element of \({\mathfrak B}\) of maximal absolute value. Let \(\mathfrak p\) be a fixed element of maximum absolute value in \({\mathfrak B}\). Such an element is called a prime element of \(K.\) Therefore, for such an ideal \({\mathfrak B}\) in \({\mathfrak D}\), we have \({\mathfrak B}= \langle \mathfrak p \rangle =\mathfrak p {\mathfrak D}.\) As it was proved in [ 34 ] , the set \({\mathfrak D}\) is compact and open. Hence, \({\mathfrak B}\) is compact and open. Therefore, the residue space \({\mathfrak D}/{\mathfrak B}\) is isomorphic to a finite field \(GF(q)\), where \(q = p^{k}\) for some prime \(p\) and \(k\in \mathbb N\).

Let \({\mathfrak D}^*= {\mathfrak D}\setminus {\mathfrak B }=\big\{ x\in K: |x|=1 \big\} \). Then, it can be proved that \({\mathfrak D}^*\) is a group of units in \(K^*\) and if \(x\not=0\), then we may write \(x=\mathfrak p^k x^\prime , x^\prime \in {\mathfrak D}^*.\) For a proof of this fact we refer to [ 18 ] . Moreover, each \({\mathfrak B}^k= \mathfrak p^k {\mathfrak D}=\big\{ x \in K: |x| {\lt} q^{-k}\big\} \) is a compact subgroup of \(K^+\) and usually known as the fractional ideals of \(K^+\). Let \({\mathcal U}= \left\{ c_i \right\} _{ i=0}^{q-1}\) be any fixed full set of coset representatives of \({\mathfrak B}\) in \({\mathfrak D}\), then every element \(x\in K\) can be expressed uniquely as \(x=\sum _{\ell =k}^{\infty } c_\ell \mathfrak p^\ell \) with \(c_\ell \in {\mathcal U}.\) Let \(\chi \) be a fixed character on \(K^+\) that is trivial on \({\mathfrak D}\) but is non-trivial on \({\mathfrak B}^{-1}\). Therefore, \(\chi \) is constant on cosets of \({\mathfrak D}\) so if \(y \in {\mathfrak B}^k\), then \(\chi _y(x)=\chi (yx), x\in K.\) Suppose that \(\chi _u\) is any character on \(K^+\), then clearly the restriction \(\chi _u|{\mathfrak D}\) is also a character on \({\mathfrak D}\). Therefore, if \(\left\{ u(n): n\in \mathbb N_0\right\} \) is a complete list of distinct coset representative of \({\mathfrak D}\) in \(K^+\), then, as it was proved in [ 34 ] , the set \(\big\{ \chi _{u(n)}: n\in \mathbb N_0\big\} \) of distinct characters on \({\mathfrak D}\) is a complete orthonormal system on \({\mathfrak D}\).

The Fourier transform \(\widehat f\) of a function \(f \in L^1(K)\cap L^2(\mathbb K)\) is defined by

It is noted that

Furthermore, the properties of Fourier transform on local field \(K\) are much similar to those of on the real line. In particular Fourier transform is unitary on \(L^2(\mathbb K)\). Also, if \(f\in L^2(\mathfrak D)\), then we define the Fourier coefficients of \(f\) as

The series \(\sum _{n\in \mathbb N_{0}} \widehat f\big( u(n)\big) \chi _{u(n)}(x)\) is called the Fourier series of \(f\). From the standard \(L^2\)-theory for compact Abelian groups, we conclude that the Fourier series of \(f\) converges to \(f\) in \(L^2(\mathfrak D)\) and Parseval’s identity holds:

We now impose a natural order on the sequence \(\{ u(n)\} _{n=0}^\infty \). We have \({\mathfrak D}/ \mathfrak B \cong GF(q) \) where \(GF(q)\) is a \(c\)-dimensional vector space over the field \(GF(p)\). We choose a set \(\left\{ 1=\zeta _0,\zeta _1,\zeta _2,\dots ,\zeta _{c-1}\right\} \subset {\mathfrak D^*}\) such that span \(\left\{ \zeta _j\right\} _{j=0}^{c-1}\cong GF(q)\). For \(n \in \mathbb N_0\) satisfying

we define

Also, for \(n=b_0+b_1q+b_2q^2+\dots +b_sq^s, ~ n\in \mathbb N_{0},~ 0\leq b_k{\lt}q,k=0,1,2,\dots ,s\), we set

This defines \(u(n)\) for all \(n\in \mathbb N_{0}\). In general, it is not true that \(u(m + n)=u(m)+u(n)\). But, if \(r,k\in \mathbb N_{0}\; \text{and}\; 0\le s{\lt}q^k\), then \(u(rq^k+s)=u(r){\mathfrak p}^{-k}+u(s).\) Further, it is also easy to verify that \(u(n)=0\) if and only if \(n=0\) and \(\{ u(\ell )+u(k):k \in \mathbb N_0\} =\{ u(k):k \in \mathbb N_0\} \) for a fixed \(\ell \in \mathbb N_0.\) Hereafter we use the notation \(\chi _n=\chi _{u(n)}, \, n\ge 0\).

Let the local field \(K\) be of characteristic \(p{\gt}0\) and \(\zeta _0,\zeta _1,\zeta _2,\dots ,\zeta _{c-1}\) be as above. We define a character \(\chi \) on \(K\) as follows:

We also denote the test function space on \(K\) by \(\Omega (K)\), that is, each function \(f\) in \(\Omega (K)\) is a finite linear combination of functions of the form \({\bf 1}_k(x-h), \; h\in K,\; k\in \mathbb Z\), where \({\bf 1}_k\) is the characteristic function of \({\mathfrak B}^k\). This class of functions can also be described in the following way. A function \(g\in \Omega (K)\) if and only if there exist integers \(k,\ell \) such that \(g\) is constant on cosets of \({\mathfrak B}^{k}\) and is supported on \({\mathfrak B}^{\ell }\). It follows that \(\Omega \) is closed under Fourier transform and is an algebra of continuous functions with compact support, which is dense in \({\mathcal C}_{0}(K)\) as well as in \(L^p(K), \; 1\le p {\lt}\infty \).

For \(j\in \mathbb N_{0}\), let \({\mathcal N}_{j}\) denote a full collection of coset representatives of \(\mathbb N_{0}/q^{j}\mathbb N_{0}\), i.e.,

Then, \(\mathbb N_{0}=\bigcup _{n\in {\mathcal N}_{j}} \left(n+q^{j}\mathbb N_{0}\right)\), and for any distinct \(n_{1}, n_{2}\in {\mathcal N}_{j}\), we have \(\left(n_{1}+q^{j}\mathbb N_{0}\right)\cap \left(n_{2}+q^{j}\mathbb N_{0}\right)=\emptyset .\) Thus, every non-negative integer \(k\) can uniquely be written as \(k=rq^{j}+s\), where \(r\in \mathbb N_{0},\; s\in {\mathcal N}_{j}\). Further, a bounded function \(W : \mathbb K\to \mathbb K\) is said to be a radial decreasing \(L^1\)-majorant of \(f(x)\in L^2(\mathbb K)\) if \(|f(x)|\le W(x),\, W\in L^{1}(\mathbb K),\) and \(W(0){\lt}\infty .\)

For \(j\in \mathbb Z\) and \(y\in \mathbb K\), we define the dilation \(\delta _j\) and the translation operators \(T_{y}\) as follows:

For an arbitrary measurable function \(f\), we define

and

In particular, we define

For a finite subset \(E\) of \(L^2(\mathbb K)\), we write

we require that \(k\) in 2.8 belongs to \(\mathcal{N}_j\) instead of \(\mathbb N_0\). Otherwise, every \(f_{j,k}^{{\rm {\rm per}}}\) with \( k \in \mathcal{N}_j\) will repeat infinitely many times since \( f_{j,k}^{{\rm {\rm per}}} = f_{j,k+\mathfrak {p}^{-j}u(s)}~ \text{for}~ s \in \mathbb N_0\), and thus we cannot create a new frame. The restrictions on \(j~ \text{and}~ k \) in 2.8 are also related to our method for the construction of frames.

Let \(\Psi = \left\{ \psi _{\ell } : 1 \le \ell \le L\right\} \) and \(\widetilde{\Psi }=\left\{ \widetilde{\psi }_{\ell } : 1 \le \ell \le L\right\} \) be two finite subsets of \(L^2(\mathbb K)\) with the same cardinality. A bi-frame for \(L^2(\mathbb K)\) of the form \(\big(X(\Psi ),X(\widetilde{\Psi })\big)\) is called a wavelet bi-frame for \(L^2(\mathbb K)\), i.e., \(X(\Psi )\) and \(X(\widetilde{\Psi })\) are two frames for \(L^2(\mathbb K)\) satisfying

In the similar manner, a bi-frame for \(L^2(\mathfrak {D})\) of the form \(\big(X^{{\rm {\rm per}}}(\Psi ),X^{{\rm {\rm per}}}(\widetilde{\Psi })\big)\) is called a periodic wavelet bi-frame.

A function \(f \in L^1(K)\cap L^2(\mathbb K)\) is said to be \(\mathfrak {p}\)-refinable if there exists a periodic measurable function \(m_f\) on \(K\) such that

where \(m_f\) is called a symbol of \(f\).

3 Periodic Bessel Sequences on Local Fields

In this section, we establish a Bessel sequence in \(L^2(\mathfrak {D})\) from a Bessel sequence in \(L^2(\mathbb K)\) of the form \(\left\{ g_{m,n}:m,n \in \mathbb N_0\right\} \). For that purpose, we first introduce a Banach space \(\mathcal{L}^{p}(\mathbb K)\) with \(1\le p\le \infty \). The space \({\mathcal L}^p\) was introduced on \(\mathbb {R}^d\) by Jia and Micchelli [ 19 ] .

For \( 1\le p \le \infty \) and a measurable \(f\) on \(\mathbb K\), we define

and we write

Then \(\mathcal{L}^p(\mathbb K)\) is a Banach space. It is well known that \(\mathcal{L}^{p_2}(\mathbb K) \subset \mathcal{L}^{p_1}(\mathbb K)\) if \( 1\le p_1 \le p_2 \le \infty .\)

Since \(|f(x)| = \mathcal{O}\left((1+|x|)^{-\alpha }\right)\) as \(|x| \rightarrow \infty \), there exists a constant \(C\) and \(N \in \mathbb N_0\) such that

It follows that

for a.e. \(x \in \mathfrak {D}\) and some \(C^{'}\) independent of x. This implies that

Since \(\bigcup _{p\in \mathcal{N}_1}(\mathfrak {D}+u(p))\) and \(\mathfrak {p^{-1}}\mathfrak {D}\) have the same measure, to finish the proof we only need to prove that \(\bigcup _{p\in \mathcal{N}_1}(\mathfrak {D}+u(p))\) is congruent to a subset of \(\mathfrak {p^{-1}}\mathfrak {D}.\) Observe that \(\left\{ \mathfrak {p}^{-1}\mathfrak {D}+\mathfrak {p}^{-1}u(r) : r \in \mathbb N_0\right\} \) is a partition of \(K\). If \(\bigcup _{p\in \mathcal{N}_1} (\mathfrak {D}+u(p))\) is not \(q\mathbb N_0\)-congruent to any subset of \(\mathfrak {p}^{-1}\mathfrak {D},\) then there exists \(E\subset \mathfrak {D}\) with \(|E|{\gt}0\) such that

It follows that

This is a contradiction due to the fact that \(E \subset \mathfrak {D}.\) This completes the proof. â–¡

Since \(\mathfrak {p}^{-j} \mathbb N_0 \subset \mathbb N_0\) and by lemma 3.3 we have

It follows that

and thus \(f_{j,k}\in \mathcal{L}^2(\mathbb K),\) and \(|f_{j,k}|_2 \le q^j|f|_2\) by 3.11. This completes the proof. â–¡

As an immediate consequences of lemma 3.4, we have

The following two lemmas are very useful in the later sections.

The proof of the lemma follows from the observation

By lemma 3.4 and lemma 3.6,

It leads to the lemma due to the fact that \(\mathbb N_0 = \mathcal{N}_j +\mathfrak {p}^j\mathbb N_0\). Thus the proof is complete. â–¡

It is clear that\( \lim \limits _{j\rightarrow \infty }\| \mathfrak {p}^{-j}\| ^{1/j} {\lt} 1.\) It follows that there exists \(J \in \mathbb N\) such that \(\| \mathfrak {p}^{-j}\| {\lt} 1~ \text{for}~ j {\gt} J.\) By setting \(C =\tfrac {1}{\max \{ \| \mathfrak {p}^{-j}\| : 0\le j \le J\} },\) then \(\| \mathfrak {p}^{-j}\| \le \tfrac {1}{C}\) for \(j \in \mathbb N_0\) and thus \(|\mathfrak {p}^{-j}x|\le \tfrac {1}{C}|x|\) for \(j \in \mathbb N_0\) and \(x \in K.\) This implies that \(|\mathfrak {p}^j x| \le C|x|\) for \(j \in \mathbb N_0\) and \( x \in K.\) â–¡

By invoking lemma 3.1 and lemma 3.4 and the fact that \(\tau {\gt} 1\), it is clear that \( g_{j,k}^{{\rm {\rm per}}} \in L^2(\mathfrak {D}).\) By a direct computation, we have \(g_{j,k}^{{\rm {\rm per}}} =g_{j,k+\mathfrak {p}^{-j}u(r)}^{{\rm {\rm per}}}\) for \( j \in \mathbb N_0,\; k,\; r\in \mathbb N_0.\) First we claim that if there exists a constant \( C {\gt} 0 \) such that

holds, then \(\sum _{r\in \mathbb N_0}\left\langle F_0, g_{j,k}(\cdot +u(r))\right\rangle \) is well defined and

It follows that

Therefore by 3.13, we have

which implies that \(\left\{ 1,g_{j,k}^{{\rm {\rm per}}} : j \in \mathbb N_0, k\in \mathcal{N}_j\right\} \) is a Bessel sequence in \(L^2(\mathfrak {D}).\)

Now we proceed to prove 3.13. By lemma 3.8, there exists \( C_1 {\gt} 0\) such that

for \( j \in \mathbb N_0, x \in \mathfrak {D},k\in \mathcal{N}_j\) and \( r\in \mathbb N_0.\) We observe that \(|k|\rightarrow \infty \) if and only if \(\rho (k)\rightarrow \infty .\) It follows that there exists \(J\in \mathbb N\) such that, if \( |r| {\gt} J,\) then

and \( \rho (r) \ge 2c_0 \max _{x \in \mathfrak {D}}\rho (x)\) for \( j \in \mathbb N_0, x \in \mathfrak {D}~ \text{and}~ k \in \mathcal{N}_j,\) where \(c_0\) is as in definition 2.3. So

for \( k \in \mathbb N_0\) with \( |r| {\gt} J,\; j\in \mathbb N_0,\; x \in \mathfrak {D}\) and \( k \in \mathcal{N}_j,\) where \(c_1\) is as in definition 2.3. Therefore, it follows that

for \( j \in \mathbb N_0, \; k \in \mathcal{N}_j\) and \( r \in \mathbb N_0\) with \( |r| {\gt} J, \) and thus

where \(C_4\) is a constant independent of \(f\). Further, we have

Therefore, it follows that

and thus we get 3.13. This completes the proof of the theorem.â–¡

4 Construction of Wavelet Bi-frames on Local Fields

We devote this section to the construction of periodic wavelet bi-frames on local fields. we start with some lemmas.

Suppose \(H\) is a symbol of \(f\). Then

Also observe that \(\widehat{f}(0) \ne 0\) and \(\widehat{f}\) is continuous by the fact that \(f\in L^1(K).\) This implies that \(H(\xi )=\tfrac {\widehat{f}(\mathfrak {p}^{-1}\xi )}{f(\xi )}\) and is continuous in some neighborhood of the origin, and thus \(H(0) = 1\) by letting \(\xi \rightarrow 0.\) So, at an arbitrary \(\beta \in \mathbb N_0, H(\xi )\) is continuous and equals \(1\) by its periodicity. Combined with 4.14, it follows that \(\widehat{f}(\mathfrak {p}^{-r}u(\beta )) = \widehat{f}(u(\beta ))\) for \(r\in \mathbb N,\beta \in \mathbb N.\) Letting \(r\rightarrow \infty ,\) we have \(\widehat{f}(u(\beta ))=0\) since \(\lim _{|\xi |\rightarrow \infty }\widehat{f}(\xi ) =0\) due to the fact \(f \in L^1(K).\) This completes the proof. â–¡

Set \(J = \max \left\{ |u(r_1)-u(r_2)| : r_1,r_2 \in S\right\} .\) Since we have \(\lim \limits _{j\rightarrow \infty }\| \mathfrak {p}^{-j}\| =0\), there exists \(j_0 \in \mathbb N\) such that \(\| \mathfrak {p}^{-j_0}\| {\lt}\frac{1}{J}.\) It follows that

and thus \( \mathfrak {p}^{-j_0}(u(r_1)-u(r_2)) \notin \mathbb N_0\). This completes the proof. â–¡

For \( r_1 \ne r_2 \in \mathcal{N}_j, u(r_1)-u(r_2) \notin \mathfrak {p}^{j}\mathbb N_0,\) which implies \(r_1 \ne r_2 \in \mathcal{N}_j, u(r_1)-u(r_2) \notin \mathfrak {p}^{j+1}\mathbb N_0.\) This completes the proof. â–¡

To prove the above theorem, we first prove the following lemma:

By invoking lemma 3.7 to \( \psi _\ell , \widetilde{\psi }_\ell \) with \(0 \le \ell \le L\), we have

for each \(0 \le \ell \le L.\) By 4.18 and \(\mathfrak {p}\)-refinable property of \(\psi _0\), we have

where

In the similar manner, we have

where

Since \(X(\Psi ), X(\widetilde{\Psi })\) are both Bessel sequences, and

We observe that \(\left\{ q^{-j/2}\chi _r(\mathfrak {p}^j \xi ) :r\in \mathbb N_0\right\} \) is an orthonormal basis for \( L^2(\mathfrak {p}^{-j}\mathfrak {D}).\) It follows that

and

So by 4.21, we have

Further, by 4.19 we observe that

Also we have

Thus by 4.22, we have

By applying lemma 3.3 to \(\mathfrak {p}^{-1}\) leads to \(\bigcup _{p \in \mathcal{N}_1}\big(\mathfrak {D}+u(p))\) being \(\mathfrak {p}^{-1}\mathbb N_0\)-congruent to \(\mathfrak {p}^{-1}\mathfrak {D}\), therefore \({\mathfrak {p}^{-j}\big(\bigcup _{p \in \mathcal{N}_1}(\mathfrak {D}+u(p))\big)}\) is \(\mathfrak {p}^{-j-1}\mathbb N_0\)-congruent to \(\mathfrak {p}^{-j-1}\mathfrak {D}.\) Combined with 4.23, it follows that

Further, we have

Similarly, we have

for \(s \in \mathbb N_0.\)

By 4.15 and lemma 3.1, we have \(\psi _0 \in \mathcal{L}^2(\mathbb K).\) This implies that
\(\{ \psi _0(\cdot -u(r)) : r\in \mathbb N_0\} \) is a Bessel sequence in \(L^2(\mathbb K)\) by lemma 3.2, and thus \(\{ \psi _{0,j+1,r} :r \in \mathbb N_0\} \) is a Bessel sequence in \(L^2(\mathbb K).\) This implies that \(\big\{ \left\langle F_0, \psi _{0,j+1,r}\right\rangle _{L^2(\mathbb K)}\big\} _{r \in \mathbb N_0} \in \ell ^2(\mathbb N_0)\) and thus

In the similar manner, we have

Therefore, by 4.24, 4.25 and 4.26, we have

However, applying 4.21 to the case of \(j+1\) and \(\ell =0\), we have

Hence

for \(k \in \mathbb N_0.\)

Now we need to check \(\psi _0^{{\rm {\rm per}}}\) and \(\widetilde{\psi }_0^{{\rm {\rm per}}}\). By 4.15, we have \(\psi _0, \widetilde{\psi }_0 \in L^2(\mathbb K),\) thus \( \widehat{\psi }(u(\alpha )) = \widehat{\widetilde{\psi }_0}(u(\alpha )) =0~ \text{for}~ \alpha \in \mathbb N\) by lemma 4.1. It follows that

Combining 4.28 and 4.29, 4.20 follows. This completes the proof of lemma 4.8. â–¡

[Proof of theorem 4.7]

By calling theorem 3.9, it is clear that \(\Big\{ \psi _{\ell ,j,k}^{{\rm {\rm per}}} :j \in \mathbb N_0, k \in \mathcal{N}_j\Big\} \) and \(\left\{ \widetilde{\psi }_{\ell ,j,k}^{{\rm {\rm per}}} :j \in \mathbb N_0, k \in \mathcal{N}_j\right\} \) are Bessel sequences in \(L^2(\mathfrak {D})\) for each \(1 \le \ell \le L.\) It follows that \(X^{{\rm {\rm per}}}(\Psi )\) and \(X^{{\rm {\rm per}}}(\widetilde{\Psi })\) are both Bessel sequences in \(L^2(\mathfrak {D})\). Also observe that the set of trigonometric polynomials is dense in \(L^2(\mathfrak {D}).\) In order to finish the proof, we only need to show that

for arbitrary trigonometric polynomials \(f\) and \(g\) by lemma 4.6. Again by lemma 4.8, it is equivalent to

for arbitrary trigonometric polynomials \(f\) and \(g\). Let us fix the trigonometric polynomials \(f\) and \(g\). Now we prove 4.30. Clearly there exists a finite set \(S\subset \mathbb N_0\) such that \(f\) and \(g\) have the form

where

By lemma 4.2, we can extend \(S\) as a full set \(\mathcal{N}_{j_0}.\) Also by lemma 4.3, we can extend \(\mathcal{N}_{j_0}\) to \(\mathcal{N}_{j_0+1}\). By repeating this procedure, we can obtain a sequence \(\{ \mathcal{N}_j\} _{j=j_0}^{\infty } \) satisfying \(\mathcal{N}_j \subset \mathcal{N}_{j+1}.\) It implies that

By using lemma 3.1 and lemma 3.5, we have \(\psi _{0,j,k}^{{\rm per}},\widetilde{\psi }_{0,j,k}^{{\rm per}} \in L^2(\mathfrak {D}).\) It follows by lemma 4.5 that

where

Combining the above with 4.33, we obtain

Applying lemma 4.4 to \(\mathfrak {p}^{-j}\), we obtain an orthonormal basis \(\Big\{ q^{-j/2}\overline{\chi _k(\mathfrak {p}^{-j}\xi )} :k \in \mathcal{N}_j\Big\} \) for \(\ell ^2(\mathcal{N}_j)\). So

Also observe that \(\psi _0,\widetilde{\psi }_0 \in \mathcal{L}^2(\mathbb K)\subset L^1(K)\) by lemma 3.1, which implies that \(\psi _0,\widetilde{\psi }_0\) are continuous, and by 4.26, we have

By letting \(j \rightarrow \infty \) in 4.35, we obtain that

This completes the proof of theorem 4.7. â–¡

We define \(\widetilde{\Gamma }\) by

Then clearly \(\widetilde{\Gamma }\) is \(\mathfrak {p}\)-refinable with the symbol

Further, we define \(\widetilde{\Gamma }_\ell \) by

Denote \(\widetilde{\mho } =\{ \widetilde{\Gamma }_\ell : 1 \le \ell \le L\} \), it is easy to check that the systems \(\Psi \cup \{ \psi _0\} \) and \(\widetilde{\mho }\cup \{ \widetilde{\Gamma }_0\} \) associated with \(\widetilde{H'}_\ell \) satisfy the conditions of theorem 4.7, and \(\widetilde{\mho } = \widetilde{\Psi }.\) Therefore, \(\big(X^{{\rm per}}(\Psi ),X^{{\rm per}}(\widetilde{\mho }\big)\) i.e, \(\big(X^{{\rm per}}(\Psi ),X^{{\rm per}}(\widetilde{\Psi }\big)\) is a periodic wavelet bi-frame by theorem 4.7. The proof is complete. â–¡