# Vanishing Moments of Wavelets on $p$-adic Fields

Athira N∗ and Lineesh M C†

###### Abstract

A study on the vanishing moments of wavelets on $p$-adic fields is carried out
in this paper. The $p$-vanishing moments and discrete $p$-vanishing moments
are defined on a $p$-adic field and the relation between them is investigated.
The $p$-vanishing moments of Haar-type and non-Haar type wavelet functions are
computed. Also, the connection between $p$-vanishing moment of non-Haar type
wavelet functions and the approximation order of the indicator function of the
compact open subgroup $B_{0}(0)$ of $\mathbb{Q}_{p}$ are established. A
characterization of the $p$-vanishing moments of nonorthogonal wavelets is
done.

Department of Mathematics
National Institute of Technology Calicut
NIT Campus P O - 673 601, India
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
keywords: $p$-adic field; pseudo-differential operator; refinable function;
p-adic wavelets; approximation order; vanishing moment; compact support.
AMS Subject Classification 2020: 11E95, 11F85, 41A10, 42C40, 43A70.

##  1 Introduction

Over the past few years, wavelets have gained immense popularity in almost all
fields of science and technology due to their unique time-frequency
localization feature. One of the most significant facts is that, in addition
to the canonical tool of representing a function by its Fourier series, there
is a different representation using wavelets which is more suitable to certain
problems in data compression and signal processing. The first wavelet
construction is due to Alfréd Haar in 1910. Then many mathematicians including
Jean Morlet, Alex Grossmann, Yves Meyer, Stephane Mallat and Ingrid Daubechies
have contributed various kinds of wavelets to theoretical and applied science.
The concept of wavelet is then extended to the Euclidean space as well as many
other topological spaces such as $p$-adic fields.

Wavelets can be generated from scaling functions as well as wavelet sets. In
their paper, Albeverio et al.[1, 2] proposed a complete characterization of
scaling functions and explained what types of scaling functions form $p$-adic
multiresolution analysis. Khrennikov et al. [3] described the procedure to
construct $p$-adic wavelets from $p$-adic scaling functions associated with an
expansive automorphism. The authors [4] discussed about all compactly
supported orthogonal wavelet bases for $L_{2}(\mathbb{Q}_{p})$ generated by
the unique p-adic multiresolution analysis, i.e.,the Haar bases of
$L_{2}(\mathbb{Q}_{p})$.

In 2008, Shelkovich and Skopina [5] constructed infinitely many different
multidimensional $p$-adic Haar orthonormal wavelet bases for
$L_{2}(\mathbb{Q}_{p}^{n})$ and in 2009, Khrennikov and Shelkovich [6]
developed infinite family of compactly supported non-Haar type $p$-adic
wavelet bases for $L_{2}(\mathbb{Q}_{p}^{n}),\,n\geq 1$. Kozyrev et al. [7]
also discussed about the one-dimensional and multi-dimensional wavelet bases
and their relation to the spectral theory of pseudo-differential operators.

Reﬁnable functions play an important role in the construction and properties
of wavelets. Basically, most of the wavelets are generated from reﬁnable
functions. In their paper Athira and Lineesh [8] discussed about the
approximation order of shift-invariant space of a reﬁnable function on p-adic
ﬁeld. The relation between the approximation order, accuracy of refinable
function and order of the Strang–Fix condition are also established.

In order to fulfill the requirements, some properties are relevant for the
wavelet bases. One of the important property among them is the vanishing
moment. Vanishing moments are essential in the context of compression of a
signal. A vanishing moment limits the wavelet’s ability to represent
polynomial behaviour or information in a signal. Higher vanishing moments of
wavelets are required for signal compression and denoising. In 1999, S. Mallat
[9] established that the number of vanishing moments of a wavelet and the
approximation order of the corresponding scaling function are equivalent in
$L_{2}(\mathbb{R})$. Then Di-Rong Chen et al. [10] obtained the the relation
between the order of sum rules and the number of vanishing moments of wavelets
on $L_{2}(\mathbb{R}^{n})$. An explicit formula for refinement masks providing
vanishing moments is explored by Skopina[11]. In 2010, Yu Liu and L. Peng [12]
proved the connection between discrete vanishing moments and sum rules on the
Heisenberg group.

Our goal is to extend the concept of vanishing moments to the $p$-adic field
$\mathbb{Q}_{p}$. Section 2 contains preliminary informations about scaling
functions, wavelet functions and accuracy of scaling functions on
$\mathbb{Q}_{p}$. In section 3, published results about the relationship
between accuracy and vanishing moments on Euclidean spaces are discussed. The
definitions of $p$-vanishing moments of compactly supported functions on
$\mathbb{Q}_{p}$ and discrete $p$-vanishing moments of finitely supported
sequences on $I_{p}$ are given in section 4. The relationship between the
$p$-vanishing moments and discrete $p$-vanishing moments are also established
in this section. In section 5, the $p$-vanishing moments of Haar-type wavelet
functions are calculated. The $p$-vanishing moments of non-Haar type wavelet
functions are computed in section 6. In this section, we proved the connection
between $p$-vanishing moment of non-Haar type wavelet functions and the
approximation order of the indicator function of the compact open subgroup
$B_{0}(0)$ of $\mathbb{Q}_{p}$. Finally in section 7, we characterized the
$p$-vanishing moments of nonorthogonal wavelets.

##  2 Preliminary

Let $p$ be a prime number. Consider the completion field of $\mathbb{Q}$ with
respect to the norm $\lvert\cdot\rvert_{p}$ defined by,

$\lvert x\rvert_{p}=\begin{cases}0&;\,x=0\\\ p^{-\gamma}&;\,x\neq
0,x=(p^{\gamma})\frac{m}{n},\end{cases}$

where $\gamma=\gamma(x)\in\mathbb{Z},\,m,n\in\mathbb{Z}$ not divisible by $p$.
Denote the above field as $G=\mathbb{Q}_{p}$. The canonical form of
$x\in\mathbb{Q}_{p}$, $x\neq 0$ is,

$x=p^{\gamma}(x_{0}+x_{1}p+x_{2}p^{2}+\cdots),$ (2.1)

where $\gamma\in\mathbb{Z},\,x_{j}\in\\{0,1,\ldots,p-1\\},\,x_{0}\neq 0$. Then
for this $x\in\mathbb{Q}_{p}$, the fractional part of $x$ is,

$\\{x\\}_{p}=\begin{cases}0&;\,\gamma(x)\geq 0\text{ or }x=0\\\
p^{\gamma}(x_{0}+x_{1}p+\cdots+x_{-\gamma-1}p^{-\gamma-1})&;\,\gamma(x)<0.\end{cases}$

The dual group of $\mathbb{Q}_{p}$ is $\mathbb{Q}_{p}$ itself and the
character on $\mathbb{Q}_{p}$ is defined as,

$\chi(x,\xi)=e^{2\pi i\\{x\xi\\}_{p}},$ (2.2)

where $\\{\cdot\\}_{p}$ is the fractional part of a number. Denote

$B_{\gamma}(a)=\\{x\in\mathbb{Q}_{p}:\lvert x-a\rvert_{p}\leq p^{\gamma}\\}.$

Then $B_{0}(0)$ is a compact open subgroup of $\mathbb{Q}_{p}$. Let $\mu$ be
the Haar measure on $\mathbb{Q}_{p}$ with $\mu(B_{0}(0))=1$. Denote $d\mu(x)$
by $dx$. Let $L_{q}(\mathbb{Q}_{p})$ be the collection of all integrable
functions $f:\mathbb{Q}_{p}\rightarrow\mathbb{C}$ such that
$\int_{\mathbb{Q}_{p}}\lvert f(x)\rvert^{q}dx<\infty$. The Fourier transform
of a complex-valued function $f$ defined on $\mathbb{Q}_{p}$ is defined as

$\widehat{f}(\xi):=\int_{\mathbb{Q}_{p}}f(x)\chi(x,\xi)dx.$

If $E$ is a measurable subset of $\mathbb{Q}_{p}$ and $1\leq q<\infty$, then
$\lVert f\rVert_{q}(E)$ denotes the quantity $(\int_{E}\lvert
f(x)\rvert^{q}dx)^{1/q}$. If $q=\infty$, then $\lVert f\rVert_{\infty}(E)$
denotes the essential supremum of $f$ over $E$.

For the $p$-adic analysis related to the mapping
$\mathbb{Q}_{p}\rightarrow\mathbb{C}$, the operation of differentiation is not
defined. An analogy of the differentiation operator is a pseudo-differential
operator. The pseudo-differential operator $D^{\alpha}:\phi\rightarrow
D^{\alpha}\phi$ is defined [13] by

$D^{\alpha}\phi=f_{-\alpha}*\phi$ (2.3)

where $f_{\alpha}(x)=\frac{\lvert x\rvert_{p}^{\alpha-1}}{\Gamma_{p}(\alpha)}$
with $\Gamma_{p}(\alpha)=\frac{1-p^{\alpha-1}}{1-p^{-\alpha}}$.

######  Remark 2.1.

[13] The derivative $D^{\alpha}\phi$, $\alpha>0$ is given by the expression

$(D^{\alpha}\phi)(x)=\int_{\mathbb{Q}_{p}}\lvert\xi\rvert_{p}\widehat{\phi}(\xi)\chi(x,-\xi)d\xi.$
(2.4)

######  Remark 2.2.

[13] For $\alpha\in\mathbb{R}$ and $a\in\mathbb{Q}_{p}\setminus\\{0\\}$,
$D^{\alpha}\chi(a,x)=\lvert a\rvert_{p}^{\alpha}\chi(a,x)$.

######  Remark 2.3.

[13] For $\alpha\in\mathbb{R},\,\gamma\in\mathbb{Z}$, let
$\Phi(x)=\delta(\lvert\xi\rvert_{p}-p^{\gamma})f(\xi)$. Then

$D^{\alpha}(\widehat{\Phi(x)})=p^{\gamma\alpha}\widehat{\Phi(x)}.$

A polynomial $r(x)$ on $\mathbb{Q}_{p}$ is given by

$r(x)=c_{0}+c_{1}x+\cdots+c_{n}x^{n},\quad x\in\mathbb{Q}_{p},$

where each $c_{j}\in\mathbb{Q}_{p}$. If $c_{n}\neq 0$, then the degree of
$r(x)$ is $n$. For a nonnegative integer $k$, we denote by $\mathscr{P}_{k}$
the linear span of $\\{r(x):\text{degree of }r(x)\leq k\\}$. Then
$\mathscr{P}=\bigcup_{k=0}^{\infty}\mathscr{P}_{k}$ is the linear space of all
polynomials.

Let us consider the set

$I_{p}=\\{a=p^{-\gamma}(a_{0}+a_{1}p+\cdots+a_{\gamma-1}p^{\gamma-1}):\gamma\in\mathbb{N},\,a_{j}\in\\{0,1,\ldots,p-1\\}\\}.$

Then there is a “natural” decomposition of $\mathbb{Q}_{p}$ into a union of
mutually disjoint discs: $\mathbb{Q}_{p}=\bigcup_{a\in I_{p}}B_{0}(a)$. So,
$I_{p}$ is a “natural set” of shifts for $\mathbb{Q}_{p}$.

We denote by $l(I_{p})$ the linear space of all sequences on $I_{p}$, and by
$l_{0}(I_{p})$ the linear space of all finitely supported sequences on
$I_{p}$.

For a compactly supported function $\phi$ on $\mathbb{Q}_{p}$ and a sequence
$b\in l(I_{p})$, the semi-convolution of $\phi$ with $b$ is defined by

$\phi*^{\prime}b(x):=\sum_{\alpha\in I_{p}}\phi(x-\alpha)b(\alpha).$ (2.5)

Let $S(\phi)$ denote the linear space $\\{\phi*^{\prime}b:b\in l(I_{p})\\}$.
We call $S(\phi)$ the shift-invariant space generated by $b$.

Define $A:\mathbb{Q}_{p}\rightarrow\mathbb{Q}_{p}$ by $A(x)=\frac{1}{p}x$.
Then, $A$ is an expansive automorphism with modulus, $\lvert
A\rvert=\mu(A(B_{0}(0)))=\mu(B_{1}(0))=p$ and $A^{*}=A$.

######  Definition 2.1.

[2] A collection of closed spaces $V_{j}\subset
L_{2}(\mathbb{Q}_{p}),\,j\in\mathbb{Z}$, is called a multiresolution
analysis(MRA) in $L_{2}(\mathbb{Q}_{p})$ if the following axioms hold:

  1. 1.

$V_{j}\subset V_{j+1}$ for all $j\in\mathbb{Z}$;

  2. 2.

$\bigcup_{j=-\infty}^{\infty}V_{j}$ is dense in $L_{2}(\mathbb{Q}_{p})$;

  3. 3.

$\bigcap_{j=-\infty}^{\infty}V_{j}=\\{0\\}$;

  4. 4.

$f(\cdot)\in V_{j}\Leftrightarrow f(A\cdot)\in V_{j+1}$ for all
$j\in\mathbb{Z}$;

  5. 5.

there exists a function $\phi\in V_{0}$ such that $\\{\phi(\cdot-a),\,a\in
I_{p}\\}$ is an orthonormal basis for $V_{0}$.

The function $\phi$ from axiom $(5)$ is called scaling function. Then we says
that a MRA is generated by its scaling function $\phi$ (or $\phi$ generates
the MRA). It follows immediately from axioms $(4)$ and $(5)$ that the
functions $p^{j/2}\phi(p^{-j}\cdot-a),\,a\in I_{p}$, form an orthonormal basis
for $V_{j},\,j\in\mathbb{Z}$. Let $\phi$ be an orthogonal scaling function for
a MRA $\\{V_{j}\\}_{j\in\mathbb{Z}}$, then

$\phi(x)=\sum_{a\in
I_{p}}\alpha(a)\phi(p^{-1}x-a),\quad\alpha(a)\in\mathbb{C}.$ (2.6)

Such equations are called refinement equations, and their solutions are called
refinable functions.

Generally, a refinement equation (2.6) does not imply the inclusion property
$V_{0}\subset V_{1}$ because the set of shifts $I_{p}$ does not form a group.
Indeed, we need all the functions $\phi(\cdot-b),b\in I_{p}$, to belong to the
space $V_{1}$, i.e., the identities $\phi(x-b)=\sum_{a\in
I_{p}}\alpha(a,b)\phi(p^{-1}x-a)$ should be fulfilled for all $b\in I_{p}$.
Since $p^{-1}b+a$ is not in $I_{p}$ in general, we cannot argue that
$\phi(x-b)$ belongs to $V_{1}$ for all $b\in I_{p}$. Thus the refinement
equation must be redefined as in the following definition.

######  Definition 2.2.

[1, 2] If $\phi\in L_{2}(\mathbb{Q}_{p})$ is a refinable function and
$\text{supp}(\phi)\subset B_{N}(0)$, $N\geq 0$, then its refinement equation
is

$\phi(x)=\sum_{k=0}^{p^{N+1}-1}h\left(\frac{k}{p^{N+1}}\right)\phi\left(\frac{x}{p}-\frac{k}{p^{N+1}}\right),\quad\forall
x\in\mathbb{Q}_{p},$ (2.7)

with $h\in l_{0}(I_{p})$ such that $h(a)=0$ for all $a\in
I_{p}\setminus\\{k/p^{N+1}:k=0,1,\ldots,p^{N+1}-1\\}$ and

$\sum_{k=0}^{p^{N+1}-1}h\left(\frac{k}{p^{N+1}}\right)=p.$ (2.8)

Taking Fourier transform on both sides of (2.7), we obtain

$\widehat{\phi}(\xi)=H(A^{-1}\xi)\widehat{\phi}(A^{-1}\xi),\quad\xi\in\mathbb{Q}_{p},$
(2.9)

where

$H(\xi)=\frac{1}{p}\sum_{k=0}^{p^{N+1}-1}h\left(\frac{k}{p^{N+1}}\right)\chi\left(\frac{k}{p^{N+1}},\xi\right),\quad\xi\in\mathbb{Q}_{p}.$
(2.10)

Denote by $\mathscr{D}_{N}^{M}$ the set of all $p^{M}$-periodic functions
supported on $B_{N}(0)$. Then we have the following results about refinable
functions [2].

######  Theorem 2.1.

A function $\phi\in\mathscr{D}_{N}^{M},\,M,N\geq 0$, with
$\widehat{\phi}(0)\neq 0$ generates a MRA if and only if

  1. 1.

$\phi$ is refinable;

  2. 2.

there exists at most $p^{N}$ integers $l$ such that $0\leq l<p^{M+N}$ and
$\widehat{\phi}(\frac{l}{p^{M}})\neq 0$.

######  Theorem 2.2.

Let $\widehat{\phi}$ be defined by (2.9), where $H$ is the trigonometric
polynomial (2.10) with $H(0)=1$. If $H(k)=0$ for all $k=1,\ldots,p^{N+1}-1$
not divisible by $p$, then $\phi\in\mathscr{D}_{N}^{0}$. If furthermore,
$\lvert H(k)\rvert=1$ for all $k=1,\ldots,p^{N+1}-1$ divisible by $p$, then
$\\{\phi(x-a),\,a\in I_{p}\\}$ is an orthonormal system. Conversely, if
$\text{supp }\widehat{\phi}\subset B_{0}(0)$ and the system
$\\{\phi(x-a),\,a\in I_{p}\\}$ is orthonormal, then $\lvert H(k)\rvert=0$
whenever $k$ is not divisible by $p$, $\lvert H(k)\rvert=1$ whenever $k$ is
divisible by $p$, $k=1,\ldots,p^{N+1}-1$, and
$\lvert\widehat{\phi}(x)\rvert=1$ for any $x\in B_{0}(0)$.

######  Theorem 2.3.

[2] There exists a unique MRA generated by an orthogonal scaling function.

######  Remark 2.4.

That is all orthogonal scaling functions generate the same Haar MRA.

Suppose we have a p-adic MRA generated using a scaling function $\phi$
satisfying the refinement equation (2.7), then the wavelet functions
$\psi_{j},\,j=1,\ldots,p-1$ are in the form

$\psi_{j}(x)=\sum_{k=0}^{p^{N+1}-1}h_{j}\left(\frac{k}{p^{N+1}}\right)\phi\left(\frac{x}{p}-\frac{k}{p^{N+1}}\right),\quad\forall
x\in\mathbb{Q}_{p},$ (2.11)

where the coefficients $h_{j}(\frac{k}{p^{N+1}})$ are chosen such that

$<\psi_{j},\phi(\cdot-a)>=0,\,\,<\psi_{j},\psi_{k}(\cdot-a)>=\delta_{j,k}\delta_{0,a},\,\,j,k=1,\ldots,p-1,$
(2.12)

for any $a\in I_{p}$.
Set

$h=\frac{1}{\sqrt{p}}\left(h(0),\ldots,h\left(\frac{p^{N+1}-1}{p^{N+1}}\right)\right),$
$h_{j}=\frac{1}{\sqrt{p}}\left(h_{j}(0),\ldots,h_{j}\left(\frac{p^{N+1}-1}{p^{N+1}}\right)\right),\,j=1,\ldots,p-1,$

and

$S=\begin{bmatrix}0&0&\cdots&0&1\\\ 1&0&\cdots&0&0\\\ 0&1&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots\\\ 0&0&\cdots&1&0\end{bmatrix}.$

In order to satisfy (2.12), we need to find $h_{j},\,j=1,\ldots,p-1$ such that
the matrix

$U=(S^{0}h,\ldots,S^{p^{N}-1}h,S^{0}h_{1},\ldots,S^{p^{N}-1}h_{1},\ldots,S^{0}h_{p-1},\ldots,S^{p^{N}-1}h_{p-1})$
(2.13)

is unitary.[3]

Let $\phi$ be a compactly supported function in $L_{q}(\mathbb{Q}_{p})(1\leq
q\leq\infty)$. The norm in $L_{q}(\mathbb{Q}_{p})$ is denoted by
$\lVert\cdot\rVert_{q}$. For an element $f\in L_{q}(\mathbb{Q}_{p})$ and a
subset $G$ of $L_{q}(\mathbb{Q}_{p})$, the distance from $f$ to $F$, denoted
by $\text{dist}_{q}(f,F)$, is defined by

$\text{dist}_{q}(f,F):=\inf_{g\in F}\lVert f-g\rVert_{q}.$

Let $S:=S(\phi)\cap L_{q}(\mathbb{Q}_{p})$. For $n\in\mathbb{Z}$, let
$S^{n}:={g(A_{n}^{-1}\cdot):g\in S}$, where
$A_{n}(x)=p^{n}x,\,\,x\in\mathbb{Q}_{p}$. For a real number $k\geq 0$, we say
that $S(\phi)$ provides approximation order $k$ if for each sufficiently
smooth function $f\in L_{q}(\mathbb{Q}_{p})$, there exists a constant $C>0$
such that

$\text{dist}_{q}(f,S^{n})\leq Cp^{-nk},\quad\forall
n\in\mathbb{N}\cup\\{0\\}.$

The relation between the approximation order provided by $S(\phi)$, the order
of the Strang–Fix condition and the accuracy of $\phi$ are established in the
following theorems.

######  Theorem 2.4.

[8] Let $1\leq q\leq\infty$, and let $\phi$ be a compactly supported function
in $L_{q}(\mathbb{Q}_{p})$ with $\widehat{\phi}(0)\neq 0$. For every positive
integer $k$, the following statements are equivalent:

  1. 1.

$S(\phi)$ provides approximation order $k$.

  2. 2.

$S(\phi)$ contains $\mathscr{P}_{k-1}$.

  3. 3.

$D^{\mu}\widehat{\phi}(p^{N}\alpha)=0,\,\forall\alpha\in
I_{p}\setminus\\{0\\},\,\lvert\mu\rvert\leq k-1$.

######  Definition 2.3.

[8] For a compactly supported function $\phi$ on $\mathbb{Q}_{p}$, define

$N(\phi)=\\{\xi\in\mathbb{Q}_{p}:\widehat{\phi}(\xi)=0\\}.$

######  Theorem 2.5.

[8] Let $h$ be a finitely supported sequence on $I_{p}$, satisfying (2.8) and
$H$ be a function given by (2.10). Let
$\Omega=\\{0,\frac{1}{p},\frac{2}{p},\ldots,\frac{p-1}{p}\\}$. If

$D^{\mu}H(p^{N}A^{-1}\omega)=0,\,\forall\omega\in\Omega\setminus\\{0\\},\,\lvert\mu\rvert\leq
k-1,$ (2.14)

then the normalized solution $\phi$ of (2.7) has accuracy $k$. Conversely, if
$\phi$ has accuracy $k$ and if $N(\phi)\cap(p^{N}A^{-1}\Omega)=\emptyset$,
then (2.14) holds true.

##  3 Vanishing Moments on Euclidean spaces

In this section we recall some facts about vanishing moments on Euclidean
spaces.

Let $\psi$ be a wavelet function on $\mathbb{R}$. Then we say that $\psi$ has
$n$ vanishing moments if

$\int_{-\infty}^{\infty}t^{k}\psi(t)dt=0,\text{ for }0\leq k<n.$

A wavelet with $n$ vanishing moments is orthogonal to each of the polynomials
of degree $n-1$ [9].

The following theorem gives the relationship between the approximation order
of the corresponding scaling function and the number of vanishing moments of
the wavelets on $\mathbb{R}$.

######  Theorem 3.1.

[Theorem 7.4 in Chapter 7 of [9]] Let $\psi$ and $\phi$ be a wavelet and a
scaling function that generate an orthogonal basis. Let $h$ be the refinement
mask for the scaling function $\phi$. Suppose that
$\lvert\psi(x)\rvert=O((1+t^{2})^{-\frac{n}{2}-1})$ and
$\lvert\phi(x)\rvert=O((1+t^{2})^{-\frac{n}{2}-1})$. Then the following
statements are equivalent:

  1. 1.

The wavelet $\psi$ has $n$ vanishing moments.

  2. 2.

$\widehat{\psi}(\omega)$ and its first $n-1$ derivatives are zero at
$\omega=0$.

  3. 3.

$\widehat{h}(\omega)$ and its first $n-1$ derivatives are zero at
$\omega=\pi$.

  4. 4.

For any $0\leq k<n$,

$q_{k}(t)=\sum_{n=-\infty}^{\infty}n^{k}\phi(t-n)\text{ is a polynomial of
degree }k.$ (3.1)

The hypothesis $(4)$ is called the Strang-Fix condition on $\mathbb{R}$.

Let $M$ be a $d\times d$ dilation matrix on $\mathbb{R}^{d}$ and denote
$m:=\lvert\det M\rvert$. Let $\Omega$ be a complete set of representatives of
the distinct cosets of the quotient group $\mathbb{Z}^{d}/M\mathbb{Z}^{d}$. It
is obvious that the cardinality of $\Omega$ is equal to $m$. Without loss of
generality we assume that $0\in\Omega$.

For any positive integer $k$, a compactly supported function $f\in
L_{2}(\mathbb{R}^{d})$ has $k$ vanishing moments if

$\int_{\mathbb{R}^{d}}p(x)f(x)dx=0,\quad\text{for all }p\in\mathscr{P}_{k},$

where $\mathscr{P}_{k}$ denotes the set of all polynomials of degree less than
$k$.

Let $l_{0}(\mathbb{Z}^{d})$ denotes the space of all finitely supported
sequences on $\mathbb{Z}^{d}$. Then a sequence $b\in l_{0}(\mathbb{Z}^{d})$
has $k$ discrete vanishing moments if it satisfies the following equalities

$\sum_{\gamma\in\mathbb{Z}^{d}}b(\gamma)\gamma^{j}=0,\quad\text{for any
}j\in\mathbb{Z}_{+}^{d},\,\,\lvert j\rvert=j_{1}+\cdots+j_{d}<k,$

where $\gamma^{j}=\gamma_{1}^{j_{1}}\ldots\gamma_{d}^{j_{d}}$.

######  Proposition 3.1.

[10] Given a multiresolution analsysis on $\mathbb{R}^{d}$, with $m-1$
wavelets $\psi_{\epsilon},\epsilon\in\Omega\setminus\\{0\\}$. Then
$\psi_{\epsilon},\epsilon\in\Omega\setminus\\{0\\}$, have $k$ vanishing
moments if and only if the corresponding wavelet filters
$\\{h_{\epsilon}(\gamma)\\},\epsilon\in\Omega\setminus\\{0\\}$, have $k$
discrete vanishing moments.

A finitely supported sequence $h$ on $\mathbb{Z}^{d}$ satisfies the sum rules
[14] of order $k$ if

$\sum_{\gamma\in\mathbb{Z}^{d}}h(\epsilon+M(\gamma))p(\epsilon+M(\gamma))=\sum_{\gamma\in\mathbb{Z}^{d}}h(M(\gamma))p(M(\gamma))$

for all $\epsilon\in\Omega,\,p\in\mathscr{P}_{k}$.

The following proposition gives the relationship between the order of sum
rules and the number of discrete vanishing moments on $\mathbb{R}^{d}$.

######  Proposition 3.2.

[10] Let $h\in l_{0}(\mathbb{Z}^{d})$ satisfies the sum rules of order $k$ and
$\sum_{\gamma\in\mathbb{Z}^{d}}h(\gamma)=2^{d}$. If $b\in
l_{0}(\mathbb{Z}^{d})$ satisfies
$\sum_{\gamma\in\mathbb{Z}^{d}}b(\gamma)h(\gamma-M(\beta))=0$, for any
$\beta\in\mathbb{Z}^{d}$, then $b$ has $k$ discrete vanishing moments.

If the refinement mask $h$ of the scaling function $\phi$ satisfies the sum
rule of order $k$, then the corresponding wavelet filters $h_{\epsilon}$
satisfies

$\sum_{\gamma\in\mathbb{Z}^{d}}h_{\epsilon}(\gamma)h(\gamma-M(\beta))=0,\quad\text{for
any }\beta\in\mathbb{Z}^{d},\,\epsilon\in\Omega\setminus\\{0\\}.$

Thus, by Propositions 3.1 and 3.2, the corresponding wavelets
$\psi_{\epsilon}$ has vanishing moments of order $k$, for each
$\epsilon\in\Omega\setminus\\{0\\}$.

##  4 $p$-Vanishing Moments on $\mathbb{Q}_{p}$

This section gives a description of vanishing moments and discrete vanishing
moments on $\mathbb{Q}_{p}$.

######  Definition 4.1.

For any positive integer $k$, a compactly supported function $f$ in
$L_{2}(\mathbb{Q}_{p})$ is said to have $k$ $p$-vanishing moments if

$\int_{\mathbb{Q}_{p}}\lvert x\rvert_{p}^{\mu}f(x)dx=0,\quad\text{for all
}0\leq\mu<k.$ (4.1)

######  Theorem 4.1.

Let $f$ be a compactly supported function in $L_{2}(\mathbb{Q}_{p})$. Suppose
that $\widehat{f}$ is $k$ times pseudo-differentiable. Then $f$ has $k$
$p$-vanishing moments if and only if $D^{\mu}\widehat{f}(0)=0,\text{ for all
}0\leq\mu<k$.

###### Proof.

From the definition of the pseudo-differential operator $D^{\mu}$ we have,

$(D^{\mu}f)(x)=\int_{\mathbb{Q}_{p}}\lvert\xi\rvert_{p}^{\mu}\widehat{f}(\xi)\chi(x,-\xi)d\xi,\qquad\mu>0.$

Thus we have,

$(D^{\mu}\widehat{f})(\xi)=\int_{\mathbb{Q}_{p}}\lvert
x\rvert_{p}^{\mu}f(x)\chi(\xi,-x)dx,\qquad\mu>0.$

Then,

$(D^{\mu}\widehat{f})(0)=\int_{\mathbb{Q}_{p}}\lvert
x\rvert_{p}^{\mu}f(x)dx,\qquad\mu>0.$

Hence, $f$ has $k$ $p$-vanishing moments if and only if
$D^{\mu}\widehat{f}(0)=0,\,\forall\,\,0\leq\mu<k$. ∎

######  Definition 4.2.

A sequence $b\in l_{0}(I_{p})$ has $k$ discrete $p$-vanishing moments if it
satisfies the equalities

$\sum_{a\in I_{p}}b(a)\lvert a\rvert_{p}^{\mu}=0,\quad\text{for all
}0\leq\mu<k.$ (4.2)

######  Theorem 4.2.

Given a multiresolution analsysis on $\mathbb{Q}_{p}$, with $p-1$ wavelets
$\psi_{j},\,j=1,\ldots,p-1$. Then $\psi_{j}$, $j=1,\ldots,p-1$, have $k$
$p$-vanishing moments if and only if the corresponding wavelet filters
$h_{j},\,j=1,\ldots,p-1$, have $k$ discrete $p$-vanishing moments.

###### Proof.

From Theorem 4.1, $\psi_{j},\,j=1,\ldots,p-1$, have $k$ $p$-vanishing moments
if and only if $D^{\mu}\widehat{\psi_{j}}(0)=0,\text{ for all
}0\leq\mu<k,\,j=1,\ldots,p-1$. By the definition of wavelets (2.11),
$\psi_{j},\,j=1,\ldots,p-1$ are of the form,

$\psi_{j}(x)=\sum_{k=0}^{p^{N+1}-1}h_{j}\left(\frac{k}{p^{N+1}}\right)\phi\left(\frac{x}{p}-\frac{k}{p^{N+1}}\right),\quad
x\in\mathbb{Q}_{p}.$

Taking the Fourier transform on both sides,

$\widehat{\psi}_{j}(\xi)=H_{j}(T^{-1}\xi)\widehat{\phi}(T^{-1}\xi),\quad\xi\in\mathbb{Q}_{p},$
(4.3)

where

$H_{j}(\xi)=\frac{1}{p}\sum_{k=0}^{p^{N+1}-1}h_{j}\left(\frac{k}{p^{N+1}}\right)\chi\left(\frac{k}{p^{N+1}},\xi\right),\quad\xi\in\mathbb{Q}_{p},\,j=1,\ldots,p-1.$
(4.4)

Thus, $D^{\mu}\widehat{\psi_{j}}(0)=0,\text{ for all
}0\leq\mu<k,\,j=1,\ldots,p-1$ if and only if

$D^{\mu}H_{j}(0)=0,\text{ for all }0\leq\mu<k,\,j=1,\ldots,p-1.$

That is, if and only if

$\sum_{k=0}^{p^{N+1}-1}h_{j}\left(\frac{k}{p^{N+1}}\right)\left\lvert\frac{k}{p^{N+1}}\right\rvert_{p}^{\mu}=0,\text{
for all }0\leq\mu<k,\,j=1,\ldots,p-1.$

i.e., if and only if each $h_{j},\,j=1,\ldots,p-1$, has $k$ discrete
$p$-vanishing moments. ∎

######  Theorem 4.3.

Let $\phi\in L_{2}(\mathbb{Q}_{p})$ be a scaling function that generates a MRA
and $\text{supp}(\phi)\subset B_{0}(0)$. Then the corresponding refinement
mask of $\phi$ is of the form $\left\\{h\left(\frac{k}{p}\right)\right\\}$
with $h\left(\frac{k}{p}\right)=1,\text{ for all }k=0,1,\ldots,p-1$. Moreover,
$\phi$ has accuracy 1.

###### Proof.

Suppose that the scaling function $\phi$ that generates an MRA is given by

$\phi(x)=\sum_{k=0}^{p-1}h\left(\frac{k}{p}\right)\phi\left(\frac{x}{p}-\frac{k}{p}\right).$

Then by (2.10),

$H(\xi)=\frac{1}{p}\sum_{k=0}^{p-1}h\left(\frac{k}{p}\right)\chi\left(\frac{k}{p},\xi\right).$

Now by theorem 2.2, $H(0)=1$ and $H(k)=0$ for all $k=1,\ldots,p-1$. Also, we
have $\sum_{k=0}^{p-1}h\left(\frac{k}{p}\right)=p$. Thus we have a system of
$p$ equations $Uh=B$, where

$U=\frac{1}{\sqrt{p}}\begin{bmatrix}1&1&1&\cdots&1\\\
1&\chi\left(\frac{1}{p},1\right)&\chi\left(\frac{2}{p},1\right)&\cdots&\chi\left(\frac{p-1}{p},1\right)\\\
1&\chi\left(\frac{1}{p},2\right)&\chi\left(\frac{2}{p},2\right)&\cdots&\chi\left(\frac{p-1}{p},2\right)\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&\chi\left(\frac{1}{p},p-1\right)&\chi\left(\frac{2}{p},p-1\right)&\cdots&\chi\left(\frac{p-1}{p},p-1\right)\\\
\end{bmatrix},$ $h=\begin{bmatrix}h(0)\\\ h\left(\frac{1}{p}\right)\\\
h\left(\frac{2}{p}\right)\\\ \vdots\\\
h\left(\frac{p-1}{p}\right)\end{bmatrix}\text{ and
}B=\begin{bmatrix}\sqrt{p}\\\ 0\\\ 0\\\ \vdots\\\ 0\end{bmatrix}$

Since $\chi\left(\frac{k}{p},l\right),\,k,l=0,1,\ldots,p-1$ are the
$p^{\text{th}}$ roots of unity, the matrix $U$ is unitary. Also, $U$ is
symmetric. Hence $U^{-1}=\bar{U}$. Then $Uh=B$ has a solution $h=U^{-1}B$.
That is,

$\displaystyle h$ $\displaystyle=\begin{bmatrix}h(0)\\\
h\left(\frac{1}{p}\right)\\\ h\left(\frac{2}{p}\right)\\\ \vdots\\\
h\left(\frac{p-1}{p}\right)\end{bmatrix}$
$\displaystyle=\frac{1}{\sqrt{p}}\begin{bmatrix}1&1&1&\cdots&1\\\
1&\overline{\chi\left(\frac{1}{p},1\right)}&\overline{\chi\left(\frac{2}{p},1\right)}&\cdots&\overline{\chi\left(\frac{p-1}{p},1\right)}\\\
1&\overline{\chi\left(\frac{1}{p},2\right)}&\overline{\chi\left(\frac{2}{p},2\right)}&\cdots&\overline{\chi\left(\frac{p-1}{p},2\right)}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&\overline{\chi\left(\frac{1}{p},p-1\right)}&\overline{\chi\left(\frac{2}{p},p-1\right)}&\cdots&\overline{\chi\left(\frac{p-1}{p},p-1\right)}\\\
\end{bmatrix}\begin{bmatrix}\sqrt{p}\\\ 0\\\ 0\\\ \vdots\\\ 0\end{bmatrix}$

Hence $h\left(\frac{k}{p}\right)=1$, for all $k=0,1,\ldots,p-1$.

That is, $H(\xi)=\frac{1}{p}\sum_{k=0}^{p-1}\chi(\frac{k}{p},\xi)$. Since
$\sum_{k=0}^{p-1}\chi(\frac{k}{p},\xi)=0,\text{ for all }\xi=1,2,\ldots,p-1$,
we have $H(j)=0,\text{ for all }\xi=1,2,\ldots,p-1$ and for $\mu>1$,

$D^{\mu}H(\xi)=\frac{1}{p}\sum_{k=0}^{p-1}\left\lvert\frac{k}{p}\right\rvert_{p}^{\mu}\chi(\frac{k}{p},\xi)=p^{\mu-1}\sum_{k=1}^{p-1}\chi(\frac{k}{p},\xi).$

But $\sum_{k=1}^{p-1}\chi(\frac{k}{p},\xi)\neq 0,\text{ for all
}\xi=1,2,\ldots,p-1$,$D^{\mu}H(T^{-1}\omega)\neq 0$, for all
$\omega\in\\{\frac{1}{p},\frac{2}{p},\ldots,\frac{p-1}{p}\\},\,\mu\geq 0$.
Thus by theorem 2.5, the accuracy of $\phi$ is 1. ∎

######  Remark 4.1.

This scaling function $\phi$ is called the Haar scaling function.

######  Theorem 4.4.

Let $\phi$ be a Haar scaling function and $\psi_{j},\,j=1,\ldots,p-1$ be the
corresponding wavelets on $\mathbb{Q}_{p}$. Then $\psi_{j}$ has only $1$
$p$-vanishing moment for each $j=1,\ldots,p-1$.

###### Proof.

Here
$\psi_{j}(x)=\sum_{k=0}^{p-1}h_{j}\left(\frac{k}{p}\right)\phi\left(\frac{x}{p}-\frac{k}{p}\right)$,
where $h_{j}$ are chosen so that the matrix $U$ in (2.13) is unitary. That is
$h_{j}$ satisfies the condition

$\sum_{k=0}^{p-1}h\left(\frac{k}{p}\right)\overline{h_{j}\left(\frac{k}{p}\right)}=0,\,\text{
for all }j=1,\ldots,p-1.$ (4.5)

Since $\phi$ is the Haar scaling function, $h\left(\frac{k}{p}\right)=1$, for
all $k=0,1,\ldots,p-1$. Thus from (4.5), we can conclude that

$\sum_{k=0}^{p-1}h_{j}\left(\frac{k}{p}\right)=0,\,\forall j=1,\ldots,p-1.$

Suppose that $\psi_{j},\,j=1,\ldots,p-1$ have $k$ $p$-vanishing moments for
$k>1$. Then
$\sum_{k=0}^{p-1}h_{j}\left(\frac{k}{p}\right)\left\lvert\frac{k}{p}\right\rvert_{p}^{\mu}=0,\,0\leq\mu<k$.
That is,

$\sum_{k=0}^{p-1}h_{j}\left(\frac{k}{p}\right)=0\text{ and
}p^{\mu}\sum_{k=1}^{p-1}h_{j}\left(\frac{k}{p}\right)=0,\,\forall
j=1,\ldots,p-1,\,0<\mu<k,$

which is possible only when $h_{j}(0)=0$, for all $j=1,\ldots,p-1$.

Since the matrix $U$ in (2.13) is unitary, its rows are orthonormal. That is
$\lvert h(0)\rvert^{2}+\sum_{j=1}^{p-1}\lvert h_{j}(0)\rvert^{2}=p$. This is a
contradiction, since $h_{j}(0)=0$, for all $j=1,\ldots,p-1$.

Hence $\psi_{j}$ has only $1$ $p$-vanishing moment for each $j=1,\ldots,p-1$.
∎

######  Remark 4.2.

  1. 1.

If $\phi$ be an orthogonal scaling function that generates a MRA and
$\text{supp}(\phi)\subset B_{N}(0),\,N>0$. Then, by theorem 2.2, $\phi$
doesn’t satisfies the sum rules and $\phi$ has no accuracy.

  2. 2.

Let $\phi$ be a scaling function that generates a MRA and
$\text{supp}(\phi)\subset B_{0}(0)$. By theorems 4.3 and 4.4, if $\phi$ has
accuracy 1 then the corresponding wavelet has 1 $p$-vanishing moment. But this
cannot be true for scaling functions having support $\subset B_{N}(0),\,N>0$.

######  Example 4.1.

Let $p=3,\,N=1$. Set $H(k)=0$ if $k$ is not divisible by $3$, and
$H(0)=1,\,H(3)=H(6)=-1$. Then

$\widehat{\phi}(\xi)=\begin{cases}1,&\lvert\xi\rvert_{3}\leq\frac{1}{3},\\\
-1,&\lvert\xi-1\rvert_{3}\leq\frac{1}{3},\\\
-1,&\lvert\xi-2\rvert_{3}\leq\frac{1}{3},\\\ 0,&\lvert\xi\rvert_{3}\geq
3.\end{cases}$

Thus, the corresponding refinement equation is
$\phi(x)=\sum_{k=0}^{8}h(k/9)\phi(x/3-k/9)$ where $h(0)=h(3/9)=h(6/9)=-1/3$,
$h(1/9)=h(2/9)=h(4/9)=h(5/9)=h(7/9)=h(8/9)=2/3$. That is

$\phi(x)=\begin{cases}\frac{-1}{3},&\lvert x\rvert_{3}\leq 1,\\\
\frac{2}{3},&\lvert x-\frac{1}{3}\rvert_{3}\leq 1,\\\ \frac{2}{3},&\lvert
x-\frac{2}{3}\rvert_{3}\leq 1,\\\ 0,&\lvert\xi\rvert_{3}\geq 9.\end{cases}$
(4.6)

The corresponding wavelet functions are

$\psi_{1}=\sqrt{\frac{3}{2}}\left(\phi\left(\frac{x}{3}\right)-\phi\left(\frac{x}{3}-\frac{1}{3}\right)\right)$

and

$\psi_{2}=\frac{1}{\sqrt{2}}\left(\phi\left(\frac{x}{3}\right)+\phi\left(\frac{x}{3}-\frac{1}{3}\right)-2\phi\left(\frac{x}{3}-\frac{2}{3}\right)\right).$

That is

$\psi_{1}(x)=\begin{cases}-\sqrt{\frac{3}{2}},&\lvert
x\rvert_{3}\leq\frac{1}{3},\\\ \sqrt{\frac{3}{2}},&\lvert
x-1\rvert_{3}\leq\frac{1}{3},\\\ -0,&\lvert x-2\rvert_{3}\leq\frac{1}{3},\\\
0,&\lvert x\rvert_{3}\geq 3\\\ \end{cases}\text{ and
}\psi_{2}(x)=\begin{cases}-\frac{1}{\sqrt{2}},&\lvert
x\rvert_{3}\leq\frac{1}{3},\\\ -\frac{1}{\sqrt{2}},&\lvert
x-1\rvert_{3}\leq\frac{1}{3},\\\ \sqrt{2},&\lvert
x-2\rvert_{3}\leq\frac{1}{3},\\\ 0,&\lvert x\rvert_{3}\geq 3\\\ \end{cases}.$

Here $\phi$ has no accuracy. But $\psi_{1}$ and $\psi_{2}$ have $1$
$p$-vanishing moment.

##  5 $p$-Vanishing Moments of Haar-type Wavelets

In contrast to the real case, the wavelet basis generated by the Haar MRA in
the $p$-adic case is not unique. In [4], the authors provides an explicit
description about the family of wavelet functions generated by the Haar MRA in
$L_{2}(\mathbb{Q}_{p})$.

######  Theorem 5.1.

[4] Let $\phi=1_{B_{0}(0)}$ be the indicator function of $B_{0}(0)$ and

$\psi_{\nu}^{(0)}(x)=\sum_{r=0}^{p-1}e^{2\pi i\nu
r/p}\phi\left(\frac{x}{p}-\frac{r}{p}\right),\,x\in\mathbb{Q}_{p},\,\nu=1,\ldots,p-1.$

Then the set of all compactly supported wavelet functions on $\mathbb{Q}_{p}$
are given by

$\psi_{j}(x)=\sum_{\nu=1}^{p-1}\sum_{k=0}^{p^{s}-1}\alpha_{\nu,k}^{j}\psi_{\nu}^{(0)}\left(x-\frac{k}{p^{s}}\right),\,x\in\mathbb{Q}_{p},\,j=1,\ldots,p-1,$
(5.1)

where $s=0,1,2,\ldots$, and

$\alpha_{\nu,k}^{j}=\begin{cases}-p^{-s}\sum_{m=0}^{p^{s}-1}e^{-2\pi
i\frac{\frac{-\nu}{p}+m}{p^{s}}k}\sigma_{jm}z_{jj},&\text{ if }j=\nu,\\\
p^{-2s}\sum_{m=0}^{p^{s}-1}\sum_{n=0}^{p^{s}-1}e^{-2\pi
i\frac{\frac{-\nu}{p}+m}{p^{s}}k}\left(\frac{1-e^{2\pi
i\frac{j-\nu}{p}}}{e^{2\pi
i\frac{\frac{j-\nu}{p}+m-n}{p^{s}}}-1}\right)\sigma_{\nu m}z_{\nu j},&\text{
if }j\neq\nu,\end{cases}$ (5.2)

$\lvert\sigma_{jm}\rvert=1$ and $z_{j\nu}$ are entries of a $(p-1)\times(p-1)$
unitary matrix $Z$.

######  Example 5.1.

Let $s=2$ and $p=2$. Then by (5.1), the wavelet function is given by

$\psi_{1}(x)=\sum_{k=0}^{3}\alpha_{1,k}^{1}\psi_{1}^{(0)}\left(x-\frac{k}{4}\right),\,x\in\mathbb{Q}_{p}.$

That is

$\begin{split}\psi_{1}(x)=\,&\alpha_{1,0}^{1}\left(\phi\left(\frac{x}{2}\right)-\phi\left(\frac{x}{2}-\frac{4}{8}\right)\right)+\alpha_{1,1}^{1}\left(\phi\left(\frac{x}{2}-\frac{1}{8}\right)-\phi\left(\frac{x}{2}-\frac{5}{8}\right)\right)\\\
&+\alpha_{1,2}^{1}\left(\phi\left(\frac{x}{2}-\frac{2}{8}\right)-\phi\left(\frac{x}{2}-\frac{6}{8}\right)\right)+\alpha_{1,3}^{1}\left(\phi\left(\frac{x}{2}-\frac{3}{8}\right)-\phi\left(\frac{x}{2}-\frac{7}{8}\right)\right).\end{split}$

Here
$h_{1}(0)=\alpha_{1,0}^{1},\,h_{1}(1/8)=\alpha_{1,1}^{1},\,h_{1}(2/8)=\alpha_{1,2}^{1},\,h_{1}(3/8)=\alpha_{1,3}^{1},\,h_{1}(4/8)=-\alpha_{1,0}^{1},\,h_{1}(5/8)=-\alpha_{1,1}^{1},\,h_{1}(6/8)=-\alpha_{1,2}^{1},\,h_{1}(7/8)=-\alpha_{1,3}^{1}$.

Then we have $\sum_{k=0}^{7}h_{1}(k/8)=0$. That is $\psi_{1}$ has $1$
$p$-vanishing moment. Also,

$\displaystyle\sum_{k=0}^{7}\left\lvert\frac{k}{8}\right\rvert_{2}^{\mu}h_{1}\left(\frac{k}{8}\right)=$
$\displaystyle
2^{3\mu}\left(h_{1}\left(\frac{1}{8}\right)+h_{1}\left(\frac{3}{8}\right)+h_{1}\left(\frac{5}{8}\right)+h_{1}\left(\frac{7}{8}\right)\right)$
$\displaystyle+2^{2\mu}\left(h_{1}\left(\frac{2}{8}\right)+h_{1}\left(\frac{6}{8}\right)\right)+2^{\mu}h_{1}\left(\frac{4}{8}\right)$
$\displaystyle=$ $\displaystyle
8^{\mu}(\alpha_{1,1}^{1}+\alpha_{1,3}^{1}-\alpha_{1,1}^{1}-\alpha_{1,3}^{1})+4^{\mu}(\alpha_{1,2}^{1}-\alpha_{1,2}^{1})-2^{\mu}\alpha_{1,0}^{1}$
$\displaystyle=$ $\displaystyle-2^{\mu}\alpha_{1,0}^{1}.$

Thus

$\sum_{k=0}^{7}\left\lvert\frac{k}{8}\right\rvert_{2}^{\mu}h_{1}\left(\frac{k}{8}\right)=0\Leftrightarrow
h_{1}(0)=0,\,\mu>0.$

That is $\psi_{1}$ has infinite number of $p$-vanishing moments if and only if
$h_{1}(0)=0$.

######  Example 5.2.

Let $s=1$ and $p=3$. Then by (5.1), the wavelet function is given by

$\psi_{j}(x)=\sum_{\nu=1}^{2}\sum_{k=0}^{2}\alpha_{\nu,k}^{j}\psi_{\nu}^{(0)}\left(x-\frac{k}{3}\right),\,x\in\mathbb{Q}_{p},$

where

$\psi_{1}^{(0)}(x)=\phi\left(\frac{x}{3}\right)+\frac{-1+i\sqrt{3}}{2}\phi\left(\frac{x}{3}-\frac{1}{3}\right)+\frac{-1-i\sqrt{3}}{2}\phi\left(\frac{x}{3}-\frac{2}{3}\right),$
$\psi_{2}^{(0)}(x)=\phi\left(\frac{x}{3}\right)+\frac{-1-i\sqrt{3}}{2}\phi\left(\frac{x}{3}-\frac{1}{3}\right)+\frac{-1+i\sqrt{3}}{2}\phi\left(\frac{x}{3}-\frac{2}{3}\right).$

That is,

$h_{j}(0)=(\alpha_{1,0}^{j}+\alpha_{2,0}^{j}),\,h_{j}\left(\frac{1}{9}\right)=(\alpha_{1,1}^{j}+\alpha_{2,1}^{j}),\,h_{j}\left(\frac{2}{9}\right)=(\alpha_{1,2}^{j}+\alpha_{2,2}^{j}),$
$h_{j}\left(\frac{3}{9}\right)=\left(\alpha_{1,0}^{j}\frac{-1+i\sqrt{3}}{2}+\alpha_{2,0}^{j}\frac{-1-i\sqrt{3}}{2}\right),$
$h_{j}\left(\frac{4}{9}\right)=\left(\alpha_{1,1}^{j}\frac{-1+i\sqrt{3}}{2}+\alpha_{2,1}^{j}\frac{-1-i\sqrt{3}}{2}\right),$
$h_{j}\left(\frac{5}{9}\right)=\left(\alpha_{1,2}^{j}\frac{-1+i\sqrt{3}}{2}+\alpha_{2,2}^{j}\frac{-1-i\sqrt{3}}{2}\right),$
$h_{j}\left(\frac{6}{9}\right)=\left(\alpha_{1,0}^{j}\frac{-1-i\sqrt{3}}{2}+\alpha_{2,0}^{j}\frac{-1+i\sqrt{3}}{2}\right),$
$h_{j}\left(\frac{7}{9}\right)=\left(\alpha_{1,1}^{j}\frac{-1-i\sqrt{3}}{2}+\alpha_{2,1}^{j}\frac{-1+i\sqrt{3}}{2}\right),$
$h_{j}\left(\frac{8}{9}\right)=\left(\alpha_{1,2}^{j}\frac{-1-i\sqrt{3}}{2}+\alpha_{2,2}^{j}\frac{-1+i\sqrt{3}}{2}\right).$

Then we have $\sum_{k=0}^{8}h_{j}(k/9)=0$. That is $\psi_{j}$ has $1$
$p$-vanishing moment. Also,

$\displaystyle\sum_{k=0}^{8}\left\lvert\frac{k}{9}\right\rvert_{3}^{\mu}$
$\displaystyle h_{j}\left(\frac{k}{9}\right)$
$\displaystyle=3^{2\mu}\left(h_{j}\left(\frac{1}{9}\right)+h_{j}\left(\frac{2}{9}\right)+h_{j}\left(\frac{4}{9}\right)+h_{j}\left(\frac{5}{9}\right)+h_{j}\left(\frac{7}{9}\right)+h_{j}\left(\frac{8}{9}\right)\right)$
$\displaystyle+3^{\mu}\left(h_{j}\left(\frac{3}{9}\right)+h_{j}\left(\frac{6}{9}\right)\right)$
$\displaystyle=$ $\displaystyle-3^{\mu}(\alpha_{1,0}^{j}+\alpha_{2,0}^{j}).$

Thus

$\sum_{k=0}^{8}\left\lvert\frac{k}{9}\right\rvert_{3}^{\mu}h_{j}\left(\frac{k}{9}\right)=0\Leftrightarrow
h_{j}(0)=0,\,\mu>0,\,j=1,2.$

That is $\psi_{j}$ has infinite number of $p$-vanishing moments if and only if
$h_{j}(0)=0$.

######  Remark 5.1.

We can rewrite (5.1) as

$\displaystyle\psi_{j}(x)$
$\displaystyle=\sum_{\nu=1}^{p-1}\sum_{k=0}^{p^{s}-1}\alpha_{\nu,k}^{j}\psi_{\nu}^{(0)}\left(x-\frac{k}{p^{s}}\right)$
$\displaystyle=\sum_{\nu=1}^{p-1}\sum_{k=0}^{p^{s}-1}\alpha_{\nu,k}^{j}\sum_{r=0}^{p-1}e^{2\pi
i\nu r/p}\phi\left(\frac{x}{p}-\frac{k}{p^{s+1}}-\frac{r}{p}\right)$
$\displaystyle=\sum_{l=0}^{p^{s+1}-1}h_{j}\left(\frac{l}{p^{s+1}}\right)\phi\left(\frac{x}{p}-\frac{l}{p^{s+1}}\right),$

where

$h_{j}\left(\frac{l}{p^{s+1}}\right)=h_{j}\left(\frac{k+rp^{s}}{p^{s+1}}\right)=\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}e^{2\pi
i\nu r/p},$

for $r=0,1,\ldots,p-1,\,k=0,1,\ldots,p^{s}-1$.

The following theorem gives a characterization for $p$-vanishing moments of
Haar-type wavelets.

######  Theorem 5.2.

Let $\psi_{j},\,j=1,\ldots,p-1$ be the wavelet finctions described in theorem
5.1. then $\psi_{j}$ always have $1$ $p$-vanishing moment. Moreover,
$\psi_{j}$ has infinite number of $p$-vanishing moments if and only if
$h_{j}(0)=0,\,j=1,\ldots,p-1$.

###### Proof.

We have

$\psi_{j}(x)=\sum_{l=0}^{p^{s+1}-1}h_{j}\left(\frac{l}{p^{s+1}}\right)\phi\left(\frac{x}{p}-\frac{l}{p^{s+1}}\right),$

where

$h_{j}\left(\frac{l}{p^{s+1}}\right)=h_{j}\left(\frac{k+rp^{s}}{p^{s+1}}\right)=\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}e^{2\pi
i\nu r/p},$

for $r=0,1,\ldots,p-1,\,k=0,1,\ldots,p^{s}-1$. Then

$\displaystyle\sum_{l=0}^{p^{s+1}-1}h_{j}\left(\frac{l}{p^{s+1}}\right)$
$\displaystyle=\sum_{k=0}^{p^{s}-1}\sum_{r=0}^{p-1}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}e^{2\pi
i\nu r/p}$
$\displaystyle=\sum_{k=0}^{p^{s}-1}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}\left(\sum_{r=0}^{p-1}e^{2\pi
i\nu r/p}\right)$
$\displaystyle=\sum_{k=0}^{p^{s}-1}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}\times
0$ $\displaystyle=0,$

for $j=1,\ldots,p-1$. That is $\psi_{j}$ always have $1$ $p$-vanishing moment.

We can write any $\frac{k}{p^{s+1}}$, for $k=0,1,2,\ldots,p^{s}-1$ in the form

$\frac{k}{p^{s+1}}=\frac{k_{0}}{p^{s+1}}+\frac{k_{1}}{p^{s}}+\cdots+\frac{k_{s-1}}{p^{2}},\,k_{0},k_{1},\ldots,k_{s-1}\in\\{0,1,\ldots,p-1\\}.$

Let $M_{\gamma}$ for $\gamma=2,\ldots,s$ be the set given by,

$\begin{split}M_{\gamma}=\\{k=k_{0}+k_{1}p&+k_{2}p^{2}+\cdots
k_{s-1}p^{s-1}:k_{0}=k_{1}=\cdots=k_{s-\gamma}=0,\\\
&k_{s-\gamma+1},k_{s-\gamma+2},\ldots,k_{s-1}=0,1,\ldots,p-1,\,k_{s-\gamma+1}\neq
0\\},\end{split}$

and

$M_{s+1}=\\{k=k_{0}+k_{1}p+k_{2}p^{2}+\cdots
k_{s-1}p^{s-1}:k_{0},k_{1},\ldots,k_{s-1}=0,1,\ldots,p-1,\,k_{0}\neq 0\\}.$

Then

$\displaystyle\sum_{l=1}^{p^{s+1}-1}\left\lvert\frac{l}{p^{s+1}}\right\rvert_{p}^{\mu}h_{j}\left(\frac{l}{p^{s+1}}\right)=$
$\displaystyle\sum_{k=0}^{p^{s}-1}\sum_{r=0}^{p-1}\left\lvert\frac{k+rp^{s}}{p^{s+1}}\right\rvert_{p}^{\mu}h_{j}\left(\frac{k+rp^{s}}{p^{s+1}}\right)$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}p^{\mu}\sum_{\nu=1}^{p-1}\alpha_{\nu,0}^{j}e^{2\pi
i\nu r/p}$
$\displaystyle+\sum_{k=1}^{p^{s}-1}\sum_{r=0}^{p-1}\sum_{\nu=1}^{p-1}\left\lvert\frac{k+rp^{s}}{p^{s+1}}\right\rvert_{p}^{\mu}\alpha_{\nu,k}^{j}e^{2\pi
i\nu r/p}$ $\displaystyle=$
$\displaystyle\,p^{\mu}\sum_{\nu=1}^{p-1}\alpha_{\nu,0}^{j}\left(\sum_{r=1}^{p-1}e^{2\pi
i\nu r/p}\right)$ $\displaystyle+p^{2\mu}\sum_{k\in
M_{2}}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}\left(\sum_{r=0}^{p-1}e^{2\pi i\nu
r/p}\right)$ $\displaystyle+p^{3\mu}\sum_{k\in
M_{3}}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}\left(\sum_{r=0}^{p-1}e^{2\pi i\nu
r/p}\right)$ $\displaystyle+\cdots+p^{(s+1)\mu}\sum_{k\in
M_{s+1}}\sum_{\nu=1}^{p-1}\alpha_{\nu,k}^{j}\left(\sum_{r=0}^{p-1}e^{2\pi i\nu
r/p}\right)$ $\displaystyle=$
$\displaystyle\,-p^{\mu}\sum_{\nu=1}^{p-1}\alpha_{\nu,0}^{j}=-h_{j}(0).$

Thus, $\psi_{j}$ has infinite number of $p$-vanishing moments if and only if
$h_{j}(0)=0,\,j=1,\ldots,p-1$. ∎

##  6 $p$-Vanishing Moments of non-Haar type Wavelets

In [6], wavelet bases different from those described above were constructed;
these bases were called non-Haar bases.

######  Theorem 6.1.

[6] Let

$\begin{split}J_{p,m}=\\{s=p^{-m}(s_{0}+s_{1}p+s_{2}p^{2}+\cdots&+s_{m-1}p^{m-1}):s_{j}=0,1,\ldots,p-1;\\\
&j=0,1,\ldots,m-1;\,s_{0}\neq 0\\},\end{split}$ (6.1)

where $m\geq 1$ is a fixed positive integer. Consider the set of
$(p-1)p^{m-1}$ functions

$\theta_{s}^{(m)}(x)=\chi(x,s)1_{B_{0}(0)}(x),\quad s\in
J_{p,m},\,x\in\mathbb{Q}_{p},$ (6.2)

and the family of functions generated by their dilations and translations:

$\theta_{s;ja}^{(m)}(x)=p^{-j/2}\theta_{s}^{(m)}(p^{j}x-a),\quad s\in
J_{p,m},\,x\in\mathbb{Q}_{p},\,j\in\mathbb{Z},\,a\in I_{p},$ (6.3)

where $1_{B_{0}(0)}$ is the indicator function of $B_{0}(0)$. Then the
functions (6.3) form an orthonormal $p$-adic wavelet basis in
$L_{2}(\mathbb{Q}_{p})$ which is non-Haar type for $m\geq 2$.

######  Theorem 6.2.

[6] For any fixed $\nu=1,2,\ldots$, the functions

$\psi_{s}^{(m),\nu}(x)=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\theta_{s}^{(m)}\left(x-\frac{k}{p^{\nu}}\right),\quad
s\in J_{p,m},$ (6.4)

are wavelet functions if and only if

$\alpha_{s,k}=p^{-\nu}\sum_{r=0}^{p^{\nu}-1}\gamma_{s,r}e^{-2\pi
i\frac{-s+r}{p^{\nu}}k},$ (6.5)

where
$\gamma_{s,k}\in\mathbb{C},\,\lvert\gamma_{s,k}\rvert=1,\,k=0,1,\ldots,p^{\nu}-1,\,s\in
J_{p,m}$.

The following two theorems gives the connection between $p$-vanishing moment
of non-Haar type wavelet functions and the approximation order of the
indicator function of the compact open subgroup $B_{0}(0)$ of
$\mathbb{Q}_{p}$.

######  Theorem 6.3.

Let $\phi=1_{B_{0}(0)}$ and $\theta_{s}^{(m)}$ be defined by (6.3). If $\phi$
has approximation order $k$, then $\theta_{s}^{(m)}$ has $k$ $p$-vanishing
moments.

###### Proof.

We have for a fixed integer $m>2$,

$\theta_{s}^{(m)}(x)=\chi(x,s)1_{B_{0}(0)}(x),\quad s\in
J_{p,m},\,x\in\mathbb{Q}_{p}.$

That is, $\theta_{s}^{(m)}(x)=\chi(x,s)\phi(x)$. Now,

$\displaystyle\widehat{\theta_{s}^{(m)}}(\xi)$
$\displaystyle=\int_{\mathbb{Q}_{p}}\chi(x,s)\phi(x)\chi(x,\xi)dx$
$\displaystyle=\int_{\mathbb{Q}_{p}}\phi(x)\chi(x,\xi+s)dx$
$\displaystyle=\widehat{\phi}(\xi+s).$

Hence $D^{\mu}\widehat{\theta_{s}^{(m)}}(0)=D^{\mu}\widehat{\phi}(s)$.

From theorem 2.4, $\phi$ has $k$ approximation order if and only if
$D^{\mu}\widehat{\phi}(\alpha)=0$ for all $\alpha\in
I_{p}\setminus\\{0\\},\,\lvert\mu\rvert\leq k-1$. Here
$I_{p}\setminus\\{0\\}=\bigcup_{m\geq 1}J_{p,m}$.

We have $\\{\theta_{s}^{(m)}\\}_{s\in J_{p,m}}$ has $k$ $p$-vanishing moments
if and only if $D^{\mu}\widehat{\theta_{s}^{(m)}}(0)=0$ for all $s\in
J_{p,m},\,0\leq\mu\leq k-1$. That is, if and only if
$D^{\mu}\widehat{\phi}(s)=0$ for all $s\in J_{p,m},\,0\leq\mu\leq k-1$.

Since $m$ is fixed, we can conclude that if $\phi$ has approximation order
$k$, then $\theta_{s}^{(m)}$ has $k$ $p$-vanishing moments. ∎

######  Theorem 6.4.

Let $\phi=1_{B_{0}(0)}$ and $\psi_{s}^{(m),\nu}$ be the wavelet functions
defined by (6.4). If $\phi$ has approximation order $k$, then
$\psi_{s}^{(m),\nu}$ has $k$ $p$-vanishing moments.

###### Proof.

For any fixed $\nu=1,2,\ldots$, let $\psi_{s}^{(m),\nu}$ be the wavelet
functions defined by (6.4),

$\psi_{s}^{(m),\nu}(x)=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\theta_{s}^{(m)}\left(x-\frac{k}{p^{\nu}}\right),\quad
s\in J_{p,m},$

with

$\alpha_{s,k}=p^{-\nu}\sum_{r=0}^{p^{\nu}-1}\gamma_{s,r}e^{-2\pi
i\frac{-s+r}{p^{\nu}}k},$

where
$\gamma_{s,k}\in\mathbb{C},\,\lvert\gamma_{s,k}\rvert=1,\,k=0,1,\ldots,p^{\nu}-1,\,s\in
J_{p,m}$. That is,

$\psi_{s}^{(m),\nu}(x)=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\chi\left(x-\frac{k}{p^{\nu}},s\right)\phi\left(x-\frac{k}{p^{\nu}}\right),\quad
s\in J_{p,m}.$

Now,

$\displaystyle\widehat{\psi_{s}^{(m),\nu}}(\xi)$
$\displaystyle=\int_{\mathbb{Q}_{p}}\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\chi\left(x-\frac{k}{p^{\nu}},s\right)\phi\left(x-\frac{k}{p^{\nu}}\right)\chi(x,\xi)dx$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\int_{\mathbb{Q}_{p}}\phi(t)\chi(t,s)\chi\left(t+\frac{k}{p^{\nu}},\xi\right)dt$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\chi\left(\frac{k}{p^{\nu}},\xi\right)\int_{\mathbb{Q}_{p}}\phi(t)\chi(t,s)\chi(t,\xi)dt$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\chi\left(\frac{k}{p^{\nu}},\xi\right)\int_{\mathbb{Q}_{p}}\phi(t)\overline{\chi(t,\xi+s)}dt$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\chi\left(\frac{k}{p^{\nu}},\xi\right)\widehat{\phi}(\xi+s).$

Let $f(\xi)=\chi\left(\frac{k}{p^{\nu}},\xi\right)$ and
$g(\xi)=\widehat{\phi}(\xi+s)$. Then applying the Leibniz formula for
diﬀerentiation,

$\displaystyle D^{\mu}\widehat{\psi_{s}^{(m),\nu}}(\xi)$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\sum_{\beta\leq\mu}{\mu\choose\beta}D^{\beta}f(\xi)D^{\mu-\beta}g(\xi)$
$\displaystyle=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\sum_{\beta\leq\mu}{\mu\choose\beta}p^{\nu\beta}\chi\left(\frac{k}{p^{\nu}},\xi\right)D^{\mu-\beta}\widehat{\phi}(\xi+s).$

That is,

$D^{\mu}\widehat{\psi_{s}^{(m),\nu}}(0)=\sum_{k=0}^{p^{\nu}-1}\alpha_{s,k}\sum_{\beta\leq\mu}{\mu\choose\beta}p^{\nu\beta}D^{\mu-\beta}\widehat{\phi}(s).$

From theorem 2.4, $\phi$ has $k$ approximation order if and only if
$D^{\mu}\widehat{\phi}(\alpha)=0$ for all $\alpha\in
I_{p}\setminus\\{0\\},\,\lvert\mu\rvert\leq k-1$. Here
$I_{p}\setminus\\{0\\}=\bigcup_{m\geq 1}J_{p,m}$.

We have $\\{\psi_{s}^{(m),\nu}\\}_{s\in J_{p,m}}$ has $k$ $p$-vanishing
moments if and only if $D^{\mu}\widehat{\psi_{s}^{(m),\nu}}(0)=0$ for all
$s\in J_{p,m},\,0\leq\mu\leq k-1$. That is, if and only if
$D^{\mu-\beta}\widehat{\phi}(s)=0$ for all $s\in J_{p,m},\,0\leq\mu\leq
k-1,\,\beta\leq\mu$.

Since $m$ is fixed, we can conclude that if $\phi$ has approximation order
$k$, then $\psi_{s}^{(m),\nu}$ has $k$ $p$-vanishing moments. ∎

######  Remark 6.1.

Let $\phi=1_{B_{0}(0)}$ and $\psi$ be a non-Haar type wavelet function. If
$\phi$ has approximation order $k$, then $\psi$ has $k$ $p$-vanishing moments.

##  7 $p$-vanishing moments of nonorthogonal wavelets

In [15], the authors developed a method for constructing MRA-based p-adic
wavelet systems that form Riesz bases in $L_{2}(\mathbb{Q}_{p})$. The explicit
construction is as follows:

For an integer $K\geq 0$, we set

$A_{K}=\\{\frac{ap^{j}-b}{p^{j+1}},\frac{p^{K+1}-b}{p^{K+1}}:\,j=1,\ldots,K,\,a,b=1,\ldots,p-1\\},$
$B_{K}=\\{0,\frac{p^{j}-b}{p^{j}}:\,j=1,\ldots,K,\,b=1,\ldots,p-1\\}$

It is easily seen that $\\#A_{K}=(p-1)\\#B_{K}$ and $\\#B_{K}=1+K(p-1)$ for
all $K$. Moreover, $A_{K},B_{K}\subset I_{p}$ and

$A_{K}\cap B_{K}=\emptyset,\quad A_{K}\cup
B_{K}=\bigcup_{j=0}^{p-1}\frac{1}{p}(j+B_{K}).$

Set $\chi_{p}(\xi)=e^{2\pi i\\{\xi\\}_{p}}$. Let us deﬁne trigonometric
polynomials $m_{0}=m_{0,K}$ and $n_{0}=n_{0,K}$ of degrees $(p-1)(1+(p-1)K)$
and $1+(p-1)K$, respectively, by

$m_{0,K}(\xi)=\frac{1}{p}\prod_{r\in
A_{K}}\left(\chi_{p}(\xi)-\chi_{p}(r)\right),\qquad
n_{0,K}(\xi)=\frac{1}{p}\prod_{r\in
B_{K}}\left(\chi_{p}(\xi)-\chi_{p}(r)\right).$ (7.1)

One can easily verify that $m_{0,K}(0)=1$ for all $K\geq 0$.

Given integers $M\geq 0$ and $N\geq 0$, we deﬁne the Fourier transform of
functions $\phi=\phi_{M,N}$ and $\psi=\psi_{M,N}$ by

$\widehat{\phi}_{M,N}(\xi)=\prod_{j=0}^{\infty}m_{0,K}(\frac{\xi}{p^{N-j}}),\qquad\xi\in\mathbb{Q}_{p},$
(7.2)
$\widehat{\psi}_{M,N}(\xi)=n_{0,K}(\frac{\xi}{p^{N}})\widehat{\phi}_{M,N}(p\xi),\qquad\xi\in\mathbb{Q}_{p}.$
(7.3)

where $K=M+N$.

######  Lemma 7.1.

[15] For all $M$ and $N$, the following statements hold:

  1. 1.

$\widehat{\phi}_{M,N}\in\mathcal{D}_{M}^{N}$ and
$\widehat{\psi}_{M,N}\in\mathcal{D}_{M+1}^{N}$,

  2. 2.

$\widehat{\phi}_{M,N}(\frac{l}{p^{M}})\neq 0$ if and only if $l\in
p^{K}B_{K},\,0\leq l\leq p^{K}-1$,

  3. 3.

$\widehat{\psi}_{M,N}(\frac{l}{p^{M+1}})\neq 0$ if and only if $l\in
p^{K+1}A_{K},\,0\leq l\leq p^{K+1}-1$.

Set

$\psi^{(\nu)}(x)=\psi(x-\nu+1),\,\nu=1,\ldots,p-1.$ (7.4)

######  Theorem 7.1.

[15] For integers $M,N\geq 0$, the function $\phi_{M,N}$ deﬁned by its Fourier
transform (7.2) generates an MRA if and only if $M\leq\frac{p^{N}-1}{p-1}-N$.
Moreover, in this case, $\psi^{(\nu)}_{M,N},\,\nu=1,\ldots,p-1$ is a set of
wavelet functions, and the corresponding wavelet system
$\\{\psi^{(\nu)}_{ja}(x):=p^{j/2}\psi^{(\nu)}(p^{-j}x-a):\nu=1,\ldots,r,\,a\in
I_{p},\,j\in\mathbb{Z}$ forms a Riesz basis for $L_{2}(\mathbb{Q}_{p})$ if and
only if $M=\frac{p^{N}-1}{p-1}-N$.

It is easy to see that if $(M,N)=(0,0)$ or $(M,N)=(0,1)$, then
$\phi_{M,N}(x)=\Omega(\lvert x\rvert_{p})$ and we obtain the Haar MRA. On the
contrary, for $N>1$ and $M=\frac{p^{N}-1}{p-1}-N$, the functions $\phi_{M,N}$
generate pairwise distinct MRAs and each of these scaling functions is not
orthogonal, which leads to nonorthogonal wavelet Riesz bases.

######  Example 7.1.

Let $p=2,\,N=2$ and $M=1$, then $K=3$ and

$A_{3}=\\{\frac{1}{4},\frac{3}{8},\frac{7}{16},\frac{15}{16}\\}\text{ and
}B_{3}=\\{0,\frac{1}{2},\frac{3}{4},\frac{7}{8}\\}.$

Then

$m_{0,K}(\xi)=\frac{1}{p}\left(\chi_{p}(\xi)-\chi_{p}(\frac{1}{4})\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{3}{8})\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{7}{16})\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{15}{16})\right),$

and

$n_{0,K}(\xi)=\frac{1}{p}\left(\chi_{p}(\xi)-1\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{1}{2})\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{3}{4})\right)\left(\chi_{p}(\xi)-\chi_{p}(\frac{7}{8})\right).$

Then the refinable function is given by (7.2) and the corresponding wavelets
function is $\psi^{(\nu)}_{M,N}$ where $\psi_{M,N}$ is given by (7.3).

The following theorem characterizes the $p$-vanishing moment of nonorthogonal
MRA wavelets.

######  Theorem 7.2.

Let $\psi^{(\nu)}_{M,N}$ be the wavelet functions defined by (7.4). Then
$\psi^{(\nu)}_{M,N}$ has $k$ $p$-vanishing moments if and only if
$D^{\mu}n_{0,K}(0)=0,\,\forall 0\leq\mu\leq k-1$.

###### Proof.

We have $\psi^{(\nu)}(x)=\psi(x-\nu+1)$. Then the Fourier transform of
$\psi^{(\nu)}_{M,N}$ is given by

$\displaystyle\widehat{\psi^{(\nu)}}_{M,n}(\xi)$
$\displaystyle=\int_{\mathbb{Q}_{p}}\psi^{(\nu)}_{M,N}(x)\chi(x,\xi)dx$
$\displaystyle=\int_{\mathbb{Q}_{p}}\psi_{M,N}(x-\nu+1)\chi(x,\xi)dx$
$\displaystyle=\int_{\mathbb{Q}_{p}}\psi_{M,N}(t)\chi(t+\nu-1,\xi)dx$
$\displaystyle=\chi(\nu-1,\xi)\int_{\mathbb{Q}_{p}}\psi_{M,N}(t)\chi(t,\xi)dx$
$\displaystyle=\chi(\nu-1,\xi)\widehat{\psi}_{M,N}(\xi).$

Then applying the Leibniz formula for diﬀerentiation,

$D^{\mu}\widehat{\psi}^{(\nu)}_{M,N}(\xi)=\sum_{\beta\leq\mu}{\mu\choose\beta}D^{\beta}\chi(\nu-1,\xi)D^{\mu-\beta}\widehat{\psi}_{M,N}(\xi).$

Also
$\widehat{\psi}_{M,N}(\xi)=n_{0,K}(\frac{\xi}{p^{N}})\widehat{\phi}_{M,N}(p\xi)$.
By the Leibniz formula for diﬀerentiation,

$D^{\mu}\widehat{\psi}_{M,N}(\xi)=\sum_{\beta\leq\mu}{\mu\choose\beta}D^{\beta}n_{0,K}(\frac{\xi}{p^{N}})D^{\mu-\beta}\widehat{\phi}_{M,N}(\xi).$

By Lemma 7.1 we have, $\phi_{M,N}(0)\neq 0$. Thus we can see that
$\psi^{(\nu)}_{M,N}$ has $k$ $p$-vanishing moments if and only if
$D^{\mu}n_{0,K}(0)=0,\,\forall 0\leq\mu\leq k-1$. ∎

##  8 Conclusion

The definitions of $p$-vanishing moments of compactly supported functions on
$\mathbb{Q}_{p}$ and discrete $p$-vanishing moments of finitely supported
sequences on $I_{p}$ are given in this work. The relationship between the
$p$-vanishing moments and discrete $p$-vanishing moments are established. The
$p$-vanishing moments of Haar-type and non-Haar type wavelet functions are
calculated. We proved the connection between $p$-vanishing moment of non-Haar
type wavelet functions and the approximation order of the indicator function
of the compact open subgroup $B_{0}(0)$ of $\mathbb{Q}_{p}$. Finally, we
characterized the $p$-vanishing moments of nonorthogonal wavelets.

## Acknowledgement

We are very grateful to the authors of the articles in the references.

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