Vanishing Moments of Wavelets on $p$-adic Fields

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.

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 $\Gamma_{p}(\alpha)=\frac{1-p^{\alpha-1}}{1-p^{-\alpha}}$.

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 $A^{*}=A$.

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.

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].

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.

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 $\mathbb{R}$.

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

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 $\mathbb{R}^{d}$.

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 $\epsilon\in\Omega\setminus\{0\}$.

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