Continuous wavelet transform on local fields

The main objective of this paper is to define the mother wavelet on local fields and study the continuous wavelet transform (CWT) and some of their basic properties. its inversion formula, the Parseval relation and associated convolution are also studied.

D = {x ∈ K : |x| ≤ 1} is called the ring of integers in K. It is the unique maximal compact subring of K. Define P = x ∈ K : |x| < 1. The set P is called the prime ideal in K. The prime ideal in K is the unique maximal ideal in D. It is principal and prime.
If K is a local field, then there is a nontrivial, unitary, continuous character χ on K + and K + is self dual.
χ is fixed character on K + that is trivial on D but is nontrivial on P −1 . It follows that χ is constant on cosets of D and that if y ∈ P k , then χ y (χ y (x)) = χ(xy) is constant on cosets of P −k Definition 1.1. The fourier transform of f ∈ L 1 (K) is denoted byf (ξ) and define by the [8]f and the inverse Fourier transform by Some important properties of the Fourier transform can prove easily : (ii) If f ∈ L 1 (K), thenf is uniformly continuous.
If the convolution of f and g is defined as The article is divided in four sections. In section 2. we proposed the definition of mother wavelet and define the continuous wavelet transform (CWT). In section 3. discus the some basic properties of CWT. In section 4. we prove the Plancherel , inversion formula and define the convolution associated with CWT.

Continuous Wavelet transform on local fields
Similar to L 2 (R) [1,3,5], we define the wavelet on local fields and define the continuous wavelet transform.

Definition 2.1. Admissible wavelet on local fields
The function ψ(x) ∈ L 2 (K) is said to be an admissible wavelet on local fields if ψ(x) satisfies the following admissibility condition: whereψ is the Fourier transform of ψ.
Remark 2.2. If |ψ(ξ)| is continuous near ξ = 0, then the existence of integral (2.1) guarantees that ,ψ(0) = 0. Since the Fourier transform of mother wavelet This means that the integral of mother wavelet is zero: 3. If ψ is a mother wavelet and φ ∈ L 1 (K), then the convolution function ψ * φ is a mother wavelet.
This completes the proof of the theorem

Definition 2.4. Continuous wavelet transform (CWT) on local fields
For ψ(x) ∈ L 2 (K) and a, b ∈ K, a = 0, we define the unitary linear operator: ψ is called mother wavelet and ψ a,b (x) are called daughter wavelets, where a is a dilation parameter, b is a translation parameter.
The Fourier transform of ψ a,b (x) is given bŷ whereψ is the Fourier transform of ψ.
The CWT on local fields of a function f ∈ L 2 (K) with respect to a mother wavelet ψ is defined by

Basic Properties of CWT on local fields
Before giving the fundamental properties of CWT, we list their basic properties.
Theorem 3.1. Let ψ and ϕ be to wavelets and f, g are two function belong to where η and ϑ are any two scalers.
(ii)Shift property where ς is any scalers.
(iii) Scaling property If σ = 0 any scaler. The CWT of the scaled function where P is a parity operator define by P f (x) = f (−x).
Proof. The proof is the straight forward application of CWT Theorem 3.2. Show that the continuous wavelet transform can also expressed as where the * is defined as Proof. From define of CWT we have where the λ is scaler .
Proof. From define of CWT we have where c ψ is given in (2.1).
Proof. By using perseval formula for Fourier we can write the wavelet transform as Now, by using above (4.2) and (4.3) we get Then we have where c ψ is given in (2.1).
Proof. Let h(x) ∈ L 2 (K) be any function, then by using above theorem, we have Hence the result follows.
Moreover the wavelet transform is isometry from L 2 (K) to L 2 (K × K) 4.1. Associated convolution for CWT on local fields. Using Pathak and Pathak techniques [5], we define the basic function D(x, y, z) , translation τ x and associated convolution # operators for CWT.
The basic function D(x, y, z) for (2.4) is define as where ψ, φ and χ are three wavelets satisfying certain conditions (2.1). Now, by using(4.6) we get, The translation τ x is defined as [5] (τ x h)(y) = h * (x, y) = K D(x, y, z)h(z)dz The associated convolution is defined as (h#g)(x) = K h * (x, y)g(y)dy