The Necessary and Sufficient Conditions for Wavelet Frames in Sobolev Space Over Local Fields

Let F be an algebraic field and topological space with the topological properties of non-discrete, complete, locally compact and totally disconnected.Let F and F are the multiplicative and additive groups of F respectively. Now we define Haar measure dξ for F . Then for β 6= 0(β ∈ F), d(βξ) is also a Haar measure. Let d(βξ) = |β|dξ and we say |β| is the absolute value or valuation of β.Let |0| = 0. The valuation or absolute value has following properties : (a) |ξ| ≥ 0 and |ξ| = 0 if and only if ξ = 0; (b) |ξη| = |ξ||η| ; (c) |ξ + η| ≤ max(|ξ|, |η|). The last property is called ultrametric inequality. The set D = {ξ ∈ F : |ξ| ≤ 1} is the ring of integers in F and is the unique maximal compact subring of F. Define P = {ξ ∈ F : |ξ| < 1} The set P is called the prime ideal in F. The prime ideal in F is the unique maximal ideal in D. Then set P is principal and prime. Let A be a measurable subset of F and |A| = ∫


Introduction
Let F be an algebraic field and topological space with the topological properties of non-discrete, complete, locally compact and totally disconnected.Let F * and F + are the multiplicative and additive groups of F respectively. Now we define Haar measure dξ for F + . Then for β = 0(β ∈ F), d(βξ) is also a Haar measure. Let d(βξ) = |β|dξ and we say |β| is the absolute value or valuation of β.Let |0| = 0. The valuation or absolute value has following properties : (a) |ξ| ≥ 0 and |ξ| = 0 if and only if ξ = 0; (b) |ξη| = |ξ||η| ; (c) |ξ + η| ≤ max(|ξ|, |η|). The last property is called ultrametric inequality. The set D = {ξ ∈ F : |ξ| ≤ 1} is the ring of integers in F and is the unique maximal compact subring of F. Define P = {ξ ∈ F : |ξ| < 1} The set P is called the prime ideal in F. The prime ideal in F is the unique maximal ideal in D. Then set P is principal and prime. Let A be a measurable subset of F and |A| = F ζ A (ξ)dξ, where ζ A is the characteristic function of A and dξ is the Haar measure of F normalized so that |D| = 1. Then we observe that |P| = q −1 and |p| = q −1 . Therefore for ξ = 0(ξ ∈ F), |ξ| = q k for some k ∈ Z. Define P k = p k D = {ξ ∈ F : |ξ| ≤ q −k , k ∈ Z} . These are called fractional ideals. Each P k is a subgroup of F + .It is to see that P k is open as well as compact. If F is a local field, then there is a nontrivial, unitary, continuous character χ on F + . It can be proved that F + is self dual. Let χ be a fixed character on F + that is trivial on D but is nontrivial on P −1 . We will define fixed character χ for a local field of positive characteristic by χ η (ξ) = χ(ηξ), for ξ, η ∈ F. Definition 1.1. If g ∈ L 1 (F), then the Fourier transform of g is the functionĝ defined byĝ The Fourier transform in L p (F), 1 < p ≤ 2, can be defined similarly as in L p (R). The inner product is defined by The "natural"order on the sequence {v(n) ∈ F} ∞ n=0 is described as follows. Recall that P is the prime ideal in D, D/P ∼ = GF (q) = τ , q = p c , p is a prime, c a positive integer and Ω : D → τ the canonical homomorphism of D on to τ . Note k=0 is a basis of GF (q) over GF (p). Definition 1.2. For k, 0 ≤ k < q, k = a 0 + a 1 p + ... + a c−1 p c−1 , 0 ≤ a i < p, i = 0, 1, ..., c − 1, we define v(k) = (a 0 + a 1 ǫ 1 + ... + a c−1 ǫ c−1 )p −1 (0 ≤ k < q).

Distributions over local fields
We denote S (F) the spaces of all finite linear combinations of characteristics functions of ball of F. The Fourier transform is homeomorphism of S (F) onto S (F). The distribution space of S (F) is denoted by S ′ (F) . The Fourier transform of g ∈ S (F) is denoted byĝ(ω) and defined bŷ

1)
The Necessary and Sufficient Conditions for Wavelet Frames in H s (F) 3 and the inverse Fourier transform defined by, The Fourier transform and inverse Fourier transforms of a distributions g ∈ S ′ (F) is defined by We equip H s (F) with the inner product which induces the norm Proof. See [15]. ✷

A necessary condition of wavelet frame for
satisfies for all g ∈ H s (F).
Since g ∈ S (F) so the ∞ l=0 contains only finite non-zero terms and χ k (v(l)) = 1 for all k, l ∈ N 0 , then we get By the convergence theorem of Fourier Series on D, we get The Necessary and Sufficient Conditions for Wavelet Frames in H s (F) 5 where ϕ l is the characteristic function of ω 0 + P l . Then for l ∈ N and j ≥ −M ,ĝ(ω)ḡ(ω + p −j v(l)) = 0. Since ω and (ω + p −j v(l)) can not be in ω 0 + P m simultaneously. Now, we have To prove the left hand inequality, where By condition of frame, T 1 ≥ C − T 2 . Since we have already show that T 1 = j>−Mν s (ω 0 )|ψ(p −j ω 0 )| 2 . So we only need to show that T 2 → 0 as M → ∞. Now, using the fact S ′ (F) is dence in H s (F) in (2.2) and Schwarz's inequality, we have Fν , so there exists a characteristic funtion ϕ r (ω − ω 0 ) of the set ω 0 + P r , where r is some integers. Nowf can be written asf (ω) = q Then summation index k is bounded by q −r−j . So using this, we get Suppose that ω 0 = 0. For any ǫ > 0, choose J < 0 enough small satisfies the following two inequalities : q J < |ω 0 | = q ρ such that J + ρ < 0 and P −J−ρν s (p −J ω)|ψ(ω)| 2 dω < ǫ. We have (2.6) Since |p −j ω 0 | = q j q ρ ≤ q J q ρ and P −j+r ⊂ P −J−ρ . Hence, T 2 → 0 as j → −∞. Therefore there exists j such that Hence we obtain required result. ✷

Sufficient conditions of wavelet frame for H s (F)
To find the sufficient conditions of wavelet frame for H s (F), we need the following Lemma Then iterated series in (3.2) is absolutely convergent.
Proof. Since g ∈ S (F) so the ∞ l=0 in (3.2) contains only finite non-zero terms. Hence, j∈Z Fν The Necessary and Sufficient Conditions for Wavelet Frames in H s (F) 7 We claim that, holds for all g ∈ S (F). We have j∈Z l∈N0 By using the condition sup{ν s (ω) j∈Z |ψ(p j ω)| 2 : ω ∈ P −1 \D} < +∞ and Levi's Lemma for integral, we get j∈Z l∈N0 Now, we show that series (3.6) is absolutely convergent.
Since g ∈ S (F), there exist a constant J > 0 such that for all |j| > Ĵ On the other hand, for each |j| > J, there exist a constant L such that for all l ≥ L g(p −j ω + p −j v(l)) = 0. (3.10) Therefore only finite number of terms of the iterated series in (3.8) are nonzero .
Hence the T 2 is absolutely convergent. The proof is complete. ✷ Now using above lemma, we establish sufficient condition of frame for H s (F). Let and We set where Then {ψ j,k : j ∈ Z, k ∈ N 0 } is wavelet frame for H s (F) with bounds ρ 1 (ψ) and ρ 2 (ψ) .
The Necessary and Sufficient Conditions for Wavelet Frames in H s (F) 9 = j∈Z Fν We derive further that Taking infimum and suprimum in above two inequality respectively, we get The proof of theorem 3.1 is complete.