aa r X i v : . [ m a t h . A P ] F e b Two-wavelet theory in Weinstein setting
Ahmed Saoudi
Northern Border University, College of Science, Arar, P.O. Box 1631, Saudi Arabia.Université de Tunis El Manar, Faculté des sciences de Tunis, Tunisie. e-mail: [email protected]
Abstract
In this paper we introduce the notion of a Weinstein two-wavelet. Then weestablish and prove the resolution of the identity formula for the Weinstein con-tinuous wavelet transform. Next, we give results on Calderón’s type reproducingformula in the context of the Weinstein two-wavelet.
Keywords . Weinstein operator; Weinstein wavelet transform; Weinsteintwo-wavelet transform; Calderón’s type reproducing formula.
Mathematics Subject Classification . Primary 43A32; Secondary 44A15
The Weinstein operator ∆ dW,α defined on R d +1+ = R d × (0 , ∞ ) , by ∆ dW,α = d +1 X j =1 ∂ ∂x j + 2 α + 1 x d +1 ∂∂x d +1 = ∆ d + L α , α > − / , where ∆ d is the Laplacian operator for the d first variables and L α is the Besseloperator for the last variable defined on (0 , ∞ ) by L α u = ∂ u∂x d +1 + 2 α + 1 x d +1 ∂u∂x d +1 . The Weinstein operator ∆ dW,α has several applications in pure and applied mathemat-ics, especially in fluid mechanics [2, 27].Very recently, many authors have been investigating the behaviour of the Wein-stein transform (2.5) with respect to several problems already studied for the clas-sical Fourier transform. For instance, Heisenberg-type inequalities [17], Littlewood-Paley g-function [19], Shapiro and HardyâĂŞLittlewoodâĂŞSobolev type inequalities[16, 18], Paley-Wiener theorem [9], Uncertainty principles [12, 21, 24], multiplierWeinstein operator [20], wavelet and continuous wavelet transform [4, 11], Wignertransform and localization operators [22, 23], and so forth...In the classical setting, the notion of wavelets was first introduced by Morlet inconnection with his study of seismic traces and the mathematical foundations were1 A. Saoudi given by Grossmann and Morlet [6]. Later, Meyer and many other mathematiciansrecognized many classical results of this theory [8, 14]. Classical wavelets have wideapplications, ranging from signal analysis in geophysics and acoustics to quantumtheory and pure mathematics [3, 5, 7].Recently, the theory of wavelets and continuous wavelet transform has been ex-tended and generalized in the context of differential-differences operators [4, 10, 11,13].Wavelet analysis has attracted attention for its ability to analyze rapidly changingtransient signals. Any application using the Fourier like transform can be formulatedusing wavelets to provide more accurately localized temporal and frequency infor-mation. The reason for the extension from one wavelet to two wavelets comes fromthe extra degree of flexibility in signal analysis and imaging when the localizationoperators are used as time-varying filters. This paper is an attempt to fill this gapby extending one wavelet to two wavelets in Weinstein setting.Using the harmonic analysis associated with the Weinstein operator (generalizedtranslation operators, generalized convolution, Weinstein transform, ...) and the sameidea as for the classical case, we define and study in this paper the notion of aWeinstein two-wavelet. For ϕ and ψ be in L α ( R d +1+ ) , the pair ( ϕ, ψ ) is said a Weinsteintwo-wavelet on R d +1+ if the following integral C ϕ,ψ = Z ∞ F W ( ψ )( aξ ) F W ( ϕ )( aξ ) daa is constant for almost all ξ ∈ R d +1+ . We prove for this Weinstein two-wavelet a Parsevaltype formula Z X Φ Wϕ ( f )( a, x )Φ Wψ ( g )( a, x ) dµ α ( a, x ) = C ϕ,ψ Z R d +1+ f ( x ) g ( x ) dµ α ( x ) , for all ϕ and ψ in L α ( R d +1+ ) where Φ Wϕ is a Weinstein continuous wavelet transformon R d +1+ defined for regular functions f on R d +1+ in [11] by ∀ ( a, x ) ∈ X , Φ Wϕ ( f )( a, x ) = Z R d +1+ f ( y ) ϕ a,x ( y ) dµ α ( y ) = h f, ϕ a,x i α, . Next, we prove for the pair ( ϕ, ψ ) an inversion formula for all f ∈ L α ( R d +1+ ) (resp. L α ( R d +1+ ) ) such that F W ( f ) belongs to f ∈ L α ( R d +1+ ) (resp. L α ( R d +1+ ) ∩ L ∞ α ( R d +1+ ) ),of the form f ( y ) = 1 C ϕ,ψ Z ∞ Z R d +1+ Φ Wϕ ( f )( a, x ) ψ a,x ( y ) µ α ( a, x ) , where for each y ∈ R d +1+ both the inner integral and the outer integral are absolutelyconvergent, but eventually not the double integral. In the end, we prove for thepair ( ϕ, ψ ) a Calderón’s type formulas. For ϕ and ψ be two Weinstein wavelets in L α ( R d +1+ ) such that ( ϕ, ψ ) be a Weinstein two-wavelet, C ϕ,ψ = 0 and F W ( ϕ ) , and F W ( ψ ) are in L ∞ α ( R d +1+ ) . Then for all f in L α ( R d +1+ ) , the function f γ,δ ( y ) = 1 C ϕ,ψ Z δγ Z R d +1+ Φ Wϕ ( f )( a, x ) ψ a,x ( y ) dµ α ( x ) daa α + d +3 , y ∈ R d +1+ wo-wavelet theory in Weinstein setting L α ( R d +1+ ) and satisfies lim ( γ,δ ) → (0 , ∞ ) k f γ,δ − f k α, = 0 . This paper is organized as follows. In Section 2, we recall some properties ofharmonic analysis for the Weinstein operators. In Section 3, we define the two-waveletin Weinstein in setting and prove for it a Parseval type formula and we establish forit an inversion formula. In last Section, we introduce a Calderón’s type reproducingformula.
For all λ = ( λ , ..., λ d +1 ) ∈ C d +1 , the system ∂ u∂x j ( x ) = − λ j u ( x ) , if ≤ j ≤ dL α u ( x ) = − λ d +1 u ( x ) ,u (0) = 1 , ∂u∂x d +1 (0) = 0 , ∂u∂x j (0) = − iλ j , if ≤ j ≤ d (2.1)has a unique solution denoted by Λ dα ( λ, . ) , and given by Λ dα ( λ, x ) = e − i
0) = 1 . (iv) For all ν ∈ N d +1 , x ∈ R d +1 and λ ∈ C d +1 we have (cid:12)(cid:12) D νλ Λ dα ( λ, x ) (cid:12)(cid:12) ≤ k x k | ν | e k x kk Im λ k A. Saoudi where D νλ = ∂ ν / ( ∂λ ν ...∂λ ν d +1 d +1 ) and | ν | = ν + ... + ν d +1 . In particular, for all ( λ, x ) ∈ R d +1 × R d +1 , we have (cid:12)(cid:12) Λ dα ( λ, x ) (cid:12)(cid:12) ≤ . (2.3)In the following we denote by(i) − λ = ( − λ ′ , λ d +1 ) (ii) C ∗ ( R d +1 ) , the space of continuous functions on R d +1 , even with respect to thelast variable.(iii) S ∗ ( R d +1 ) , the space of the C ∞ functions, even with respect to the last variable,and rapidly decreasing together with their derivatives.(iv) S ∗ ( R d +1 × R d +1 ) , the Schwartz space of rapidly decreasing functions on R d +1 × R d +1 even with respect to the last two variables.(v) D ∗ ( R d +1 ) , the space of C ∞ -functions on R d +1 which are of compact support,evenwith respect to the last variable.(vi) L pα ( R d +1+ ) , ≤ p ≤ ∞ , the space of measurable functions f on R d +1+ such that k f k α,p = Z R d +1+ | f ( x ) | p dµ α ( x ) ! /p < ∞ , p ∈ [1 , ∞ ) , k f k α, ∞ = ess sup x ∈ R d +1+ | f ( x ) | < ∞ , where dµ α ( x ) is the measure on R d +1+ = R d × (0 , ∞ ) given by dµ α ( x ) = x α +1 d +1 (2 π ) d α Γ ( α + 1) dx. For a radial function ϕ ∈ L α ( R d +1+ ) the function ˜ ϕ defined on R + such that ϕ ( x ) = ˜ ϕ ( | x | ) , for all x ∈ R d +1+ , is integrable with respect to the measure r α + d +1 dr ,and we have Z R d +1+ ϕ ( x ) dµ α ( x ) = a α Z ∞ ˜ ϕ ( r ) r α + d +1 dr, (2.4)where a α = 12 α + d Γ( α + d + 1) . The Weinstein transform generalizing the usual Fourier transform, is given for ϕ ∈ L α ( R d +1+ ) and λ ∈ R d +1+ , by F W ( ϕ )( λ ) = Z R d +1+ ϕ ( x )Λ dα ( x, λ ) dµ α ( x ) , (2.5)We list some known basic properties of the Weinstein transform are as follows.For the proofs, we refer [1, 15]. wo-wavelet theory in Weinstein setting ϕ ∈ L α ( R d +1+ ) , the function F W ( ϕ ) is continuous on R d +1+ and we have kF W ϕ k α, ∞ ≤ k ϕ k α, . (2.6)(ii) The Weinstein transform is a topological isomorphism from S ∗ ( R d +1 ) onto itself.The inverse transform is given by F − W ϕ ( λ ) = F W ϕ ( − λ ) , for all λ ∈ R d +1+ . (2.7)(iii) For all f in D ∗ ( R d +1 ) (resp. S ∗ ( R d +1 ) ), we have the following relations ∀ λ ∈ R d +1+ , F W ( ϕ )( λ ) = F W ( e ϕ )( λ ) , (2.8) ∀ λ ∈ R d +1+ , F W ( ϕ )( λ ) = F W ( e ϕ )( − λ ) , (2.9)where e ϕ is the function defined by ∀ λ ∈ R d +1+ , e ϕ ( λ ) = ϕ ( − λ ) . (iv) Parseval’s formula: For all ϕ, φ ∈ S ∗ ( R d +1 ) , we have Z R d +1+ ϕ ( x ) φ ( x ) dµ α ( x ) = Z R d +1+ F W ( ϕ )( x ) F W ( φ )( x ) dµ α ( x ) . (2.10)(v) Plancherel’s formula: For all ϕ ∈ L α ( R d +1+ ) , we have kF W ϕ k α, = k ϕ k α, . (2.11)(vi) Plancherel Theorem: The Weinstein transform F W extends uniquely to an iso-metric isomorphism on L α ( R d +1+ ) . (vii) Inversion formula: Let ϕ ∈ L α ( R d +1+ ) such that F W ϕ ∈ L α ( R d +1+ ) , then we have ϕ ( λ ) = Z R d +1+ F W ϕ ( x )Λ dα ( − λ, x ) dµ α ( x ) , a.e. λ ∈ R d +1+ . (2.12)Using relations (2.6) and (2.11) with Marcinkiewicz’s interpolation theorem [25]we deduce that for every ϕ ∈ L pα ( R d +1+ ) for all ≤ p ≤ , the function F W ( ϕ ) ∈ L qα ( R d +1+ ) , q = p/ ( p − , and kF W ϕ k α,q ≤ k ϕ k α,p . (2.13) Definition 2.1.
The translation operator τ αx , x ∈ R d +1+ associated with the Weinsteinoperator ∆ dW,α , is defined for a continuous function ϕ on R d +1+ , which is even withrespect to the last variable and for all y ∈ R d +1+ by τ αx ϕ ( y ) = C α Z π ϕ (cid:16) x ′ + y ′ , q x d +1 + y d +1 + 2 x d +1 y d +1 cos θ (cid:17) (sin θ ) α dθ, with C α = Γ( α + 1) √ π Γ( α + 1 / . A. Saoudi
By using the Weinstein kernel, we can also define a generalized translation, for afunction ϕ ∈ S ∗ ( R d +1 ) and y ∈ R d +1+ the generalized translation τ αx ϕ is defined by thefollowing relation F W ( τ αx ϕ )( y ) = Λ dα ( x, y ) F W ( ϕ )( y ) . (2.14)In the following proposition, we give some properties of the Weinstein translationoperator: Proposition 2.2.
The translation operator τ αx , x ∈ R d +1+ satisfies the following prop-erties.i). For ϕ ∈ C ∗ ( R d +1 ) , we have for all x, y ∈ R d +1+ τ αx ϕ ( y ) = τ αy ϕ ( x ) and τ α ϕ = ϕ. (2.15) ii). Let ϕ ∈ L pα ( R d +1+ ) , ≤ p ≤ ∞ and x ∈ R d +1+ . Then τ αx ϕ belongs to L pα ( R d +1+ ) andwe have k τ αx ϕ k α,p ≤ k ϕ k α,p . (2.16) Proposition 2.3.
Let ϕ ∈ L α ( R d +1+ ) . Then for all x ∈ R d +1+ , Z R d +1+ τ αx ϕ ( y ) dµ α ( y ) = Z R d +1+ ϕ ( y ) dµ α ( y ) . (2.17) Proof.
The result comes from combination identities (2.12) and (2.14).By using the generalized translation, we define the generalized convolution product ϕ ∗ ψ of the functions ϕ, ψ ∈ L α ( R d +1+ ) as follows ϕ ∗ ψ ( x ) = Z R d +1+ τ αx ϕ ( − y ) ψ ( y ) dµ α ( y ) . (2.18)This convolution is commutative and associative, and it satisfies the following prop-erties. Proposition 2.4. i) For all ϕ, ψ ∈ L α ( R d +1+ ) , (resp. ϕ, ψ ∈ S ∗ ( R d +1 ) ), then ϕ ∗ ψ ∈ L α ( R d +1+ ) , (resp. ϕ ∗ ψ ∈ S ∗ ( R d +1 ) ) and we have F W ( ϕ ∗ ψ ) = F W ( ϕ ) F W ( ψ ) . (2.19) ii) Let p, q, r ∈ [1 , ∞ ] , such that p + q − r = 1 . Then for all ϕ ∈ L pα ( R d +1+ ) and ψ ∈ L qα ( R d +1+ ) the function ϕ ∗ ψ belongs to L rα ( R d +1+ ) and we have k ϕ ∗ ψ k α,r ≤ k ϕ k α,p k ψ k α,q . (2.20) iii) Let ϕ, ψ ∈ L α ( R d +1+ ) . Then ϕ ∗ ψ = F − W ( F W ( ϕ ) F W ( ψ )) . (2.21) iv) Let ϕ, ψ ∈ L α ( R d +1+ ) . Then ϕ ∗ ψ belongs to L α ( R d +1+ ) if and only if F W ( ϕ ) F W ( ψ ) belongs to L α ( R d +1+ ) and we have F W ( ϕ ∗ ψ ) = F W ( ϕ ) F W ( ψ ) . (2.22) v) Let ϕ, ψ ∈ L α ( R d +1+ ) . Then k ϕ ∗ ψ k α, = kF W ( ϕ ) F W ( ψ ) k α, , (2.23) where both sides are finite or infinite. wo-wavelet theory in Weinstein setting In the following, we denote by X = (cid:8) ( a, x ) : x ∈ R d +1+ and a > (cid:9) . L pα ( X ) , p ∈ [1 , ∞ ] the space of measurable functions ϕ on X such that k ϕ k L pα ( X ) := (cid:18)Z X | ϕ ( a, x ) | p dµ α ( a, x ) (cid:19) p < ∞ , ≤ p < ∞ , k ϕ k L ∞ α ( X ) = ess sup ( a,x ) ∈X | ϕ ( a, x ) | < ∞ , where the measure µ α ( a, x ) is defined on X by dµ α ( a, x ) = dµ α ( x ) daa α + d +3 Definition 3.1. [4] A classical wavelet on R d +1+ is a measurable function ϕ on R d +1+ satisfying for almost all ξ ∈ R d +1+ , the condition < C ϕ = Z ∞ |F W ( ϕ )( aξ ) | daa < ∞ . (3.1)We extend the notion of the wavelet to the two-wavelet in Weinstein setting asfollows Definition 3.2.
Let ϕ and ψ be in L α ( R d +1+ ) . We say that the pair ( ϕ, ψ ) is aWeinstein two-wavelet on R d +1+ if the following integral C ϕ,ψ = Z ∞ F W ( ψ )( aξ ) F W ( ϕ )( aξ ) daa (3.2) is constant for almost all ξ ∈ R d +1+ and we call the number C ϕ,ψ the Weinstein two-wavelet constant associated to the functions ϕ and ψ . It is to highlight that if u is a Weinstein wavelet then the pair ( ϕ, ψ ) is a Weinsteintwo-wavelet, and C ϕ,ψ coincides with C ϕ .Let a > and ϕ be a measurable function. We consider the function ϕ a definedby ∀ x ∈ R d +1+ , ϕ a ( x ) = 1 a α + d +2 ϕ (cid:16) xa (cid:17) . (3.3) Proposition 3.3.
1. Let a > and ϕ ∈ L pα ( R d +1+ ) , p ∈ [1 , ∞ ] . The function ϕ a belongs to L pα ( R d +1+ ) and we have k ϕ a k α,p = a (2 α + d +2)( p − k ϕ k α,p . (3.4)
2. Let a > and ϕ ∈ L α ( R d +1+ ) ∪ L α ( R d +1+ ) . Then, we have F W ( ϕ a )( ξ ) = F W ( ϕ )( aξ ) , ξ ∈ R d +1+ . (3.5) A. Saoudi
For a > and ϕ ∈ L α ( R d +1+ ) , we consider the family ϕ a,x , x ∈ R d +1+ of Weinsteinwavelets on R d +1+ in L α ( R d +1+ ) defined by ∀ y ∈ R d +1+ , ϕ a,x = a α +1+ d τ αx ϕ a ( y ) . (3.6) Remark 3.4.
1. Let ϕ be a function in L α ( R d +1+ ) , then we have ∀ ( a, x ) ∈ X , k ϕ a,x k α, ≤ k ϕ k α, . (3.7)
2. Let p ∈ [1 , ∞ ] and ϕ be a function in L pα ( R d +1+ ) , then we have ∀ ( a, x ) ∈ X , k ϕ a,x k α,p ≤ a (2 α + d +2)( p − ) k ϕ k α,p . (3.8) Definition 3.5. [11] Let ϕ be a Weinstein wavelet on R d +1+ in L α ( R d +1+ ) . The Wein-stein continuous wavelet transform Φ Wϕ on R d +1+ is defined for regular functions f on R d +1+ by ∀ ( a, x ) ∈ X , Φ Wϕ ( f )( a, x ) = Z R d +1+ f ( y ) ϕ a,x ( y ) dµ α ( y ) = h f, ϕ a,x i α, . (3.9)This transform can also be written in the form Φ Wϕ ( f )( a, x ) = a α +1+ d ˇ f ∗ ϕ a ( x ) . (3.10) Remark 3.6.
1. Let ϕ be a function in L pα ( R d +1+ ) , and let f be a function in L qα ( R d +1+ ) , with p ∈ [1 , ∞ ] , we define the Weinstein continuous wavelet trans-form Φ Wϕ ( f ) by the relation (3.10).2. Let ϕ be a Weinstein wavelet on R d +1+ in L α ( R d +1+ ) . Then from the relations(3.7) and (3.9), we have for all f ∈ L α ( R d +1+ ) k Φ Wϕ ( f ) k α, ∞ ≤ k f k α, k ϕ k α, . (3.11)
3. Let ϕ be a function in L pα ( R d +1+ ) , with p ∈ [1 , ∞ ] , then from the inequality (2.20)and the identity (3.10), we have for all f ∈ L qα ( R d +1+ ) k Φ Wϕ ( f ) k α, ∞ ≤ k f k α,q k ϕ k α,p . (3.12)The following Theorem generalizes the Parseval’s formula for the continuous We-instein wavelet transform proved by Mejjaoli [11]. Theorem 3.7.
Let ( ϕ, ψ ) be a Weinstein two-wavelet. Then for all f and g in L α ( R d +1+ ) , we have the following Parseval type formula Z X Φ Wϕ ( f )( a, x )Φ Wψ ( g )( a, x ) dµ α ( a, x ) = C ϕ,ψ Z R d +1+ f ( x ) g ( x ) dµ α ( x ) , (3.13) where C ϕ,ψ is the Weinstein two-wavelet constant associated to the functions ϕ and ψ given by the identity (3.2). wo-wavelet theory in Weinstein setting Proof.
According to Fubini’s Theorem, the relation (3.10) and Parseval’s formula forthe Weinstein transform (2.10), we obtain Z X Φ Wϕ ( f )( a, x )Φ Wψ ( g )( a, x ) dµ α ( a, x )= Z ∞ Z R d +1+ ˇ f ∗ ϕ a ( x )ˇ g ∗ ψ a ( x ) a α + d +2 dµ α ( a, x )= Z ∞ Z R d +1+ ˇ f ∗ ϕ a ( x )ˇ g ∗ ψ a ( x ) dµ α ( x ) daa = Z ∞ Z R d +1+ F W ( ˇ f )( ξ ) F W (ˇ g )( ξ ) F W ( ϕ a )( ξ ) F W ( ψ a )( ξ ) dµ α ( ξ ) daa = Z R d +1+ F W ( ˇ f )( ξ ) F W (ˇ g )( ξ ) (cid:18)Z ∞ F W ( ϕ )( aξ ) F W ( ψ )( − aξ ) daa (cid:19) dµ α ( ξ ) . Moreover, using the relations (2.8) and (2.9), we conclude that Z X Φ Wϕ ( f )( a, x )Φ Wψ ( g )( a, x ) dµ α ( a, x )= Z R d +1+ F W ( f )( ξ ) F W ( g )( ξ ) (cid:18)Z ∞ F W ( ϕ )( aξ ) F W ( ψ )( aξ ) daa (cid:19) dµ α ( ξ )= C ϕ,ψ Z R d +1+ F W ( f )( ξ ) F W ( g )( ξ ) dµ α ( ξ ) . Finally, we get the desired result using the Parseval’s formula for the Weinstein trans-form (2.10).In the particular case of the previous theorem when ϕ = ψ and f = g , we obtainthe following PlancherelâĂŹs formula for the Weinstein continuous wavelet transformprovided in [11] Z X (cid:12)(cid:12) Φ Wϕ ( f )( a, x ) (cid:12)(cid:12) dµ α ( a, x ) = C ϕ Z R d +1+ | f ( x ) | dµ α ( x ) , (3.14)where C ϕ = C ϕ,ϕ = Z ∞ |F W ( ϕ )( aξ ) | daa . (3.15)From the Parseval type formula in Theorem 3.7, we deduce the following orthogonalityresult. Corollary 3.8.
Let ( ϕ, ψ ) be a Weinstein two-wavelet. Then we have the followingassertion: If the Weinstein two-wavelet constant C ϕ,ϕ = 0 , then Φ Wϕ (cid:0) L α ( R d +1+ ) (cid:1) and Φ Wψ (cid:0) L α ( R d +1+ ) (cid:1) are orthogonal. Theorem 3.9. (Inversion formula) Let ( ϕ, ψ ) be a Weinstein two-wavelet. For all f ∈ L α ( R d +1+ ) (resp. L α ( R d +1+ ) ) such that F W ( f ) belongs to f ∈ L α ( R d +1+ ) (resp. L α ( R d +1+ ) ∩ L ∞ α ( R d +1+ ) ), we have f ( y ) = 1 C ϕ,ψ Z ∞ Z R d +1+ Φ Wϕ ( f )( a, x ) ψ a,x ( y ) µ α ( a, x ) , a.e. (3.16)0 A. Saoudi where for each y ∈ R d +1+ both the inner integral and the outer integral are absolutelyconvergent, but eventually not the double integral.Proof. Using similar ideas as in the proof in [26, Theorem 6.III.3], we obtain therelation (3.16).
The main result of this section is to establish a Calderón’s type formulas for theWeinstein two-wavelet transform under the following assumptions: • ( A ) Let ϕ and ψ be two Weinstein wavelets in L α ( R d +1+ ) such that ( ϕ, ψ ) be aWeinstein two-wavelet, and F W ( ϕ ) and F W ( ψ ) are in L ∞ α ( R d +1+ ) . • ( A ) The Weinstein two-wavelet constant C ϕ,ϕ = 0 . Theorem 4.1. (Calderón’s type formulas) Let ϕ and ψ be two-Weinstein waveletssatisfying the assumptions ( A ) and ( A ) and < γ < δ < ∞ . Then for all f in L α ( R d +1+ ) , the function f γ,δ ( y ) = 1 C ϕ,ψ Z δγ Z R d +1+ Φ Wϕ ( f )( a, x ) ψ a,x ( y ) dµ α ( x ) daa α + d +3 , y ∈ R d +1+ (4.1) belongs to L α ( R d +1+ ) and satisfies lim ( γ,δ ) → (0 , ∞ ) k f γ,δ − f k α, = 0 . (4.2)In the order to prove this theorem we need the following Lemmas. Lemma 4.2.
Let ϕ and ψ be two-Weinstein wavelets satisfying the assumptions ( A )and ( A ) and f in L α ( R d +1+ ) . Then we have the following assertions:(i) The functions ( ˇ f ∗ ϕ a ˇ) and ( ˇ f ∗ ϕ a ˇ) ∗ ψ a belongs to L α ( R d +1+ ) and we have F W (( ˇ f ∗ ϕ a ˇ) ∗ ψ a )( ξ ) = F W ( f )( ξ ) F W ( ϕ a )( ξ ) F W ( ψ a )( ξ ) , ξ ∈ R d +1+ . (4.3) (ii) We have the following inequality kF W ( ˇ f ∗ ϕ a ˇ) ∗ ψ a k α, ≤ k f k α, kF W ( ϕ ) k α, ∞ kF W ( ψ ) k α, ∞ . (4.4) Proof. (i) According to the relations (2.8), (2.9) and Proposition 2.4 ( v ) we obtain F W (( ˇ f ∗ ϕ a ˇ))( ξ ) = F W ( ˇ f ∗ ϕ a )( − ξ )= F W ( ˇ f )( − ξ ) F W ( ϕ a )( − ξ )= F W ( f )( ξ ) F W ( ˇ ϕ a )( − ξ ) . Hence, we have F W (( ˇ f ∗ ϕ a ˇ))( ξ ) = F W ( f )( ξ ) F W ( ϕ a )( ξ ) . (4.5) wo-wavelet theory in Weinstein setting G defined on R d +1+ by G ( ξ ) = F W (( ˇ f ∗ ϕ a ˇ))( ξ ) . Therefore F W (( ˇ f ∗ ϕ a ˇ) ∗ ψ a )( ξ ) = F W ( G ∗ ψ a )( ξ ) , ξ ∈ R d +1+ . According to Proposition 2.4 ( v ), we deduce that the function G belongs to L α ( R d +1+ ) and we have F W ( G ∗ ψ a )( ξ ) = F W ( G )( ξ ) F W ( ψ a )( ξ ) , ξ ∈ R d +1+ . (4.6)Finally, we obtain the result by combining the relations (4.5) and (4.6).(ii) From the assertion (i), we have Z R d +1+ |F W (( ˇ f ∗ ϕ a ˇ) ∗ ψ a )( ξ ) | dµ α ( x ) = Z R d +1+ |F W ( f )( ξ ) | |F W ( ϕ a )( ξ ) | |F W ( ψ a )( ξ ) | dµ α ( x ) . Therefore, according to Plancherel formula for Weinstein transform (2.11) and thefact that F W ( ϕ a ) and F W ( ψ a ) belongs to L ∞ α ( R d +1+ ) , we deduce that kF W ( ˇ f ∗ ϕ a ˇ) ∗ ψ a k α, ≤ k f k α, kF W ( ϕ a ) k α, ∞ kF W ( ψ a ) k α, ∞ . In the end, we conclude the result from the relation (3.5).
Lemma 4.3.
Let ϕ and ψ be two-Weinstein wavelets satisfying the assumptions ( A )and ( A ) . Then the function K γ,δ defined as follows K γ,δ ( ξ ) = 1 C ϕ,ψ Z δγ F W ( ϕ a )( ξ ) F W ( ψ a )( ξ ) daa , ξ ∈ R d +1+ (4.7) satisfies, for almost all ξ ∈ R d +1+ : < K γ,δ ( ξ ) ≤ p C ϕ C ψ C ϕ,ψ , (4.8) and lim ( γ,δ ) → (0 , ∞ ) K γ,δ ( ξ ) = 1 . (4.9) Proof.
According to the Cauchy-Schwarz inequality and the identity (3.15), we havefor almost all ξ ∈ R d +1+ | K γ,δ ( ξ ) | ≤ | C ϕ,ψ | (cid:18)Z δγ |F W ( ϕ a )( ξ ) | daa (cid:19) (cid:18)Z δγ |F W ( ψ a )( ξ ) | daa (cid:19) ≤ | C ϕ,ψ | (cid:18)Z ∞ |F W ( ϕ a )( ξ ) | daa (cid:19) (cid:18)Z ∞ |F W ( ψ a )( ξ ) | daa (cid:19) = p C ϕ C ψ C ϕ,ψ . On the other hand, it’s clear that for almost all ξ ∈ R d +1+ lim ( γ,δ ) → (0 , ∞ ) K γ,δ ( ξ ) = 1 . This completes the proof.2
A. Saoudi
Theorem 4.4.
Let ϕ and ψ be two-Weinstein wavelets satisfying the assumptions( A ) and ( A ) . Then the function f γ,δ defined by the relation (4.1) belongs to L α ( R d +1+ ) and satisfies F W ( f γ,δ )( ξ ) = F W ( f )( ξ ) K γ,δ ( ξ ) , ξ ∈ R d +1+ , (4.10) where K γ,δ is the function given by the relation (4.7).Proof. In a first step, we try to show that f γ,δ belongs to L α ( R d +1+ ) . From, thedefinition of the Weinstein continuous wavelet transform (3.9) and the relations (2.15)and (3.6) we get f γ,δ ( y ) = 1 C ϕ,ψ Z δγ Z R d +1+ ( ˇ f ∗ ϕ a )( x ) τ αy ψ a ( x ) dµ α ( x ) daa . On the other hand, from the definition of the convolution product associated to theWeinstein transform given by the relation (2.18), we have Z R d +1+ ( ˇ f ∗ ϕ a )( x ) τ αy ψ a ( x ) dµ α ( x ) = Z R d +1+ ( ˇ f ∗ ϕ a ˇ)( x ) τ αy ψ a ( − x ) dµ α ( x )= ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) . So, we have f γ,δ ( y ) = 1 C ϕ,ψ Z δγ ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) daa . (4.11)By using HÃűlder’s inequality for the measure daa , we obtain | f γ,δ ( y ) | = 1 | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) (cid:18)Z δγ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) (cid:12)(cid:12)(cid:12) daa (cid:19) . Aapplying Fubuni-Tonelli’s theorem, after integrating the pervious inequality, we get Z R d +1+ | f γ,δ ( y ) | dµ α ( y ) ≤ | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) Z δγ Z R d +1+ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) (cid:12)(cid:12)(cid:12) dµ α ( y ) ! daa . According to the Parseval’s formula for the Weinstein transform (2.10) and the as-sertion ( i ) of the Lemma 4.2, we deduce that Z R d +1+ | f γ,δ ( y ) | dµ α ( y ) ≤ | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) Z R d +1+ |F W ( f )( ξ ) | × (cid:18)Z δγ |F W ( ϕ a )( ξ ) | |F W ( ψ a )( ξ ) | daa (cid:19) dµ α ( ξ ) . On the other hand, from the identity (3.15) and the relation (3.5), we have Z δγ |F W ( ϕ a )( ξ ) | |F W ( ψ a )( ξ ) | daa ≤ C ψ kF W ( ϕ ) k α, ∞ . wo-wavelet theory in Weinstein setting Z R d +1+ | f γ,δ ( y ) | dµ α ( y ) ≤ C ψ | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) kF W ( ϕ ) k α, ∞ kF W ( f ) k α, . Finally, from the Plancherel’s formula for the Weinstein transform (2.11), we get Z R d +1+ | f γ,δ ( y ) | dµ α ( y ) ≤ C ψ | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) kF W ( ϕ ) k α, ∞ k f k α, ≤ ∞ , which implies that f γ,δ belongs to L α ( R d +1+ ) .In the second step, We prove the relation (4.10). Let h ∈ S ∗ ( R d +1 ) , then F − W ( h ) belongs to mathcalS ∗ ( R d +1 ) and from the relation (4.11), we have Z R d +1+ f γ,δ ( y ) F − W ( h )( y ) dµ α ( y )= Z R d +1+ (cid:18) C ϕ,ψ Z δγ ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) daa (cid:19) F − W ( h )( y ) dµ α ( y ) . (4.12)Consider that | C ϕ,ψ | Z R d +1+ Z δγ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) (cid:12)(cid:12)(cid:12) daa dµ α ( y )= 1 | C ϕ,ψ | Z δγ Z R d +1+ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) dµ α ( y ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) daa . By applying Hölder’s inequality to right hand side of the previous equality, we obtain | C ϕ,ψ | Z δγ Z R d +1+ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) dµ α ( y ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) daa ≤ | C ϕ,ψ | Z δγ Z R d +1+ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) dµ α ( y ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) daa . From the assertion ( ii ) of Lemma 4.2 and Plancherel’s formula for Weinstein transform(2.11), we get | C ϕ,ψ | Z δγ Z R d +1+ (cid:12)(cid:12)(cid:12) ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) dµ α ( y ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) daa ≤ | C ϕ,ψ | (cid:18)Z δγ daa (cid:19) kF W ( ϕ ) k α, ∞ kF W ( ψ ) k α, ∞ k h k α, k f k α, . Then, from Fubini’s theorem, the right hand side of the relation (4.12) can also bewritten in the form C ϕ,ψ Z δγ Z R d +1+ ( ˇ f ∗ ϕ a ˇ) ∗ ψ a ( y ) F − W ( h )( y ) dµ α ( y ) ! daa . A. Saoudi
Moreover, according to the Parseval’s formula for the Weinstein transform (2.10) andthe assertion ( i ) of the Lemma 4.2, the pervious integral becomes C ϕ,ψ Z δγ Z R d +1+ F W ( f )( ξ ) F W ( ϕ a )( ξ ) F W ( ψ a )( ξ ) h ( ξ ) dµ α ( ξ ) ! daa . By applying Fubini’s theorem to this last integral, we have Z R d +1+ F W ( f )( ξ ) (cid:18) C ϕ,ψ Z δγ F W ( ϕ a )( ξ ) F W ( ψ a )( ξ ) daa (cid:19) h ( ξ ) dµ α ( ξ )= Z R d +1+ F W ( f )( ξ ) K γ,δ ( ξ ) h ( ξ ) dµ α ( ξ ) . (4.13)On the other hand, by applying the Parseval’s formula for the Weinstein transform(2.10) to the left hand side of the relation (4.12), it takes the integral form Z R d +1+ F W ( f γ,δ )( ξ ) h ( ξ ) dµ α ( ξ ) . (4.14)Finally, from the relations (4.13) and (4.14), we obtain for all h in S ∗ ( R d +1 ) Z R d +1+ ( F W ( f γ,δ )( ξ ) − F W ( f )( ξ ) K γ,δ ( ξ )) h ( ξ ) dµ α ( ξ ) = 0 , and we deduce that F W ( f γ,δ )( ξ ) = F W ( f )( ξ ) K γ,δ ( ξ ) , ξ ∈ R d +1+ . We now return to the proof of the Theorem 4.1.
Proof. of Theorem 4.1 . At first, from Theorem 4.4, the function f γ,δ belongs to L α ( R d +1+ ) . Next, according to Plancherel’s formula for Weinstein transform (2.11)and Theorem 4.4, we get k f γ,δ − f k α, = Z R d +1+ |F W ( f γ,δ − f )( ξ ) | α ( ξ )= Z R d +1+ |F W ( f )( K γ,δ ( ξ ) − | α ( ξ )= Z R d +1+ |F W ( f ) | | (1 − K γ,δ ( ξ )) | α ( ξ ) . Another time, according to Theorem 4.4 we have for almost all ξ ∈ R d +1+ lim ( γ,δ ) → (0 , ∞ ) |F W ( f ) | | (1 − K γ,δ ( ξ )) | = 0 , and there exists a positive constant C such that lim ( γ,δ ) → (0 , ∞ ) |F W ( f ) | | (1 − K γ,δ ( ξ )) | ≤ C |F W ( f ) | , with |F W ( f ) | is in L α ( R d +1+ ) . Thus, we conclude the relation (4.2) from the domi-nated convergence theorem. wo-wavelet theory in Weinstein setting References [1] Z. Ben Nahia and N. Ben Salem. Spherical harmonics and applications associatedwith the Weinstein operator. In
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