Certain properties of continuous fractional wavelet transform on Hardy space and Morrey space
aa r X i v : . [ m a t h . F A ] F e b Certain properties of continuous fractional wavelet transform on Hardyspace and Morrey space
Amit K. Verma a ∗ , Bivek Gupta b a,b Department of Mathematics, IIT Patna, Bihta, Patna 801103.
February 10, 2021
Abstract
In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its propertiesin Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.
Keywords : Fractional Fourier Transform; Continuous Fractional Wavelet Transform; Hardy Space; Morrey Space
AMS Subject Classification : 42B10, 42C40, 46E30
Even though the classical wavelet transform (WT) serves as a powerful tool in signal processing and analysis, itsanalyzing capability is limited to the time-frequency plane. Fractional Fourier transform (FrFT)([4],[14],[16]) givesthe fractional Fourier domain (FrFD) frequency content of the signal, but it fails in giving the local information ofthe signal. Mendlovic et al. ([6]), first introduced the FrWT to deal with the optical signals. They first derive thefractional spectrum of the signal by using the FrFT and performed the WT of the fractional spectrum. But thetransform defined in such a way, fails in giving the information about the local property of the signal, since the FrFTgives the fractional frequency of the signal during the entire duration of the signal rather than for a particular time,and the fractional spectrum of the signal cannot be ascertained when those fractional frequencies exist.The novel Fractional wavelet transform (FrWT) based on fractional convolution was proposed by Shi et al. ([12]).They studied basic properties of the FrWT like inner product theorem, Parseval’s relation and inversion formula forthe function in L ( R ) . Prasad et al. ([2]) studied some properties of FrFT such as Riemann-Lebesgue lemma. Also,they extended the inner product theorem of the CFrWT, studied in [12], in the context of two fractional wavelets.Dai et al. ([9]) proposed a new type of FrWT and obtained the associated multiresolution analysis (MRA). This ismore general than the transforms defined in [2] and [12]. It displays the time and FrFD-frequency information jointlyin the time-FrFD-frequency plane.Luchko et al. ([28]) introduced a new FrFT and implemented this theory on the Lizorkin space, and also discussedmany important results involving fractional derivatives. To know more about the FrFT reader may follow [1],[11]. In([10, 13]), authors studied the new theory of FrWT, associated with the FrFT given in [1, 11, 28], and obtained someof its properties like inner product relation, inversion formula, etc. They also discussed MRA associated with thisFrWT, along with the construction of the orthogonal fractional wavelets. This theory can also be used in the studyof quantum mechanics, signal processing and other areas of science and engineering.Several important function spaces like Besov, Sobolev, Holder, Zygmund, BMO, etc are given characterization interms of wavelets involved in the classical WT ([7],[15]). WT has also been studied in various function spaces and thespaces of distributions ([25],[17],[19]). Chuong et al. [5] studied the boundedness of the WT on the Besov, BMO andHardy spaces. Furthermore, for the compactly supported basic wavelet, the boundedness of the WT is also establishedon the weighted Besov space and weighted BMO space associated with the tempered weight function. In the recentyears, Prasad and Kumar ([20],[21],[22]) discussed the CFrWT on the generalized weighted Sobolev spaces and somefunction spaces and obtained its boundedness. Not only that, the WT and CFrWT have also been studied by manyauthors on some spaces of test functions, like Gelfand-Shilov spaces ([18],[26],[23],[24]). Based on the convolutionof linear canonical transform (LCT)([27]), Guo et al. ([8]) proposed a linear canonical wavelet transform (LCWT),which is a generalization of the transform studied in [2]. The authors also proved the continuity of this transform onsome space of test functions and the generalized Sobolev space. To know more about the literature, reader can readthe references and the references therein. ∗ Corresponding author email: [email protected] H ( R ) and L ,νM ( R ) and also study the dependence of the CFrWT on its wavelet and the argumentfunction by determining the H ( R ) and L ,νM ( R ) − distance of two CFrWTs with different fractional wavelets andargument functions.The organization of the paper is as follows: In section 2, we recall some basic definitions and results. In section 3,we have derived the orthogonality relation, the reconstruction formula and characterized the range of the transformin the context of two fractional wavelets. Also, we have derived the formulas for the CFrWT when the argumentfunction or fractional wavelet is a convolution or correlation of two functions. Section 4 is further divided into twosubsections. In each of these two subsections the boundedness of CFrWT on Hardy space H ( R ) and Morrey space L ,νM ( R ) along with its approximation properties are studied. Finally, we end this paper by the conclusions in section5. In this section we recall some existing definitions and results that we be used in this paper.
Definition 2.1.
The convolution of complex-valued measurable functions f and g defined on R , is given by( f ⋆ g )( x ) = Z R f ( u ) g ( x − u ) du, x ∈ R (1)whenever the integral is well-defined. Definition 2.2.
The correlation of complex-valued measurable functions f and g defined on R , is given by( f ◦ g )( x ) = Z R f ( u ) g ( x + u ) du, x ∈ R (2)whenever the integral is well-defined. Definition 2.3. [10] The fractional Fourier transform (FrFT), of real order θ (0 < θ ≤ , of a function f ∈ L ( R ) isdefined by ( F θ f )( ξ ) = Z R e − i (sgn ξ ) | ξ | θ t f ( t ) dt, ξ ∈ R . (3)For θ = 1 , the fractional Fourier transform defined in (3) reduces to the classical Fourier transform.The corresponding inverse fractional Fourier transform is defined as follows: f ( t ) = 12 πθ Z R e i (sgn ξ ) | ξ | θ t ( F θ f )( ξ ) | ξ | θ − dξ. Lemma 2.1.
Let ψ ∈ L ( R ) , then ( F θ ψ a,b,θ )( ξ ) = | a | θ e − i (sgn ξ ) | ξ | θ b ( F θ ψ )( aξ ) , (4)where ψ a,b,θ ( t ) = 1 | a | θ ψ (cid:18) t − b (sgn a ) | a | θ (cid:19) , a, b ∈ R . (5) Proof.
Refer [10, Page 7].
Before we begin with the definition of the CFrWT we recall the definition of the fractional wavelet given by Srivastavaet al. ([10]). We then prove a theorem that helps in constructing a family of fractional wavelets from a given one.2 efinition 3.1.
A fractional wavelet is a non-zero function ψ ∈ L ( R ) ∩ L ( R ) , satisfying C ψ,θ := Z R | F θ ψ ( ξ ) | | ξ | dξ < ∞ . (6)Now, we prove the following theorem which indicate the construction of a family of fractional wavelets from agiven one. Theorem 3.1.
Let ψ be a fractional wavelet and φ be a function in L ( R ) , then ψ ⋆ φ and ψ ◦ φ are also fractionalwavelets. Proof.
Since ψ ∈ L ( R ) ∩ L ( R ) and φ ∈ L ( R ) , ψ ⋆ φ ∈ L ( R ) ∩ L ( R ) . Now, Z R | F θ ( ψ ⋆ φ )( ξ ) | | ξ | dξ = Z R | ( F θ ψ )( ξ ) | | ( F θ φ )( ξ ) | | ξ | dξ, since F θ ( ψ ⋆ φ )( ξ ) = ( F θ ψ )( ξ )( F θ φ )( ξ ) ≤ k φ k L ( R ) Z R | ( F θ ψ )( ξ ) | | ξ | dξ, since k F θ φ k L ∞ ( R ) ≤ k φ k L ( R ) = k φ k L ( R ) C ψ,θ . Since ψ is a fractional wavelet and φ ∈ L ( R ) , Z R | F θ ( ψ ⋆ φ )( ξ ) | | ξ | dξ < ∞ . Hence by definition 3.1, ψ ⋆ φ is a fractional wavelet. Similarly, it can be shown that ψ ◦ φ is also a fractional wavelet.This completes the proof. Definition 3.2. [10] The CFrWT of f with respect to a fractional wavelet ψ is defined by (cid:0) W θψ f (cid:1) ( b, a ) = Z R f ( t ) ψ a,b,θ ( t ) dt, a, b ∈ R , (7)provided the integral is well-defined. Here ψ a,b,θ is given by equation (5).We derive some new results of the CFrWT and also generalized some existing results in the context of two fractionalwavelets. We omit the proof which are similar as in [10] and [13].If f, g ∈ L ( R ) are orthogonal then the image W θψ f and W θψ g are also orthogonal in L (cid:18) R × R , dbda | a | θ +1 (cid:19) . This factis observed by the orthogonality relation for the CFrWT given in [10]. But this relation is not enough to conclude theorthogonality of W θψ f and W θφ g for two different fractional wavelets ψ and φ . So in this regard we introduce a moregeneral version of orthogonality relation. We also derive reconstruction formula and characterized its range. For thecase ψ = φ , our results coincide with the results in [10]. Theorem 3.2. (Orthogonality relation) If the fractional wavelets φ and ψ satisfies Z R | ( F θ φ )( u ) | | ( F θ ψ )( u ) | | u | du < ∞ , (8)then for f, g ∈ L ( R ) , Z R Z R (cid:0) W θφ f (cid:1) ( b, a ) (cid:16) W θψ g (cid:17) ( b, a ) dbda | a | θ +1 = C φ,ψ,θ h f, g i L ( R ) , where C φ,ψ,θ = Z R ( F θ φ )( u )( F θ ψ )( u ) 1 | u | du. (9) Proof.
The proof is similar as in [10, 13].
Corollary 3.1.
Let φ, ψ be two fractional wavelets and are such that they satisfies the hypothesis of theorem 3.2. Iffurther C φ,ψ,θ = 0 , where C φ,ψ,θ is given by equation (9). Then W θφ (cid:0) L ( R ) (cid:1) and W θψ (cid:0) L ( R ) (cid:1) are orthogonal. Proof.
Proof follows from results of theorem 3.2. 3 heorem 3.3. (Reconstruction formula) Let f ∈ L ( R ) and φ, ψ be two fractional wavelets satisfying (8) and C φ,ψ,θ , as defined in (9), is non-zero. Then f ( t ) = 1 C φ,ψ,θ Z R Z R ψ a,b,θ ( t ) (cid:0) W θφ f (cid:1) ( b, a ) dbda | a | θ +1 . Proof.
The proof is similar as in [10, 13].
Theorem 3.4. (Characterization of the range) Let C φ,ψ,θ as defined in (9), for two fractional wavelets satisfying (8),is non-zero. Then F ∈ L (cid:18) R × R , dbda | a | θ +1 (cid:19) is a CFrWT, with respect to φ, of some f ∈ L ( R ) iff F ( b , a ) = Z R Z R F ( b, a ) K φ,ψ,θ ( b , a ; b, a ) dbda | a | θ +1 , ( b , a ) ∈ R × R , (10)where K φ,ψ,θ is the reproducing kernel given by K φ,ψ,θ ( b , a ; b, a ) = 1 C φ,ψ,θ Z R ψ a,b,θ ( t ) φ a ,b ,θ ( t ) dt. (11)Moreover, in such a case the kernel is pointwise bounded: | K φ,ψ,θ ( b , a ; b, a ) | ≤ C φ,ψ,θ k φ k L ( R ) k ψ k L ( R ) . Proof.
The proof is similar as in [10, 13].Now, we prove the theorem that gives the formula for the wavelet transform of the convolution and correlation oftwo functions.
Theorem 3.5.
Let f ∈ L ( R ) , g ∈ L ( R ) and ψ be a fractional wavelet, then (cid:0) W θψ ( f ⋆ g ) (cid:1) ( b, a ) = (cid:0) f ( · ) ⋆ ( W θψ g )( · , a ) (cid:1) ( b )and (cid:0) W θψ ( f ◦ g ) (cid:1) ( b, a ) = (cid:0) f ( · ) ◦ ( W θψ g )( · , a ) (cid:1) ( b ) . Proof.
We have (cid:0) W θψ ( f ⋆ g ) (cid:1) ( b, a ) = Z R ( f ⋆ g )( t ) ψ a,b,θ ( t ) dt = Z R (cid:26)Z R f ( y ) g ( t − y ) dy (cid:27) | a | θ ψ (cid:18) t − b (sgn a ) | a | θ (cid:19) dt, using definition 2.1= Z R f ( y ) (cid:26)Z R g ( t ) ψ a,b − y,θ dt (cid:27) dy = Z R f ( y )( W θψ g )( b − y, a ) dy. Therefore, (cid:0) W θψ ( f ⋆ g ) (cid:1) ( b, a ) = (cid:0) f ( · ) ⋆ ( W θψ g )( · , a ) (cid:1) ( b ) . Similarly, it can be shown that (cid:0) W θψ ( f ◦ g ) (cid:1) ( b, a ) = (cid:0) f ( · ) ◦ ( W θψ g )( · , a ) (cid:1) ( b ) . This completes the proof.The following theorem gives the expression of the CFrWT when the fractional wavelet associated with the transformis the convolution or the correlation of two functions.
Theorem 3.6.
Let f ∈ L ( R ) g ∈ L ( R ) and ψ be a fractional wavelet, then (cid:0) W θf⋆ψ g (cid:1) ( b, a ) = 1 | a | θ (cid:18) f (cid:18) · (sgn a ) | a | θ (cid:19) ◦ ( W θψ g )( · , a ) (cid:19) ( b )and (cid:0) W θf ◦ ψ g (cid:1) ( b, a ) = 1 | a | θ (cid:18) f (cid:18) · (sgn a ) | a | θ (cid:19) ⋆ ( W θψ g )( · , a ) (cid:19) ( b ) . roof. Since f ∈ L ( R ) and ψ is a fractional wavelet, by theorem 3.1, f ⋆ ψ is a wavelet.Now, (cid:0) W θf⋆ψ g (cid:1) ( b, a ) = Z R g ( t )( f ⋆ ψ ) a,b,θ ( t ) dt = Z R g ( t ) ( | a | θ Z R f ( y ) ψ (cid:18) t − b (sgn a ) | a | θ − y (cid:19) dy ) dt = Z R f ( y ) Z R g ( t ) 1 | a | θ ψ t − ( b + y (sgn a ) | a | θ )(sgn a ) | a | θ ! dt dy = Z R f ( y )( W θψ g )( b + (sgn a ) | a | θ y, a ) dy. Therefore, (cid:0) W θf⋆ψ g (cid:1) ( b, a ) = 1 | a | θ (cid:18) f (cid:18) · (sgn a ) | a | θ (cid:19) ◦ ( W θψ g )( · , a ) (cid:19) ( b ) . Again by theorem 3.1, f ◦ ψ is a wavelet. Proceeding similarly as above it can be shown that (cid:0) W θf ◦ ψ g (cid:1) ( b, a ) = 1 | a | θ (cid:18) f (cid:18) · (sgn a ) | a | θ (cid:19) ⋆ ( W θψ g )( · , a ) (cid:19) ( b ) . This completes the proof.
Theorem 3.7.
Let f, g ∈ L ( R ) and φ, ψ be two fractional wavelets, then Z R | b | θ − ( W θφ f )( b, a )( W θψ g )( b, a ) db = | a | θ π θ h P θ , Q θ i L ( R ) , where P θ ( ξ ) = | ξ | θ − ( F θ f )( ξ )( F θ φ )( aξ ) and Q θ ( ξ ) = | ξ | θ − ( F θ g )( ξ )( F θ ψ )( aξ ) . Proof. Z R | b | θ − ( W θφ f )( b, a )( W θψ g )( b, a ) db = Z R | b | θ − h f, φ a,b,θ i L ( R ) h g, ψ a,b,θ i L ( R ) db Using Parseval’s formula [10, Theorem 1], we have Z R | b | θ − ( W θφ f )( b, a )( W θψ g )( b, a ) db = (cid:18) πθ (cid:19) Z R | b | θ − D | · | θ − ( F θ f )( · ) , ( F θ φ a,b,θ )( · ) E L ( R ) D | · | θ − ( F θ g )( · ) , ( F θ ψ a,b,θ )( · ) E L ( R ) db = (cid:18) πθ (cid:19) Z R | b | θ − (cid:18)Z R | ξ | θ − ( F θ f )( ξ )( F θ φ a,b,θ )( ξ ) dξ (cid:19) (cid:18)Z R | ω | θ − ( F θ f )( ω )( F θ ψ a,b,θ )( ω ) dω (cid:19) db = (cid:18) πθ (cid:19) | a | θ Z R | b | θ − (cid:18)Z R | ξ | θ − ( F θ f )( ξ ) e − i (sgn ξ ) | ξ | θ b ( F θ φ )( aξ ) dξ (cid:19) × (cid:18)Z R | ω | θ − ( F θ g )( ω ) e − i (sgn( ω )) | ω | θ b ( F θ ψ )( aω ) dω (cid:19) db = (cid:18) πθ (cid:19) | a | θ Z R | b | θ − (cid:18)Z R e − i (sgn ξ ) | ξ | θ b | ξ | θ − ( F θ f )( ξ )( F θ φ )( aξ ) dξ (cid:19) × (cid:18)Z R e − i (sgn( ω )) | ω | θ b | ω | θ − ( F θ g )( ω )( F θ ψ )( aω ) dω (cid:19) db = (cid:18) πθ (cid:19) | a | θ Z R | b | θ − (cid:18)Z R e − i (sgn ξ ) | ξ | θ b P θ ( ξ ) dξ (cid:19) (cid:18)Z R e − i (sgn( ω )) | ω | θ b Q θ ( ω ) dω (cid:19) db = (cid:18) πθ (cid:19) | a | θ Z R | b | θ − (cid:0) F θ P θ (cid:1) ( b ) (cid:0) F θ Q θ (cid:1) ( b ) db. (12)5sing [10, Theorem 1] in equation (12), we get Z R | b | θ − ( W θφ f )( b, a )( W θψ g )( b, a ) db = (cid:18) πθ (cid:19) | a | θ D | · | θ − (cid:0) F θ Q θ (cid:1) ( · ) , (cid:0) F θ P θ (cid:1) ( · ) E L ( R ) = (cid:18) πθ (cid:19) | a | θ (cid:10) Q θ , P θ (cid:11) L ( R ) = (cid:18) πθ (cid:19) | a | θ h P θ , Q θ i L ( R ) . This completes the proof. & Morrey spaces
In this section we consider the normalized form of the operator W θψ i.e., L θψ := √ C ψ,θ W θψ and study some of itsproperties on Hardy space and Morrey space. The purpose of this section is to establish the boundedness of theCFrWT on these spaces and to study the dependence of the CFrWT on its fractional wavelet and its argumentfunction via H ( R ) and L ,νM ( R ) − distance estimate of two CFrWTs with different fractional wavelets of differentargument functions. Definition 4.1. ([5]) The Hardy space H ( R ) , defined by H ( R ) = (cid:26) f ∈ L ( R ) : Z R sup t> | ( f ⋆ η t )( x ) | dx < ∞ (cid:27) , is a Banach space normed by k f k H ( R ) = Z R sup t> | ( f ⋆ η t )( x ) | dx, (13)where η is a function in Schwartz space such that R R η ( x ) dx = 0 and η t ( x ) = t η ( xt ) , t > , x ∈ R . We now study some properties of the CFrWT on the Hardy space H ( R ) . To establish the boundedness of theCFrWT on the Hardy space we need to prove the following lemma.
Lemma 4.1.
Let a ∈ R − { } , f ∈ H ( R ) and ψ be a fractional wavelet, then ( L θψ f )( · , a ) ∈ L ( R ) , where L θψ f denotesthe CFrWT of f. Proof.
For fix a ∈ R − { } , ( L θψ f )( b, a ) is a function of b and is such that | ( L θψ f )( b, a ) | ≤ p C ψ,θ Z R | f ( u ) || ψ a,b,θ ( u ) | dt = 1 p C ψ,θ Z R | f ( u ) | | a | θ (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:18) u − b (sgn a ) | a | θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) du = | a | θ p C ψ,θ Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ x + b (cid:17)(cid:12)(cid:12)(cid:12) | ψ ( x ) | dx. Therefore, Z R | ( L θψ f )( b, a ) | db ≤ | a | θ p C ψ,θ Z R | ψ ( x ) | (cid:18)Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ x + b (cid:17)(cid:12)(cid:12)(cid:12) db (cid:19) dx = | a | θ p C ψ,θ k ψ k L ( R ) k f k L ( R ) . Hence, it follows that ( L θψ f )( · , a ) ∈ L ( R ) . Theorem 4.1.
Let a ∈ R − { } , then the operator L θψ : H ( R ) → H ( R ) defined by f → ( L θψ f )( · , a ) is bounded.Furthermore, k ( L θψ f )( · , a ) k H ( R ) ≤ | a | θ p C ψ,θ k ψ k L ( R ) k f k H ( R ) . roof. From definition of L θψ , we get( L θψ f )( b, a ) = | a | θ p C ψ,θ Z R f (cid:16) (sgn a ) | a | θ x + b (cid:17) ψ ( x ) dx. Now, (( L θψ f )( · , a ) ⋆ η t ( · ))( b ) = Z R ( L θψ f )( b − y, a ) η t ( y ) dy = Z R | a | θ p C ψ,θ (cid:18)Z R f (cid:16) (sgn a ) | a | θ x + b − y (cid:17) ψ ( x ) dx (cid:19) η t ( y ) dy = | a | θ p C ψ,θ Z R ψ ( x ) (cid:18)Z R f (cid:16) (sgn a ) | a | θ x + b − y (cid:17) η t ( y ) dy (cid:19) dx = | a | θ p C ψ,θ Z R ( f ⋆ η t ) (cid:16) sgn a | a | θ x + b (cid:17) ψ ( x ) dx. Therefore, k ( L θψ f )( · , a ) k H ( R ) = Z R sup t> | (cid:0) ( L θψ f )( · , a ) ⋆ η t ( · ) (cid:1) ( b ) | db ≤ | a | θ p C ψ,θ Z R | ψ ( x ) | (cid:18)Z R sup t> | ( f ⋆ η t )((sgn a ) | a | θ x + b ) | db (cid:19) dx. = | a | θ p C ψ,θ k ψ k L ( R ) k f k H ( R ) . This completes the proof.
Corollary 4.1. If a ∈ R − { } , f ∈ H ( R ) and ψ is a fractional wavelet, then k ( L θψ f )( · , a ) k H ( R ) = O (cid:16) | a | θ (cid:17) . Proof.
By using the theorem 4.1, we get the result.We will now determine the H ( R ) − distance of two CFrWTs with different fractional wavelets and different argu-ment functions to study the dependence of the transform on its fractional wavelet and its argument. Theorem 4.2.
Let f, g ∈ H ( R ) and φ, ψ be two fractional wavelets then, for a ∈ R − { } , k ( L θφ f )( · , a ) − ( L θψ g )( · , a ) k H ( R ) ≤ | a | θ k f k H ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ p C φ,θ − ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) + k f − g k H ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . Proof.
We have k ( L θφ f )( · , a ) − ( L θψ g )( · , a ) k H ( R ) ≤ k ( L θφ f )( · , a ) − ( L θψ f )( · , a ) k H ( R ) + k ( L θψ f )( · , a ) − ( L θψ g )( · , a ) k H ( R ) . (14)Now, ( L θφ f )( b, a ) − ( L θψ f )( b, a ) = 1 | a | θ Z R f ( t ) ( p C φ,θ φ (cid:18) t − b (sgn a ) | a | θ (cid:19) − p C ψ,θ ψ (cid:18) t − b (sgn a ) | a | θ (cid:19)) dt = | a | θ Z R f (cid:16) (sgn a ) | a | θ x + b (cid:17) φ ( x ) p C φ,θ − ψ ( x ) p C ψ,θ ! dx. (cid:8)(cid:0) ( L θφ f )( · , a ) − ( L θφ f )( · , a ) (cid:1) ⋆ η t ( · ) (cid:9) ( b ) = Z R (cid:8)(cid:0) ( L θφ f )( b − y, a ) − ( L θφ f )( b − y, a ) (cid:1)(cid:9) η t ( y ) dy = Z R | a | θ Z R f (cid:16) (sgn a ) | a | θ x + b − y (cid:17) φ ( x ) p C φ,θ − ψ ( x ) p C ψ,θ ! dx ! η t ( y ) dy = | a | θ Z R φ ( x ) p C φ,θ − ψ ( x ) p C ψ,θ ! (cid:18)Z R f (cid:16) (sgn a ) | a | θ x + b − y (cid:17) η t ( y ) dy (cid:19) dx = | a | θ Z R ( f ⋆ η t ) (cid:16) (sgn a ) | a | θ x + b (cid:17) φ ( x ) p C φ,θ − ψ ( x ) p C ψ,θ ! dx. Therefore, k ( L θφ f )( · , a ) − ( L θψ f )( · , a ) k H ( R ) = Z R sup t> (cid:12)(cid:12)(cid:8)(cid:0) ( L θφ f )( · , a ) − ( L θφ f )( · , a ) (cid:1) ⋆ η t ( · ) (cid:9) ( b ) (cid:12)(cid:12) db ≤ | a | θ Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ( x ) p C φ,θ − ψ ( x ) p C ψ,θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18)Z R sup t> | ( f ⋆ η t ) (cid:16) (sgn a ) | a | θ x + b (cid:17) | db (cid:19) dx = | a | θ k f k H ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ p C φ,θ − ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . (15)Similarly, it can be shown that k ( L θψ f )( · , a ) − ( L θψ g )( · , a ) k H ( R ) ≤ | a | θ k f − g k H ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) (16)From equations (14),(15) and (16) the result follows. Remark 4.1.
For θ = 1 the theorem 4.1 and theorem 4.2 coincide with those studied in [5]. Definition 4.2. ([3]) The Morrey space L p,νM ( R ) , with 1 ≤ p < ∞ and 0 ≤ ν ≤ , defined by L p,νM ( R ) = f ∈ L ploc ( R ) : sup x ∈ R r> r ν Z B ( x,r ) | f ( t ) | p dt ! < ∞ , is a Banach space normed by k f k L p,νM ( R ) = sup x ∈ R r> r ν Z B ( x,r ) | f ( t ) | p dt ! p . We now study some properties of the CFrWT on the Morrey space L ,νM ( R ) . Before we establish the boundednessof the CFrWT on the Morrey space we prove the following lemma.
Lemma 4.2.
Let a ∈ R − { } , f ∈ L ,νM ( R ) and ψ be a compactly supported fractional wavelet, then ( L θψ f )( · , a ) is in L loc ( R ) . Proof.
We have ( L θψ f )( b, a ) = 1 p C ψ,θ Z R f ( t ) ψ a,b,θ ( t ) dt. This implies, | ( L θψ f )( b, a ) | ≤ p C ψ,θ Z R | f ( t ) || ψ a,b,θ ( t ) | dt. | ( L θψ f )( b, a ) | ≤ | a | θ p C ψ,θ Z R | f ( t ) || ψ (cid:18) t − b (sgn a ) | a | θ (cid:19) | dt = | a | θ p C ψ,θ Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ x + b (cid:17)(cid:12)(cid:12)(cid:12) | ψ ( x ) | dx. (17)Let K be a compact set in R . We have Z K | ( L θψ f )( b, a ) | db ≤ | a | θ p C ψ,θ Z K Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ x + b (cid:17)(cid:12)(cid:12)(cid:12) | ψ ( x ) | dxdb = | a | θ p C ψ,θ Z R | ψ ( x ) | (cid:18)Z K (cid:12)(cid:12)(cid:12) f (cid:16) sgn a | a | θ x + b (cid:17)(cid:12)(cid:12)(cid:12) db (cid:19) dx = | a | θ p C ψ,θ Z R | ψ ( x ) | (cid:18)Z A | f ( y ) | dy (cid:19) dx, where A = (sgn a ) | a | θ x + K ⊂ (sgn a ) | a | θ supp ψ + K. Since f ∈ L loc ( R ) and A is bounded, we have Z K | ( L θψ f )( b, a ) | db ≤ | a | θ p C ψ,θ C k ψ k L ( R ) , for some C ≥ . Thus, it follow that ( L θψ f )( · , a ) is in L loc ( R ) . Theorem 4.3.
Let a ∈ R − { } and ψ be a compactly supported fractional wavelet, then the operator L θψ : L ,νM ( R ) → L ,νM ( R ) defined by f → ( L θψ f )( · , a ) is bounded. Furthermore, k ( L θψ f )( · , a ) k L ,νM ( R ) ≤ | a | θ p C ψ,θ k ψ k L ( R ) k f k L ,νM ( R ) . Proof.
We have k ( L θψ f )( · , a ) k L ,νM ( R ) = sup x ∈ R r> r ν Z B ( x,r ) | ( L θψ f )( b, a ) | db ! . (18)Now using equation (17), we get1 r ν Z B ( x,r ) | ( L θψ f )( b, a ) | db ≤ | a | θ r ν p C ψ,θ Z B ( x,r ) (cid:18)Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ u + b (cid:17)(cid:12)(cid:12)(cid:12) | ψ ( u ) | du (cid:19) db = | a | θ r ν p C ψ,θ Z R | ψ ( u ) | Z B ( x,r ) (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ u + b (cid:17)(cid:12)(cid:12)(cid:12) db ! du = | a | θ p C ψ,θ Z R | ψ ( u ) | r ν Z B (cid:16) (sgn a ) | a | θ u + x,r (cid:17) | f ( z ) | d z ! du. (19)Also 1 r ν Z B (cid:16) (sgn a ) | a | θ u + x,r (cid:17) | f ( z ) | d z ≤ k f k L ,νM ( R ) . (20)From equations (19) and (20), we have1 r ν Z B ( x,r ) | ( L θψ f )( b, a ) | db ≤ | a | θ p C ψ,θ k f k L ,νM ( R ) k ψ k L ( R ) , which gives sup x ∈ R r> r ν Z B ( x,r ) | ( L θψ f )( b, a ) | db ! ≤ | a | θ p C ψ,θ k f k L ,νM ( R ) k ψ k L ( R ) . (21)9rom (18) and (21), we have k ( L θψ f )( · , a ) k L ,νM ( R ) ≤ | a | θ p C ψ,θ k ψ k L ( R ) k f k L ,νM ( R ) . This completes the proof.
Corollary 4.2.
Let a ∈ R − { } , f ∈ L ,νM ( R ) , and ψ be a compactly supported fractional wavelet, then k ( L θψ f )( · , a ) k L ,νM ( R ) = O (cid:16) | a | θ (cid:17) . Proof.
Follows from theorem 4.3.We will now determine the L ,νM ( R ) − distance of two CFrWTs with different fractional wavelets and differentargument functions to study the dependence of the transform on its fractional wavelet and its argument. Theorem 4.4.
Let f, g ∈ L ,νM ( R ) and φ, ψ be two compactly supported fractional wavelets. Then for a ∈ R − { } , k ( L θφ f )( · , a ) − ( L θψ g )( · , a ) k L ,νM ( R ) ≤ | a | θ k f k L ,νM ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ p C φ,θ − ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) + k f − g k L ,νM ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . Proof.
We have k ( L θφ f )( · , a ) − ( L θψ g )( · , a ) k L ,νM ( R ) ≤ k ( L θφ f )( · , a ) − ( L θψ f )( · , a ) k L ,νM ( R ) + k ( L θψ f )( · , a ) − ( L θψ g )( · , a ) k L ,νM ( R ) . (22)Here, k ( L θφ f )( · , a ) − ( L θψ f )( · , a ) k L ,νM ( R ) = sup x ∈ R r> r ν Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R p C φ,θ f ( y ) φ a,t,θ ( y ) − p C ψ,θ f ( y ) ψ a,t,θ ( y ) ! dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ! . (23)Now, 1 r ν Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R p C φ,θ f ( y ) φ a,t,θ ( y ) − p C ψ,θ f ( y ) ψ a,t,θ ( y ) ! dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ r ν Z B ( x,r ) Z R | f ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ a,t,θ ( y ) − p C ψ,θ ψ a,t,θ ( y ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dydt = 1 r ν Z B ( x,r ) (cid:18) Z R | a | θ | f ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ (cid:18) y − t (sgn a ) | a | θ (cid:19) − p C ψ,θ ψ (cid:18) y − t (sgn a ) | a | θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy (cid:19) dt = | a | θ r ν Z B ( x,r ) Z R (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ z + t (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ ( z ) − p C ψ,θ ψ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz ! dt = | a | θ r ν Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ ( z ) − p C ψ,θ ψ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ( x,r ) (cid:12)(cid:12)(cid:12) f (cid:16) (sgn a ) | a | θ z + t (cid:17)(cid:12)(cid:12)(cid:12) dt ! dz = | a | θ Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ ( z ) − p C ψ,θ ψ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ν Z B (cid:16) (sgn a ) | a | θ z + x,r (cid:17) | f ( u ) | du ! dz. Using equation (20), we get1 r ν Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R p C φ,θ f ( y ) φ a,t,θ ( y ) − p C φ,θ f ( y ) ψ a,t,θ ( y ) ! dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ | a | θ k f k L ,νM ( R ) Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C φ,θ φ ( z ) − p C ψ,θ ψ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz. Therefore,sup x ∈ R r> r ν Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:0) f ( y ) φ a,t,θ ( y ) − f ( y ) ψ a,t,θ ( y ) (cid:1) dy (cid:12)(cid:12)(cid:12)(cid:12) dt ! ≤ | a | θ k f k L ,νM ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ p C φ,θ − ψ p C φ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . (24)10rom equation (23) and equation (24), it follows that k ( L θφ f )( · , a ) − ( L θψ f )( · , a ) k L ,νM ( R ) ≤ | a | θ k f k L ,νM ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ p C φ,θ − ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . (25)Similarly, it can be shown that k ( L θψ f )( · , a ) − ( L θψ g )( · , a ) k L ,νM ( R ) ≤ | a | θ k f − g k L ,νM ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ p C ψ,θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) (26)From equations (22), (25) and (26) the theorem follows immediately. In this paper, we have studied CFrWT which as a generalization of the classical wavelet transform, reduces to classicalwavelet transform for θ = 1 . In section 2, we have introduced some basic definitions and results. In section 3, we havegeneralized the existing result like orthogonality relation, reconstruction formula and the range theorem, in [10, 13], inthe context of two fractional wavelets. Also we have derived the formulas for the CFrWT when the argument functionor fractional wavelet is a convolution or correlation of two functions. Lastly, in section 4, the boundedness of CFrWTon Hardy space H ( R ) and Morrey space L ,νM ( R ) along with its approximation property are established. The work is partially supported by UGC File No. 16-9(June 2017)/2018(NET/CSIR), New Delhi, India.
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