# Some Properties of Fractional Boas Transforms of Wavelets

Authors: Nikhil Khanna; A. Zothansanga; S. K. Kaushik; Dilip Kumar
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce fractional Boas transforms and discuss some of their properties. We also introduce the notion of wavelets associated with fractional Boas transforms and give some results related to their vanishing moments. Finally, a comparative study of Hilbert transforms and fractional Boas transforms is done.


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## Body

## 1. Introduction

The ordinary transformations have been replaced with the fractional ones, which play a significant role in information processing. This transition has occurred naturally due to its various applications in quantum mechanics and optics and purvey us a tool, to characterize a signal completely, in the form of the fractional order, which happens to be the new degree of freedom or an appended parameter for encoding. Among all fractional transforms, the fractional Fourier transform (FRFT), a generalization of the Fourier transform, has been widely studied. In the last three decades, the fractional Fourier transform (FRFT) has played a substantial role in signal processing, optical systems, and quantum physics [1–3]. Another important variation of FRFT is canonical fractional FT [4], which is very effective in optical information processing, since it is easily achievable using simple optical setups and it renders a mere rotation of the two important phase-space distributions: the Wigner distribution and the ambiguity function. The canonical fractional FT was first introduced in [5] more than 90 years ago, which was later improvised by various researchers for applications in quantum mechanics [3], optics [6], and signal processing [2]. Another fractional transform, the complex fractional FT, closely related to the canonical fractional FT has been introduced in [7]. The generalization of Legendre transformation to the fractional Legendre transformation was formulated on the lines of FRFT in [8]. Based on the approach of eigenfunction kernel decomposition similar to the one given in [9], some new fractional integral transforms, including the fractional Mellin transform, a fractional transform associated with the Jacobi polynomials, a Riemann-Liouville fractional derivative operator, and a fractional Riemann-Liouville integral, have been proposed in [10]. In the analogy with canonical fractional Fourier and Hankel transforms, the fractional Laplace and Barut-Girardello transforms have been introduced in [11]. The applications of these transforms in science and engineering are still subject of research.In order to process one-sided signals, fractional cosine (CT) and sine transforms (ST) were employed. Their digital application along with that of fractional Hartley transforms (HaT) was discussed in [9, 12]. One may refer to [13] for image watermarking scheme classified on the basis of variant fractional transforms such as fractional discrete FT, HaT, CT, and ST.Gabor [14] introduced the Hilbert transform (HT), an important tool in optics, by constructing an analytic signal from a one-dimensional signal. In 1950, its optical implementation was performed in two different approaches, when Kastler [15] employed it for image processing, primarily for edge enhancement, and Wolter [16] utilized it for spectroscopy. Further advancements in HT can be seen in [17]. The efficacy of HT was raised with the origination of fractional Hilbert transform (FRHT) by Lohmann et al. [18] in 1996, which proffered an additional degree of freedom in the form of a fractional order. Two ways of fractional HT were proposed, which resulted in increase of fractional order and provided improvements in image processing. The first method was based on a spatial filter with a fractional parameter and the other was based on the FRFT. For details, one may see [6, 19]. The FRHT for two-dimensional objects was presented in [20]. Later, Tseng and Pei [13] formulated an SSB modulation by considering the parameter of the fractional phase in FRHT as a secret key. Zayed [21] generalized the HT in a distinctive way and suppressed the negative frequency component of the signal in the FRF domain to obtain the analytic part of a signal. Using FRHT, Cusmariu [22] proposed three possible versions of fractional analytic signals. Tao [23] employed FRFT and FRHT and presented a secured SSB modulation system.Paley and Wiener [24] studied a class of square integrable functions whose Fourier transforms vanish outside the intervals −κ,κ in great details. This class denoted by Bκ2 was later named by Paley-Weiner class of entire functions and a member of this class is a function band limited to −κ,κ. Contrary to this study, Boas was curious in examining the properties of square integrable functions whose Fourier transforms vanish on −κ,κ, that is, the class Bℝ−−κ,κ. Boas noticed that these properties were not trivial and led to the introduction of Boas transforms (BT) in [25]. Later, BT was studied by Goldberg [26], Heywood [27], and Zaidi [28] who played a substantial role in outlining the properties and the results. For complete review of BT, one may read [29]. In a roundabout way, it was employed in the theory of filters in electrical engineering. Recall from [29] that any finite energy signal f on passing through a high pass filter whose system transfer function is given by Hw=Aeitow,ifw≥1;0,if otherwise gives an output g such that g^w=Hwf^w. Thus, g^ vanishes on −1,1. Using Boas’ theorems, one can characterize the output of the high-pass filter in two ways: (i) A signal g is the output of a high-pass filter if and only if BBg=−g. (ii) If g is an output of high-pass filter, then Bg=ℌg. Not much research has been done about it until 2019, when Khanna et al. introduced the notion of BT of wavelets (in a preprint form), which was later published in [30]. During the same year, Khanna and Kathuria [31] studied convolution of Boas transforms of wavelets. The motivation behind this study was the relationship between Boas and Fourier transforms of wavelets and the observation that wavelets ψx for which the Fourier transform ψ^η vanishes almost everywhere on −1,1 can be characterized by the Boas transform of wavelets Bψx. Since Boas transforms are closely related to Hilbert transforms, readers must be interested in reading Hilbert transforms of wavelets. For more details, see [30–40].


### 1.1. Plan of the Work

The paper is organized as follows: In Section2, we introduce the notion of fractional Boas transforms (FRBT) and give some properties in the form of observations. Titchmarsh-type and Tricomi-type results are established and a relationship between FRHT and FRBT is given followed by an inversion formula of FRBT and fractional Boas transform product theorem. In Section 3, wavelets associated with FRBT are introduced and a relationship between two wavelets in terms of Bθ−Hθ operator is given. We give a necessary and sufficient condition under which FRBT of a given wavelet is multiple of the first-order derivative of the given wavelet. We further give sufficient condition for the higher vanishing moments of FRBT of wavelets. Finally, we give a sufficient condition on two wavelets to obtain a two-dimensional wavelet and the number of vanishing moments of their convolution is given.


## 1.1. Plan of the Work

The paper is organized as follows: In Section2, we introduce the notion of fractional Boas transforms (FRBT) and give some properties in the form of observations. Titchmarsh-type and Tricomi-type results are established and a relationship between FRHT and FRBT is given followed by an inversion formula of FRBT and fractional Boas transform product theorem. In Section 3, wavelets associated with FRBT are introduced and a relationship between two wavelets in terms of Bθ−Hθ operator is given. We give a necessary and sufficient condition under which FRBT of a given wavelet is multiple of the first-order derivative of the given wavelet. We further give sufficient condition for the higher vanishing moments of FRBT of wavelets. Finally, we give a sufficient condition on two wavelets to obtain a two-dimensional wavelet and the number of vanishing moments of their convolution is given.


## 2. Fractional Boas Transforms

The Boas transform of a functionf∈L2ℝ, denoted by Bfx in terms of principal value integral, is defined as(1)Bfx=1πp.v.∫0∞fx+u−fx−uu2sinudu=1πp.v.∫−∞∞fx+uu2sinudu,for any x for which the integral exists.The relationship between the Boas transformB and the Hilbert transform ℌ of a function f is given by(2)Bfx=ℌfx−ℌf∗gx,where(3)gx=2π1/21−cosxx2.The FRHT off∈L2ℝ is defined as ℌθfx=cosθfx+sinθℌfx,−π/2≤θ≤π/2. It can be easily verified that the operator ℌθ satisfies the properties of linearity, translation-invariance, dilation-invariance, orthogonality, unitary nature, and linear independence. The linear independence property endorses one to induce a novel base from a given set of linearly independent functions.Now, we define an operatorBθ on L2ℝ by(4)Bθ=cosθI+sinθB,−π2≤θ≤π2=ℌθ−sinθℌ∗g,where Bθ is called a fractional Boas transform. For θ=π/2,Bθ=B.Observations.
(i)
Bθ is translation-invariant; that is, if eθx=Bθftx, then for translation operator Tc, we have(5)Tceθx=eθx−c=Bθft−cx=BθTcfx.(ii)
Bθ is dilation-invariant. Let Dα denote the dilation operator defined as Dαfx=fαx,α∈ℝ+. Then BθDαfx=DαBθfx=eθαx. Thus, fractional Boas transform operator commutes with Dα.(iii)
The transformationBθ:L2ℝ⟶L2ℝ is a nonsurjective bounded linear transform on L2ℝ. In fact, we have(6)Bθf2≤ℌθf2+sinθℌf∗g2≤ℌθf2+sinθℌf2g1.Now,ℌθf^η=e−iθsgnηf^η. Since f∈L2ℝ, it follows from Parseval’s identity that ℌθf2=f2. Thus,(7)Bθf2≤f21+2πsinθ<+∞.(iv)
Leth,f∈L2ℝ. Then ∫ℝhxBθfxdx=∫ℝB−θhxfxdx. In particular, if h=f and θ≠0, then ∫ℝfxBθfxdx=0. Further, ∫ℝBθfx2dx=∫ℝB−θfx2dx.Indeed, we have(8)∫ℝhxBθfxdx=cosθ∫ℝhxfxdx+sinθ∫ℝhxBfxdx=∫ℝB−θhxfxdx.Ifh=f, then(9)∫ℝfxBθfx−B−θfxdx=0,which gives∫ℝfxBfxdx=0. Further, if hx=Bθfx, then(10)∫ℝBθfx2dx=∫ℝB−θBθfxfxdx=∫ℝcos2θf2xdx−sin2θ∫ℝB2fxfxdx=∫ℝB−θfx2dx.(v)
For−π/2≤θ1,θ2≤π/2, we have(11)Bθ12fx=B2θ1fx+sin2θ12f∗gx−f∗g∗gx.Indeed, we have(12)Bθ1Bθ2fx=Bθ1ℌθ2fx−sinθ2ℌf∗gx=ℌθ1ℌθ2fx−sinθ2ℌθ1ℌf∗gx−sinθ1ℌℌθ2f∗gx+sinθ2f∗g∗gx=ℌθ1+θ2fx−sinθ1+θ2ℌf∗gx+2sinθ1sinθ2f∗gx−sinθ1sinθ2f∗g∗gx∵ℌθ1ℌθ2fx=ℌθ1+θ2fx=Bθ1+θ2fx+2sinθ1sinθ2f∗gx−sinθ1sinθ2f∗g∗gx=Bθ1+θ2fx+sinθ1sinθ2B2fx+fx.Then, the iteration property of the fractional Boas transform (11) can be easily obtained by taking θ2=θ1.(vi)
Letf∈L2ℝ be a function such that xnfx∈L2ℝ, for n∈ℕ. Then,(13)Bθxnfx=xnℌθfx−sinθ1π∑k=0n−1xk∫ℝtn−1−kftdt+−12πin∫−11f^nηe2πiηxsgnη−ηdη.Indeed, we have(14)Bθxnfx=cosθxnfx+sinθxnfx−1π∑k=0n−1xk∫ℝtn−1−kftdt−∫ℝℌtnftgx−tdt=xnℌθfx−sinθ1π∑k=0n−1xk∫ℝtn−1−kftdt+∫ℝsgnηtnft^ηExg^−ηdη=xnℌθfx−sinθ1π∑k=0n−1xk∫ℝtn−1−kftdt+∫ℝ−12πinf^nηe2πiηxsgnη1−ηdη=xnℌθfx−sinθ1π∑k=0n−1xk∫ℝtn−1−kftdt+−12πin∫−11f^nηe2πiηxsgnη−ηdη.(vii)
Letℛ denote the reflection operator, defined by ℛfx=f−x,x∈ℝ. Then, Bθℛfx=ℛBθfx−2sinθℛBfx.We have(15)Bθℛfx=ℌθℛfx−sinθℌℛf∗gx=cosθℛfx−sinθℛℌfx+sinθℌℛf∗gx=cosθℛfx−sinθℛℌfx−sinθℛℌf∗gx=ℛBθfx−2sinθℛℌfx.(viii)
It is easy to verify that ifhx=Bθfx, then(16)ddxBθfx=Bθdfxdx.(ix)
The fractional Boas transform of a convolution of two functionsf and h can be expressed as a convolution of one of the functions with the fractional Boas transform of the other function; that is,(17)Bθf∗hx=Bθf∗hx=f∗Bθhx.
Next, we give a Titchmarsh-type result for the fractional Boas transform.Proposition 1.
Iff,ϑ∈L2ℝ, then(18)∫ℝBθfxBθϑxdx≤∫ℝfxϑxdx.Proof.
We compute(19)∫ℝBθfxBθϑxdx=∫ℝcosθfx+sinθBfxcosθϑx+sinθBϑxdx=∫ℝcos2θfxϑx+cosθsinθfxBϑx+sinθcosθBfxϑx+sin2θBfxBϑxdx=cos2θ∫ℝfxϑxdx+cosθsinθ∫ℝfxBϑxdx−sinθcosθ∫ℝfxBϑxdx+sin2θ∫ℝℌfxℌϑxdx−sin2θ∫ℝℌfxℌϑ∗gxdx−sin2θ∫ℝℌf∗gxℌϑxdx+sin2θ∫ℝℌf∗gxℌϑ∗gxdx=∫ℝfxϑxdx−2sin2θ∫ℝf^ηϑ^ηg^ηdη+sin2θ∫ℝf^ηϑ^ηg^η2dx=∫ℝfxϑxdx+sin2θ∫−11f^ηϑ^η−21−η+1−η2dη≤∫ℝfxϑxdx.
Next, we give a Tricomi-type result for the fractional Boas transform.Proposition 2.
Letf,ϑ be functions such that(i)
f,ϑ,f^,ϑ^∈L1ℝ(ii)
f^0=0(iii)
f^η vanishes for η>1 and ϑ^η vanishes for η≤1
Then,BθfxBθϑx−ϑxBθfx=sinθBϑxB−θfx−ϑxBπ/2−θfx.Proof.
We have(20)BθfxBθϑx−ϑxBθfx=Bθfxcosθϑx+sinθBϑx−ϑxcosθfx+sinθBfx=BθsinθfxBϑx−ϑxBfx=sinθcosθfxBϑx−ϑxBfx+sinθBfxBϑx−ϑxBfx=sinθcosθfxBϑx−sinθcosθϑxBfx+sin2θBfxBϑx−sin2θBϑxBfx.
In view of Boas transform product theorem (Theorem 3.9, [31]), we have(21)BθfxBθϑx−ϑxBθfx=sinθcosθfxBϑx−sinθcosθϑxBfx+sin2θfxB2ϑx−sin2θBϑxBfx=sinθBϑxcosθfx−sinθBfx−ϑxcosθBfx+sinθfx=sinθBϑxB−θfx−ϑxBπ/2−θfx.
Next, we discuss fractional Boas transform of product of analytic functions (or signals).Proposition 3.
For analytic functionsf1x=g1x+ih1x and f2x=g2x+ih2x, we have(22)Bθf1xf2x=f1f2∗e−iθδ−isinθgx.Proof.
We compute(23)Bθf1xf2x=cosθf1xf2x+sinθℌf1xf2x−ℌf1xf2x∗gx=cosθf1xf2x+sinθ−if1xf2x−if1f2∗gx=f1xf2xe−iθ−isinθf1f2∗gx=f1f2∗e−iθδ−isinθgx.
In particular, iff1=f2=f, then(24)Bθf2x=f2∗e−iθδ−isinθgx.
The generalization to arbitrary powers is(25)Bθfnx=fn∗e−iθδ−isinθgx.
In the following result, we give a relationship between fractional Hilbert transform and fractional Boas transform.Proposition 4.
Letf,f^∈L1ℝ and let f^0=0. Then,(26)ℌθfx=Bθfx+∑j=1∞Bθf−cosθf∗gjx,where g1=g and gj=gj−1∗g, for j=2,3,4,….Proof.
We have(27)Bθf^η=cosθf^η+sinθℌf^η1−g^η=Bθf^η−cosθf^η∑j=0∞g^ηj=Bθfx−cosθfx∧η+∑j=1∞Bθf−cosθf∗gj∧η.
Taking inverse Fourier transform, we have(28)ℌθfx=Bθfx+∑j=1∞Bθf−cosθf∗gjx.
Note that, for anyM>N>0, we have(29)∑j=NMBθf−cosθf∗gj2=∑j=NMBθf^−cosθf^g^j2≤∑j=N∞Bθf^−cosθf^g^j2=g^j∑j=0∞Bθf^−cosθf^g^j2.
Since∑j=0∞Bθf^−cosθf^g^j∈L2ℝ, it follows by Lebesgue convergence theorem that(30)limN⟶∞g^N∑j=0∞Bθf^−cosθf^g^j2=0.
Thus, the series in (26) converges in L2 norm.
Next, we discuss the inversion of the fractional Boas transform.Proposition 5.
Iff∈L2ℝ, then f can be retrieved from its fractional Boas transform by means of the formula(31)fx=−12πi∫ℝBθf−B−θfx−zφzdz,where φx=∑j=0∞−1jx2j−1/2j!+Six with sine integral Si⋅.Proof.
Letτθx=Bθf−B−θf∗φx, where(32)φx=∑j=0∞−1jx2j−12j!+Six.
Then, we have(33)τ^θη=Bθf^η−B−θf^ηφ^η=2sinθBf^ηφ^η.
Now, observe that(34)φ^η=∫ℝe−iηx∑j=0∞−1jx2j−12j!+Sixdx=−πi2∫ℝsinη+1x+sinη−1xxdx−πi∫ℝδx−η−δx+ηxdx=0,ifη<1−πisgnη,ifη>1+−πip.v.1η,ifη<1,0,ifη>1,=−πip.v.1η,ifη<1,−πisgnη,ifη>1.
Therefore,τ^θη=−2πif^η. Also, Bθf^η−B−θf^ηφ^η=−2πif^η.
Taking the inverse Fourier transform, we obtain(35)fx=−12πi∫ℝBθf−B−θfx−zφzdz.
Towards the end, we give fractional Boas transform product theorem.Proposition 6.
Letf,ϑ be functions such that(i)
f,ϑ,f^1,f^2∈L1ℝ,(ii)
f^1η vanishes for η>1 and f^2η vanishes for η≤1.
ThenBθf1xf2x=f1xBθf2x.Proof.
In view of Theorem 3.1 in [31], we have(36)Bθf1xf2x=cosθf1xf2x+sinθf1xBf2x=f1xBθf2x.


## 3. Fractional Boas Transforms of Wavelets

The wavelet theory operates with the general properties of the wavelets and the wavelet transform. A wavelet function is chosen according to the application; for example, for space-frequency analysis, a wavelet that is localized in terms of both spatial width and frequency bandwidth is preferred, whereas a smooth wavelet is more appropriate in dealing with smooth signals. In case of analysis of a signal with certain discontinuities, wavelets with good spatial localization to scrupulously track swift changes in the signal are required. For more details on wavelets, one may read [41–45].Now, we give a sufficient condition under which fractional Boas transform of a wavelet is again a wavelet.Theorem 1.
Letψ∈L1ℝ be a wavelet such that ψ^∈L1ℝ and ψ^0=0. Then, Bθψ is again a wavelet.Proof.
In view of Theorem 2.1 in [30], Bθψ∈L2ℝ. To verify the admissibility condition, we have(37)∫ℝBθψ^η2ηdη≤cosθ2∫ℝψ^η2ηdη+sinθ2∫ℝBψ^η2ηdη+sin2θ∫ℝψ^ηBψ^ηηdη.
Now, since(38)∫ℝψ^ηBψ^ηηdη=∫η≤1ψ^η−isgnηηψ^ηηdη+∫η>1ψ^η−isgnηψ^ηηdη≤∫η≤1ψ^η2ηdη+∫η>1ψ^η2ηdη=∫ℝψ^η2ηdη,we deduce that(39)∫ℝBθψ^η2ηdη<+∞.Remark 1.
The condition thatψ,ψ^∈L1ℝ such that ψ^0=0 is not necessary for Bθψ to be a wavelet. Indeed, let ψ be a Haar wavelet defined by(40)ψx=1,if0≤x<12,−1,if12≤x<1,0,if otherwise.
SinceBθψ2≤lψ2<+∞, where l=cosθ+sinθ+2π and ∫ℝBθψ^η2/ηdη<+∞, we conclude that Bθψ is a wavelet. However, ψ^∉L1ℝ.
In the next result, we give a relationship between two wavelets in terms ofBθ−ℌθ operator.Theorem 2.
Letψ be a wavelet such that ψ,ψ^∈L1ℝ and let ρ be a function such that ρ,ρ1,ρ^1∈L1ℝ. If ψ^0=0 and ψ^η=g^η+ηρ^η, then ρ is a wavelet such that(41)Bθ−ℌθψx=Bθ−ℌθρx.Proof.
Note thatBθ−ℌθψx=sinθB−ℌψx. Thus, in view of Theorem 2.3 in [30], ρ is a wavelet such that Bθ−ℌθψx=Bθ−ℌθρx.
The following result gives a necessary and sufficient condition under which fractional Boas transform of a given wavelet is multiple of the first-order derivative of the given wavelet.Theorem 3.
Letψx be a wavelet such that ψ,ψ^∈L1ℝ and ψ^0=0. Then Bθψx=cosθψ1x if and only if ψ^η=0 for every η>1.Proof.
LetBθψx=cosθψ1x. Then, taking Fourier transform on both sides, we have(42)ψ^ηcosθ1+iη−isinθsgnη1−g^η=0.
Ifη>1, then ψ^ηSϕη=0, where(43)Sϕη=e−iϕ+iηcosϕ,ifη>1,eiϕ+iηcosϕ,ifη<−1.
SinceSϕx≠0, we get ψ^η=0.
The proof of the converse part is straightforward.
A waveletψx is said to have n vanishing moments if ∫ℝxqψxdx=0,for0≤q≤n−1. This property actually represents the regularity of the wavelet function and ability of wavelet transform to capture the localized information. If a wavelet with large number of vanishing moments is employed, then the corresponding wavelet series of a smooth function will converge very rapidly to the function. Thus, only few wavelet coefficients are required in order to obtain a good approximation. During image compression, it requires only to keep a few wavelet coefficients, where the image is smooth and, in contrary to this, more coefficients are needed at the edges. For more details, see [33–40].
Next, we define the notion ofG-function of order n.Definition 1.
Letf be a function such that f,f1,f^∈L1ℝ. Then, f is said to be a G-function of order n if ∫ℝxqGxdx=0, for 0≤q≤n, where Gx=∫−111−1/ηe−2πiηxf1^−ηdη.
Recall from [44] that a function f is said to have fast decay with decay rate l∈ℕ, if there exists a constant Cl such that fx≤Cl/1+xl, for all x∈ℝ.
In the following result, we give a sufficient condition for the higher vanishing moments of fractional Boas transform of wavelets.Theorem 4.
Ifψj,kj,k∈ℤ is an orthonormal system on ℝ, then(44)∫ℝxsBθψxdx=0,for alls=0,1,2,…,p;s+1<l,where ψ such that ψ,ψ^,ψ1∈L1ℝ is a G-function of order p, and ψ∈Cpℝ has fast decay with decay exponent l∈ℕ such that ψs∈L∞ℝ, s=1,2,…,p.Proof.
We have(45)∫ℝxsBθψxdx=∫ℝxscosθψx+sinθBψxdx=cosθ∫ℝxsψxdx+sinθ∫ℝxsBψxdx=cosθ∫ℝxsψxdx+sinθ∫ℝxsℌψxdx−12π∫ℝxs∫−111−1ηe−2πiηxψ1^−ηdηdx∵ψ,ψ1,ψ^∈L1ℝ=cosθ∫ℝxsψxdx+sinθ∫ℝxsℌψxdx−12π∫ℝxsGxdx=cosθ∫ℝxsψxdx+sinθ∫ℝxsℌψxdx,where Gx=∫−111−1/xe−2πiηxψ1^−ηdη.
In view of Theorem 3.1 in [30], we have ∫ℝxsψxdx=0, for all s=0,1,…,p.
Also, sinceψx,xpψx∈L2ℝ and xsψx∈L2ℝ for s=0,1,…,p, using the moment formula for the Hilbert transform, we have(46)∫ℝxsBθψxdx=sinθ∫ℝℌxsψxdx+1π∑j=0s−1∫ℝxj∫ℝzs−1−jψzdzdx.
Nowxsψx∈L2ℝ, for s=0,1,…,p. Hence, it follows that ∫ℝxsBθψxdx=0 for s=0,1,…,p. □
Recall from [46] that a two-dimensional function Ψ∈L2ℝ2 is called an admissible wavelet if it satisfies the admissibility condition(47)CΨ=2π2∫ℝ2Ψ^η2η2dη<∞,where η2=η12+η22.
In the following result, we give a sufficient condition on two waveletsψ1 and ψ2 such that the product cosθψi1uψiv,i=1,2 forms a two-dimensional wavelet.Theorem 5.
Letψ1,ψ2 be wavelets such that(i)
ψi,ψi1,ψ^i∈L1ℝ(ii)
ψ^i0=0 and ψ^iη=0,i=1,2 for η>1
Then,Ψiu,v=cosθψi1uψiv,i=1,2 are admissible wavelets in L2ℝ2.Proof.
Note that(48)Ψiu,v=Bθψiuψiv,i=1,2=cosθψiuψiv+sinθBψiuψiv.
Sinceψ^i∈L1ℝ, for i∈1,2, it follows that ψi is bounded. Also, ψi∈L1ℝ,i=1,2 is bounded and so it must be in L2ℝ. Thus, clearly Ψi∈L2ℝ2 for i=1,2.
Also, we have(49)CΨi=2π2∫ℝ2Ψ^iη2η2dη≤2π2cosθ2∫ℝψ^iη12η1dη1∫ℝψ^iη22η2dη2+2π2sinθ2∫ℝBψi^η12η1dη1∫ℝψ^iη22η2dη2+2π2sin2θ∫ℝψ^iη12η1dη1∫ℝψ^iη22η2dη2<∞.
Then, fori=1,2, Ψi is an admissible wavelet in L2ℝ2.
Next, we give a sufficient condition on two waveletsψ1 and ψ2 such that the convolution of two-dimensional wavelets Ψi,i=1,2 again forms a wavelet with 4p−2 vanishing moments.Theorem 6.
Letψ1,ψ2 be wavelets as defined in Theorem 5 such that(i)
upψiu∈L2ℝ,i=1,2, and ψ1∗ψ2∗g∗2δ−gx=0, where δ is Dirac delta function and g is given by (3)(ii)
ψi,i=1,2 are G-function of order p(iii)
ψ1∗ψ2^0=0, and ψi,i=1,2 have p vanishing moments
Then,Ψiu,v=cosθψi1uψiv,i=1,2 have 2p vanishing moments. Further, if Ψ3u,v=Ψ1∗Ψ2u,v, then Ψ3 is admissible wavelet with 4p−2 vanishing moments.Proof.
We have(50)∫ℝ2urvsΨiu,vdudv=cosθMrMn−r+sinθMr˜Mn−rr+s=n,where  Mr=∫ℝzrψizdz and Mr˜=∫ℝzrBψizdz.
Sinceψi,i=1,2 have p vanishing moments, using the arguments given in Theorem 3.1 in [30], we conclude that Bψi,i=1,2 have p+1 vanishing moments.
Assume thatn≤2p−1. If n−r≤p−1, then Mn−r=0; otherwise, r≤p, which gives Mr=M˜r=0. Thus, the number of vanishing moments of Ψi,i=1,2 is 2p.
Also, we compute(51)Ψ3u,v=Ψ1u,v∗Ψ2u,v=∫ℝ∫ℝΨ1t1,t2Ψ2u−t1,v−t2dt1dt2=∫ℝ∫ℝcosθψ1t1ψ1t2+sinθBψ1t1ψ1t2cosθψ2u−t1ψ2v−t2+sinθBψ2u−t1ψ2v−t2dt1dt2=cos2θ∫ℝ∫ℝψ1t1ψ2u−t1ψ1t2ψ2v−t2dt1dt2+cosθsinθ∫ℝ∫ℝψ1t1Bψ2u−t1ψ1t2ψ2v−t2dt1dt2+cosθsinθ∫ℝ∫ℝBψ1t1ψ2u−t1ψ1t2ψ2v−t2dt1dt2+sin2θ∫ℝ∫ℝBψ1t1Bψ2u−t1ψ1t2ψ2v−t2dt1dt2=cos2θψ1∗ψ2uψ1∗ψ2v+cosθsinθψ1∗Bψ2uψ1∗ψ2v+cosθsinθBψ1∗ψ2uψ1∗ψ2v+sin2θBψ1∗Bψ2uψ1∗ψ2v=cos2θψ1∗ψ2uψ1∗ψ2v+sin2θψ1∗Bψ2uψ1∗ψ2v.
Further, note that(52)∫ℝ2Ψ3u,v2dudv=∫ℝ∫ℝcos2θψ1∗ψ2uψ1∗ψ2v+sin2θψ1∗Bψ2uψ1∗ψ2v2dudv≤cos2θ2∫ℝψ1∗ψ2u2du∫ℝψ1∗ψ2v2dv+sin2θ2∫ℝψ1∗Bψ2u2du∫ℝψ1∗ψ2v2dv+sin4θ∫ℝψ1∗ψ2uψ1∗Bψ2udu∫ℝψ1∗ψ2v2dv.
Now, sinceψ1,ψ^1,ψ2∈L1ℝ and ψ1∗ψ2^0=0, it follows that ∫ℝ2Ψ3u,v2dudv<∞.
Again, sinceΨi∈L1ℝ2 for i = 1, 2, we have(53)Ψ^3η1,η2=Ψ^1η1,η2Ψ^2η1,η2=cosθψ^1η1ψ^1η2+sinθBψ^1η1ψ^1η2cosθψ^2η1ψ^2η2+sinθBψ^2η1ψ^2η2=cos2θψ^1η1ψ^1η2ψ^2η1ψ^2η2+sin2θψ^1η1ψ^1η2Bψ^2η1ψ^2η2+sin2θBψ^1η1ψ^1η2Bψ^2η1ψ^2η2.
Therefore, we obtain(54)CΨ3=2π2∫ℝ2Ψ^3η1,η22η2dη1dη2≤2π2cosθ4∫ℝψ^1η1ψ^2η12η1dη1∫ℝψ^1η2ψ^2η22η2dη2+2π2sin2θ2∫ℝψ^1η1Bψ2^η12η1dη1∫ℝψ^1η2ψ^2η22η2dη2+2π2sinθ2∫ℝBψ1^η1Bψ2^η12η1dη1∫ℝψ^1η2ψ^2η22η2dη2+22π2cos2θsin2θ∫ℝψ^1η12ψ^2η1Bψ2^η12η1dη1︷I1∫ℝψ^1η2ψ^2η22η2dη2+22π2sin2θsin2θ∫ℝψ^1η1Bψ1^η1Bψ2^η12η1dη1︷I2∫ℝψ^1η2ψ^2η22η2dη2+22π2cosθsinθ2∫ℝψ^1η1Bψ1^η1ψ2^η1Bψ2^η1η1dη1︷I3∫ℝψ^1η2ψ^2η22η2dη2.
Sinceψ2∈L1ℝ and ψ^20=0, it follows that ψ^2,Bψ2^ are bounded. Thus, we have(55)I1=∫ℝψ^1η12ψ^2η1Bψ2^η12η1dη1≤K1K2∫ℝψ^1η12η12dη1,where K1 and K2 are constants. A similar argument works for I2 and I3. Therefore, we have CΨ3<+∞. Now, observe that(56)∫ℝ2urvsΨ3u,vdudv=cos2θ∫ℝurψ1∗ψ2udu∫ℝvsψ1∗ψ2vdv+sin2θ∫ℝurψ1∗Bψ2udu∫ℝvsψ1∗ψ2vdv=cos2θMr∗Ms∗+sin2θM˜r∗Ms∗,where Mi∗=∫ℝziψ1∗ψ2zdz and M˜i∗=∫ℝziψ1∗Bψ2zdz.
Takingr+s=t, we obtain(57)∫ℝ2urvsΨ3u,vdudv=cos2θMr∗Mt−r∗+sin2θM˜r∗Mt−r∗.
Thus, in view of Theorem 4.3 in [33], ψ1∗ψ2 and ψ1∗Bψ2 have, respectively, 2p−1 and 2p vanishing moments.
Ifr≤2p−1, then Mr∗=0 and Mr˜∗=0. If not, then t−r≤2p−2. Thus, Mt−r∗=0.
Therefore,Mr∗,Mr˜∗, and Mt−r∗ all vanish if r+s≤4p−3.
Hence, the number of vanishing moments ofΨ3u,v is 4p−2.


## 4. Conclusion

In this paper, we define and study the notion of fractional Boas transforms (FRBT) with the aim of obtaining better comparative results. Various properties of FRBT are discussed, and several results are obtained. A comparative study is done to show that the FRBT of wavelets gives better results as compared to the usual wavelets of the classical Boas transform. We illustrate this study by considering the HT and the FRBTs of Daubechies wavelet and Mexican Hat wavelet, respectively, through Figures1 and 2 and show that the FRBT of wavelet at θ=0 is the wavelet itself, and the FRBT of wavelet at θ=π/2 is the BT of wavelet. It is easy to conclude that, after applying FRBT on a wavelet, the resulting wavelet is approximately equal to the original wavelet. Hence, it is better to employ FRBT of the wavelet instead of using BT of the wavelet, or HT of the wavelet.Figure 1
HT of Daubechies wavelet and FRBTs of Daubechies wavelet atθ=0,π/6,π/4,π/3,π/2. (a) HT of Daubechies wavelet. (b) FRBT of Daubechies wavelet with θ=0. (c) FRBT of Daubechies wavelet with θ=π/6. (d) FRBT of Daubechies wavelet with θ=π/4. (e) FRBT of Daubechies wavelet with θ=π/3. (f) FRBT of Daubechies wavelet with θ=π/2.
(a)(b)(c)(d)(e)(f)Figure 2
HT of Mexican Hat wavelet and FRBTs of Mexican Hat wavelet atθ=0,π/6,π/4,π/3,π/2. (a) HT of Mexican Hat wavelet. (b) FRBT of Mexican Hat wavelet with θ=0. (c) FRBT of Mexican Hat wavelet with θ=π/6. (d) FRBT of Mexican Hat wavelet with θ=π/4. (e) FRBT of Mexican Hat wavelet with θ=π/3. (f) FRBT of Mexican Hat wavelet with θ=π/2.
(a)(b)(c)(d)(e)(f)



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