Some Properties of Fractional Boas Transforms of Wavelets

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.


Introduction
e ordinary transformations have been replaced with the fractional ones, which play a significant role in information processing. is 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][2][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. e 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]. e 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]. e 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. eir 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]. e 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. e 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]. e 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. is class denoted by B 2 κ 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 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 H(w) � Ae it o w , if |w| ≥ 1; 0, if otherwise gives an output g such that us, g vanishes on (− 1, 1). Using Boas' theorems, one can characterize the output of the highpass filter in two ways: (i) A signal g is the output of a highpass filter if and only if B(Bg) � − 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 BTof 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. e motivation behind this study was the relationship between Boas and Fourier transforms of wavelets and the observation that wavelets ψ(x) for which the Fourier transform ψ(η) vanishes almost everywhere on (− 1, 1) can be characterized by the Boas transform of wavelets Bψ(x). Since Boas transforms are closely related to Hilbert transforms, readers must be interested in reading Hilbert transforms of wavelets. For more details, see [30][31][32][33][34][35][36][37][38][39][40].

Plan of the Work.
e paper is organized as follows: In Section 2, 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 (B θ − H θ ) operator is given. We give a necessary and sufficient condition under which FRBT of a given wavelet is multiple of the firstorder 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.

Fractional Boas Transforms
e Boas transform of a function f ∈ L 2 (R), denoted by Bf(x) in terms of principal value integral, is defined as for any x for which the integral exists. e relationship between the Boas transform B and the Hilbert transform H of a function f is given by where It can be easily verified that the operator H θ satisfies the properties of linearity, translation-invariance, dilation-invariance, orthogonality, unitary nature, and linear independence. e linear independence property endorses one to induce a novel base from a given set of linearly independent functions. Now, we define an operator B θ on L 2 (R) by where B θ is called a fractional Boas transform. For Observations , then for translation operator T c , we have In fact, we have Now, which Indeed, we have Journal of Mathematics en, the iteration property of the fractional Boas transform (11) can be easily obtained by taking θ 2 � θ 1 .
(vi) Let f ∈ L 2 (R) be a function such that x n f(x) ∈ L 2 (R), for n ∈ N. en, Indeed, we have (vii) Let R denote the reflection operator, defined by We have (ix) e fractional Boas transform of a convolution of two functions f and h can be expressed as a convolution of one of the functions with the fractional Boas transform of the other function; that is, Next, we give a Titchmarsh-type result for the fractional Boas transform.
Proof. We compute Next, we give a Tricomi-type result for the fractional Boas transform. □ Proposition 2. Let f, ϑ be functions such that In view of Boas transform product theorem ( eorem 3.9, [31]), we have Journal of Mathematics Next, we discuss fractional Boas transform of product of analytic functions (or signals).

Proposition 3. For analytic functions
Proof. We compute In particular, if e generalization to arbitrary powers is In the following result, we give a relationship between fractional Hilbert transform and fractional Boas transform.

Proof.
We have Taking inverse Fourier transform, we have Note that, for any M > N > 0, we have Since ∞ j�0 |B θ f − cos(θ)f|(g) j ∈ L 2 (R), it follows by Lebesgue convergence theorem that 6 Journal of Mathematics us, the series in (26) converges in L 2 norm. Next, we discuss the inversion of the fractional Boas transform. □ Proposition 5. If f ∈ L 2 (R), then f can be retrieved from its fractional Boas transform by means of the formula en, we have Now, observe that Taking the inverse Fourier transform, we obtain Towards the end, we give fractional Boas transform product theorem. □ Proposition 6. Let f 1 , f 2 be functions such that

Fractional Boas Transforms of Wavelets
e 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 spacefrequency 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][42][43][44][45]. Now, we give a sufficient condition under which fractional Boas transform of a wavelet is again a wavelet. Theorem 1. Let ψ ∈ L 1 (R) be a wavelet such that ψ ∈ L 1 (R) and ψ(0) � 0. en, B θ ψ is again a wavelet.
In the next result, we give a relationship between two wavelets in terms of (B θ − H θ ) operator.

Theorem 2.
Let ψ be a wavelet such that ψ, ψ ∈ L 1 (R) and let ρ be a function such that ρ, ρ (1) , ρ (1) ∈ L 1 (R). If ψ(0) � 0 and ψ(η) � (g(η)t + n|η|)ρ(η), then ρ is a wavelet such that us, in view of eorem 2.3 in [30], ρ is a wavelet such that e 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.
en, taking Fourier transform on both sides, we have is 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. us, 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][34][35][36][37][38][39][40].
Next, we define the notion of G-function of order n. □ Definition 1. Let f be a function such that f, f (1) Recall from [44] that a function f is said to have fast decay with decay rate l ∈ N, if there exists a constant C l such that |f(x)| ≤ (C l /1 + |x| l ), for all x ∈ R.
Also, we have en, for i � 1, 2, Ψ i is an admissible wavelet in L 2 (R 2 ). Next, we give a sufficient condition on two wavelets ψ 1 and ψ 2 such that the convolution of two-dimensional wavelets Ψ i , i � 1, 2 again forms a wavelet with (4p − 2) vanishing moments. □ Journal of Mathematics Theorem 6. Let ψ 1 , ψ 2 be wavelets as defined in eorem 5 such that and [((ψ 1 * ψ 2 ) * g) * (2δ − g)](x) � 0, where δ is Dirac delta function and g is given by (3) Proof. We have where M r � R z r ψ i (z)dz and M r � R z r Bψ i (z)dz.
us, the number of vanishing moments of Ψ i , i � 1, 2 is 2p.
Taking r + s � t, we obtain

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 Figures 1 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.

Data Availability
No data were used to support this study.