# Fourier–Boas-Like Wavelets and Their Vanishing Moments

Authors: Leena Kathuria; Shashank Goel; Nikhil Khanna
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we propose Fourier–Boas-Like wavelets and obtain sufficient conditions for their higher vanishing moments. A sufficient condition is given to obtain moment formula for such wavelets. Some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are also given. Finally, we formulate a variation diminishing wavelet associated with a Fourier–Boas-Like wavelet.


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## Body

## 1. Introduction

Fourier analysis was initiated with the reconstruction and analysis of periodic functions with the help of Fourier series. It was later extended to Fourier transform in order to study the nonperiodic signals. The Fourier transform is an effective analytical tool to study the continuous and discrete-time signals, but there is a limitation in employing this transform as it does not exhibit the temporal information of the signal. Thus, to overcome this limitation, Gabor [1] introduced the concept of short-time Fourier transform, where an analysis window of fixed length slides over the time axis to give time-localized frequency information. The inherent limitation of short-time Fourier transform lies in the fact that the window width remains the same for all the frequencies, thereby making the localization extent to be constant for different frequencies. The resolution to this problem was given by Grossmann and Morlet [2] who introduced the notion of wavelet transforms in 1984. A function ψ with finite energy, Eψ, i.e., ψ∈L2ℝ, is said to be a wavelet if it satisfies the admissibility condition given by Cψ=∫ℝψ^η2/ηdη<+∞, where ψ^ denotes the Fourier transform of a wavelet ψ. The most significant work that has driven the progress of wavelets to great heights was attributed to Mallat [3] and Meyer [4] who both were responsible in the development of the concept of multiresolution analysis (MRA), another way of constructing wavelets.An MRA consists of a sequence of closed subspacesVj,j∈ℤ of L2ℝ satisfying the following conditions: (i) Vj⊂Vj+1,for allj∈ℤ; (ii) f∈Vjif and only iff2⋅∈Vj+1,for allj∈ℤ; (iii) ∩j∈ℤVj=0; (iv) ∪j∈ℤVj¯=L2ℝ; (v) if there exists ϕ∈V0 so that ϕ⋅−k:k∈ℤ constitutes an orthonormal basis for V0, then there exists an orthonormal wavelet basis ψj,k such that Pjf=Pj−1f+∑k∈ℤf,ψj,kψj,k holds, where Pj denotes the orthogonal projection operator onto Vj. The function ϕ is called a scaling function of the given MRA. It solves the dilation equation ϕx=2∑k∈ℤakϕ2x+k with ϕ^0=1 and ψ is a function associated to ϕ, which is defined by ψx=2∑k∈ℤbkϕ2x+k. The functions ϕ and ψ are usually known as father and mother wavelets, respectively. It is well established in [5] that a pair of quadrature mirror filter coefficients, akk∈ℤ,bkk∈ℤ∈l2ℤ, is associated to the MRA and the following relations ϕ^η=m0η/2ϕ^η/2,ψ^η=m1η/2ϕ^η/2 are satisfied for η∈ℝ, where m0 and m1 are given by m0η=∑k∈ℤakeikη,m1η=∑k∈ℤbkeikη=eiηm0η+π¯. For more details on wavelets, one may refer to [6–14].Boas [15] proposed an integral transform related to the Hilbert transform, which arose due to the study of the class of functions having Fourier transforms vanishing on a finite interval. This transform is known as Boas transform, which finds an application in the theory of filters in electrical engineering. Goldberg [16], in 1960, analyzed this transform in detail and gave some significant results and properties. Subsequently, it was further examined by Heywood [17] in 1963 and Zaidi [18] in 1976. For various details pertaining to Boas transform, one may refer to [19].Khanna et al. [9] introduced Boas transforms of wavelets and furnished various results related to their higher vanishing moments. Later, Khanna and Kathuria [8] studied convolution of these resulting wavelets to analyze Boas transform of convolution of signals. Recently, Khanna et al. [20] introduced fractional Boas transforms and the associated wavelets. These new wavelets appeared to be more prominent than the Boas transforms of wavelets due to an additional degree of freedom in terms of fractional order.


### 1.1. Framework

This work is streamlined as follows. In Section2, Fourier–Boas-Like wavelets are introduced and studied. Some results related to higher vanishing moments of such wavelets are obtained. Also, the moment formula for such wavelets by enforcing sufficient condition on the wavelet is derived. Further, some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are given. Finally, a variation diminishing wavelet associated with Fourier–Boas-Like wavelet is constructed.


## 1.1. Framework

This work is streamlined as follows. In Section2, Fourier–Boas-Like wavelets are introduced and studied. Some results related to higher vanishing moments of such wavelets are obtained. Also, the moment formula for such wavelets by enforcing sufficient condition on the wavelet is derived. Further, some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are given. Finally, a variation diminishing wavelet associated with Fourier–Boas-Like wavelet is constructed.


## 2. Fourier–Boas-Like Wavelets

Letf∈L2ℝ. Then, the Boas transform of f in terms of principal value integral is defined as(1)Bfx=1πp.v.∫0∞fx+z−fx−zz2sinzdz=1πp.v.∫−∞∞fx+zz2sinzdz,for any x for which the integral exists.The relationship between the Boas transform and the Hilbert transform of a function is given by(2)Bfx=ℌfx−ℌf∗gx,where(3)gx=2π1/21−cosxπx2,whereas the Fourier transform and the Hilbert transform share a relationship, specified by(4)ℌf^η=−if^η,ifη>0,if^η,ifη<0,0,ifη=0.Taking Fourier transform on both the sides of (2), we have(5)Bf^η=ℌf^x−Fℌf∗gη.Ifℌfx∈L1ℝ, then using (2), (4), and (5), we obtain Bf^η=−isgnηf^η1−g^η, where(6)g^η=0,ifη>1,1−η,ifη≤1.Soares et al. [21] introduced Fourier-like wavelets, defined by FLikeψx=1/2ψx−iℌψx, using the concept of the Fourier kernel eiη. Further, a factor of 1/2 was imposed on function ψx−iℌψx in order to aver the same energy and admissibility coefficient of its generating wavelet. Later, Khanna et al. [12] defined an improved and natural version of such wavelets by employing Riesz projectors on wavelets. The main idea behind these wavelets was to perlustrate both even and odd symmetries of an asymmetric signal.One may note thatBsinx=cosx and Bcosx=−sinx, and thus the Fourier kernel can be written as eix=cosx−iBcosx. This observation persuaded us to analyze Boas transform further and we define Fourier–Boas-Like wavelets as BLikeFψx=ψx−iBψx. There are two main reasons for defining these new wavelets: (i) to characterize the wavelets whose Fourier transform vanishes a.e. on −1,1 and (ii) to reinforce the incompetency of wavelets to study both the symmetries of an asymmetric signal.Next, we give a sufficient condition under whichBLikeFψ forms a wavelet.Proposition 1.
Letψ∈L1ℝ be a wavelet such that ψ^∈L1ℝ and ψ^0=0. Then, BLikeFψ is again a wavelet.Proof.
Note that(7)EBLikeFψ=∫ℝBLikeFψx2dx≤∫ℝψx2+Bψx2dx<+∞.
Also, we have(8)CBLikeFψ=∫ℝFBLikeFψxη2ηdη=∫ℝψ^η−sgnη1−g^ηψ^η2ηdη<+∞.
Clearly,BLikeFψ has finite energy, and it also satisfies the admissibility condition.Vanishing moments bestow orthogonality relative to subspaces of polynomials and thus perform a significant task in signal processing. For many applications of wavelets such as reconstruction of a signal, compression of images and to examine the regularity of the analyzed signal, a large number of vanishing moments of a wavelet are needed. Theoretically, a large number of vanishing moments insinuate the competency of the scaling function in representing more complex signals scrupulously. A functionfx is said to have s vanishing moments if ∫ℝxufxdx=0,0≤u≤s−1, where the given integral is called the uth moment of fx. For more details, see [10].Daubechies [22] constructed compactly supported orthonormal wavelets having smoothness of a fixed degree. She observed that if ϕ,ψ∈Csℝ, then the low-pass filter m0 takes the form m0η=1+e−iη/2s+1ℱη, where ℱ∈Csℝ is 2π-periodic function. This can be construed using the below stated result which also helps us in deducing that there does not exist any compactly supported orthonormal wavelet ψ∈C∞ℝ.Theorem 1.
(see [5]). Let ψ∈Csℝs∈ℕ0 be a function such that ψx≤ℳ1+x−s−1−ϱfor someϱ>0, and that ψu∈L∞ℝ for u=1,2,…,s. If ψj,k:j,k∈ℤ is an orthonormal system in L2ℝ, then ∫ℝxuψxdx=0, for all u=0,1,…,s.Recall from [13] that a function f is said to have fast decay with decay rate p∈ℕ, if there exists a constant Cp such that fx≤Cp/1+xp∀x∈ℝ. The moment formula for the Hilbert transform of f is given by(9)ℌxsfx=xsℌfx−1π∑m=0s−1xm∫ℝqs−1−mfqdq,s≥0.Note that the above formula holds ifxsfx∈Lqℝ,1<q<∞.In the following result, we give the relationship between the higher vanishing moments of Fourier–Boas-Like wavelets and the fast decay of waveletψ.Theorem 2.
Letψ, ψ1, ψ^∈L1ℝ such that ψ∈Csℝ is having a fast decay with decay exponent p∈ℕandψu∈L∞ℝ, u=1,2,…,s. Also, let xsψx∈L2ℝ, and ∫ℝxuGxdx=0, for u=0,1,…,s, where Gx=∫−111−1/ηe−2πiηxψ1^−ηdη. If ψj,kj,k∈ℤ constitutes an orthonormal system in L2ℝ, then ∫ℝxuBLikeFψxdx=0, for all u=0,1,…,s, where u+1<p.Proof.
We compute(10)∫ℝxuBLikeFψxdx=∫ℝxuψx−iBψxdx=∫ℝxuψx−iℌψx−ℌψ∗gxdx=∫ℝxuψx−iℌψx−∫ℝT−xℌψ−tgtdtdx=∫ℝxuψx−iℌψx−∫ℝFℌT−xψ−ηg^ηdηdx=∫ℝxuψx−iℌψx+∫−11sgn−ηFT−xψ−η1−ηdηdx.
Sinceψ, ψ1, ψ^∈L1ℝ, we have(11)∫ℝxuBLikeFψxdx=∫ℝxuψx−iℌψx−12π∫−111−1ηe−2πiηxψ1^−ηdηdx=∫ℝxuψx−iℌψx−12πGxdx.
Now, sinceψ, ψ^∈L1ℝ and xsψx∈L2ℝ, it follows that xuψx∈L2ℝ, foru=0,1,…,s. Using the moment formula for Hilbert transform, (11) can be written as(12)∫ℝxuBLikeFψxdx=∫ℝxuψx−iℌxuψx+1π∑j=0u−1xj∫ℝqu−1−jψqdqdx.
In view of Theorem1 and since xuψx∈L2ℝ, for u=0,1,…,s, it follows that ∫ℝxuBLikeFψxdx=0 for u=0,1,2,…,s.In [14], a sufficient condition was presented in order to obtain higher vanishing moments of wavelets. The result is stated as the following.Theorem 3.
Letψx be such that for some s∈ℕ, xsψx,ηs+1ψ^η∈L1ℝ. If ψj,kxj,k∈ℤ is an orthogonal system on ℝ, then(13)∫ℝxuψxdx=0,for0≤u≤s.Next result generalizes Theorem3 for Fourier–Boas-Like wavelets.Theorem 4.
Letψ, ψ1, ψ^∈L1ℝ be such that for some s∈ℕ,xsψx∈L1ℝ∩L2ℝ,ηs+1ψ^η∈L1ℝ, and ∫ℝxuGxdx=0,for0≤u≤s, where Gx=∫−111−1/ηe−2πiηxψ1^−ηdη. If ψj,kj,k∈ℤ is an orthogonal system on ℝ, then ∫ℝxuBLikeFψxdx=0,for0≤u≤s.Proof.
We have(14)∫ℝxuBLikeFψxdx=∫ℝxuψxdx−i∫ℝℌxuψxdx−iπ∑j=0u−1∫ℝxj∫ℝqu−1−jψqdqdx.
Sincexuψx∈L2ℝ for u=0,1,…,s, by Theorem 3, it follows that ∫ℝxuBLikeFψxdx=0,foru=0,1,2,…,s.In [6], regularity of orthonormal wavelet bases was studied and a relationship between the regularity of wavelet and the multiplicity of the zero at η=0 of ψ^ was observed. This observation can be seen and deduced using the following result which is given in more generalized form.Theorem 5.
Letf,f˜ be two functions (not identically constant) such that(i)
fj,k,f˜j′,k′=δjj′δkk′, where fj,kx=2j/2f2jx−k,  f˜j,kx=2j/2f˜2jx−k.(ii)
f˜x≤M1+x−γ, with γ>s+1.(iii)
f∈Cs, with fu bounded for u≤s.
Then,∫ℝxuf˜xdx=0 for  u=0,1,…,s.The next result generalizes Theorem5 and depicts the relationship among the regularity of orthonormal wavelets ψ and vanishing moments of Fourier–Boas-Like wavelets.Theorem 6.
Letψ, ψ1, ψ^∈L1ℝ and let the system ψj,k=2j/2ψ2jx−k, j, k∈ℤ form an orthonormal set in L2ℝ such that ψx≤M1+x−s−1−ϱ, where ϱ>0 and ψ∈Csℝ such that ψu is bounded for u≤s. Also, if xsψx∈L2ℝ and ∫ℝxuGxdx=0, for 0≤u≤s, where Gx=∫−111−1/ηe−2πiηxψ1^−ηdη, then ∫ℝxuBLikeFψxdx=0foru=0,1,2,…,s.Proof.
Note that(15)∫ℝxuBLikeFψxdx=∫ℝxuψxdx−i∫ℝℌxuψxdx−iπ∑j=0u−1∫ℝxj∫ℝqu−1−jψqdqdx.
Therefore, in view of Theorem5 and the fact that xuψx∈L2ℝ, foru=0,1,…,s, it follows that ∫ℝxuBLikeFψxdx=0,foru=0,1,2,…,s.Recall from [12] that wavelets associated with projection operators ℙ+ and ℙ− involving the Hilbert transform are defined as ℙ+=1/2ℑ+iℌ and ℙ−=1/2ℑ−iℌ, where ℑ and ℌ represent the identity operator and Hilbert transform operator, respectively. These projection operators are also known as Riesz projectors.In the following result, the moment formula for Fourier–Boas-Like wavelets is given.Theorem 7.
Letψ∈L1ℝ be a wavelet such that ψ^∈L1ℝ and xsψx∈L1ℝ∩L2ℝ, for some s∈ℕ. Then,(16)BLikeFxsψx=2xsℙ−ψx+iπ∑p=0s−1xp∫ℝqs−1−pψqdq+−12πis∫−11ψ^sηe−2πiηxsgnη−ηdη,where ℙ−ψ is a wavelet associated with Riesz projectors.Proof.
We compute(17)BLikeFxsψx=xsψx−iBxsψx=2xsℙ−ψx+iπ∑p=0s−1xp∫ℝqs−1−pψqdq+i∫ℝFℌtsψtηFTxgtηdη=2xsℙ−ψx+iπ∑p=0s−1xp∫ℝqs−1−pψqdq+i∫ℝ−isgnηFtsψtηE−xg^ηdη=2xsℙ−ψx+iπ∑p=0s−1xp∫ℝqs−1−pψqdq+−12πis∫−11ψ^sηe−2πiηxsgnη−ηdη.A continual filter is defined byφfx=ℙ−fx, where f∈L2ℝ and φ:L2ℝ⟶L2ℝ. Note that φ is said to be a convolution filter if φx=w∗x for any x, where w∈L1∩L2ℝ is a weight function and w^η is known as the transfer function. For more details, one may see [12].In the next result, we give a sufficient condition under whichℙ−BLikeFψx, i.e., Fourier–Boas-Like wavelets associated with Riesz projectors form a convolution filter and Fourier transform of ℙ−BLikeFψx vanishes for all positive frequencies.Theorem 8.
Letψ∈L1ℝ be a wavelet such that ψ^∈L1ℝandψ^0=0.Then, ℙ−BLikeFψx is a convolution filter with transfer function(18)2,ifη<−1,1−η,if−1≤η<0,12,ifη=0,0,ifη>0,and Fourier transform of ℙ−BLikeFψx vanishes for all positive frequencies.Proof.
Note that(19)φBLikeFψx=ℙ−BLikeFψx=12BLikeFψx−i1πp.v.1x∗BLikeFψx=ℙ−δ∗BLikeFψx=ℙ−δ∗ψx−iℙ−δ∗ℌψ−ℌψ∗gx=ℙ−δ∗ψx−iBℙ−δ∗ψx.
Thus,ℙ−BLikeFψx is a convolution filter. Clearly, we calculate(20)FBLikeFℙ−δη=Fℙ−δη1−sgnη1−g^η=2,ifη<−1,1−η,if−1≤η<0,12,ifη=0,0,ifη>0.
Also, we evaluate(21)Fℙ−BLikeFψη=2ψ^η,ifη<−1,1−ηψ^η,if−1≤η<0,ψ^η2,ifη=0,0,ifη>0.Letαx∈L1ℝ and let hx=α∗ψx=∫ℝαx−tψtdt, where ψt is continuous and bounded. The kernel αx is variation diminishing if ηα∗ψ≤ηψ, where ηψ denotes number of changes of sign of ψ on ℝ. It was found that αx is variation diminishing if and only if(22)α^η=∫ℝαxe−iηxdx=elη2+imη∏n∈ℕ1−iηdneiη/dn−1,where l,m≠0anddnn∈ℕ are real numbers with ∑n∈ℕdn−2<∞.This gives(23)αx=12π∫ℝeiηxelη2+imη∏n∈ℕ1−iηdneiη/dn−1dη,(24)αsx=12π∫ℝiηseiηxelη2+imη∏n∈ℕ1−iηdneiη/dn−1dη.Letψ be a wavelet. Then, for a variation diminishing kernel given by (23) such that ∑n∈ℕdn−2<∞,ψ∗α is known as wavelet, a wavelet with specific changes of sign. For further details, one may read [23, 24].In the given result, we show thatBLikeFψ∗α is also a variation diminishing wavelet.Theorem 9.
Letψ∈L1ℝ be a wavelet such that ψ^∈L1ℝandψ^0=0. Let α∈L1ℝ be a variation diminishing kernel. Then, BLikeFψ∗α is a variation diminishing wavelet.Proof.
SinceBLikeFψ∈L2ℝandα∈L1ℝ, it follows that BLikeFψ∗α∈L2ℝ.
Also note that(25)BLikeFψ1=ψ−iBψ1=ψ−iℌψ+iℌψ∗g1≤ψ1+ℌψ1+ℌψ∗g1.
Sinceψ^∈L1ℝandψ^0=0, it follows that ℌψ∈L1ℝ. Thus,(26)BLikeFψ1≤ψ1+ℌψ1+ℌψ1g1<+∞.
Hence,(27)CBLikeFψ∗α=∫ℝFBLikeFψ∗αη2ηdη,=∫ℝFBLikeFψη2α^η2ηdη≤4DCψ,ifη<−1,D∫ℝψη1−η2ηdη,ifη≤1,η≠0,DCψ,ifη=0,0,ifη>1,≤4ACψ,ifη<−1,4ACψ,ifη≤1,η≠0,ACψ,ifη=0,0,ifη>1,<+∞,where D=supη|eiη2+imη∏n∈ℕ1−iη/dneiη/dn|−2 and A=supη1/∏n∈ℕ1−η2/dn2.


## 3. Conclusions

New wavelet functions, called Fourier–Boas-Like wavelets, have been established. These wavelets have been found to be better than the earlier proposed wavelets in [12, 21] derived from the Riesz projectors and Fourier kernels, respectively. Various results related to higher vanishing moments of Fourier–Boas-Like wavelets have been given, and it is observed that regularity and fast decay are significant attributes for the vanishing moments of Fourier–Boas-Like wavelets. It has also been investigated that under some conditions, Fourier–Boas-Like wavelets associated with Riesz projectors form a convolution filter. Further, using Schoenberg’s theory of variation diminishing integral operators of convolution type, variation diminishing wavelet associated with Fourier–Boas-Like wavelet is constructed.



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