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The Continuous Wavelet Transform and Window Functions

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The Continuous Wavelet Transform and Window Functions PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 11, November 2015, Pages 4759–4773 http://dx.doi.org/10.1090/proc/12590 Article electronically published on July 24, 2015 THE CONTINUOUS WAVELET TRANSFORM AND WINDOW FUNCTIONS J. N. PANDEY AND S. K. UPADHYAY (Communicated by Ken Ono) Abstract. We define a window function ψ as an element of L2(Rn)satis- fying certain boundedness properties with respect to the L2(Rn)normand prove that it satisfies the admissibility condition if and only if the integral of ψ(x1,x2, ··· ,xn) with respect to each of the variables x1,x2, ··· ,xn along the real line is zero. We also prove that each of the window functions is an element of L1(Rn). A function ψ ∈ L2(Rn) satisfying the admissibility con- dition is a wavelet. We define the wavelet transform of f ∈ L2(Rn)(which is a window function) with respect to the wavelet ψ ∈ L2(Rn) and prove an inversion formula interpreting convergence in L2(Rn). It is also proved that at a point of continuity of f the convergence of our wavelet inversion formula is in a pointwise sense. 1. Introduction The necessity for the inversion formula for the wavelet transform in dimensions higher than 1 has long been felt. Before the inversion formula for a two-dimensional wavelet transform was known, workers in image processing used to separate two- dimensional wavelets into the product of two one-dimensional wavelets, thereby making the job very simple [12]. But even two-dimensional wavelets are sometimes hard to separate and are sometimes impossible. Consider the wavelet ψ(x1,x2)in R2: −(x2+1)(x2+1) 2 ψ(x1,x2)=x1x2e 1 2 ∈ S(R ). This cannot be separated into the product of two one-dimensional wavelets so the use of the inversion formula for the wavelets in dimensions higher than 1 was needed by workers in image processing. Consider another wavelet in R2 given by − 1 1−(x2+x2) x1x2e 1 2 , |x| < 1, ψ(x1,x2)= 0, |x|≥1, | | 2 2 x = x1 + x2. Received by the editors April 17, 2014 and, in revised form, July 2, 2014. 2010 Mathematics Subject Classification. Primary 46F12; Secondary 46F05, 46F10. Key words and phrases. Continuous wavelet transform, Fourier inversion theory, inverse wavelet transform. c 2015 American Mathematical Society 4759 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4760 J. N. PANDEY AND S. K. UPADHYAY This wavelet is a wavelet of compact support in R2 and cannot be separated. Now consider ⎧ ⎨ ≤ ≤ 1 1, 0 x 2 , (1.1) ψ(x)= −1, 1 <x≤ 1, ⎩ 2 0, elsewhere, where ψ(x) is a continuous wavelet. Using this wavelet we can define a two-dimen- sional wavelet of separable type by (1.2) ψ(x1,x2)=ψ(x1)ψ(x2). So one can work with the one-dimensional wavelet inversion formula in this case but not in the aforesaid previous two cases, as those are not the wavelets of separable type [12, p. 331]. Many authors worked on the inversion formula for the n-dimensional wavelet transform such as Daubechies [5], Meyer [7], Pathak [11], Keinert [6] and others. But the most notable amongst them are the works of Dauchechies [5, pp. 33–34] and Meyers [7, pp. 125–126]. We have proved the following wavelet inversion formula for f ∈ L2(Rn). Our wavelet is a window function whose integral along each of the axes is zero, and our inversion formula is as follows: − 1 | |−1/2 x b da db a ψ( )Wf (a, b) 2 = f, Cψ Rn Rn a |a| where 1 t − b Wf (a, b)= f(t),ψ( ) |a| a and |a| = |a1 · a2 ···an|. If we take x =(x1,x2,...,xn), Λ=(λ1,λ2,...,λn)and |Λ| = |λ1 · λ2 ···λn|,then | ˆ |2 n ψ(Λ) ∞ Cψ =(2π) | | dΛ < (admissibility condition), Λn Λ where ˆ ˆ 2 n ψ(λ1,λ2, ··· ,λn)=ψ(Λ), the Fourier transform of ψ(x) ∈ L (R ). The convergence is interpreted in the L2(Rn) sense and at a point of continuity of f the convergence is in the pointwise sense. Note that in our derivation as shown later, we take a =(a1,a2,...,an) whereas Daubechies and Meyer take a =(a,a,...,a) and we do not choose ψ as spherically symmetric. In our case a1,a2,...,an are all non-zero real numbers, whereas in their cases a>0. We have used the Fourier inversion theorem to prove the wavelet inversion the- orem. The advantage of this method is that it is simple and to prove the pointwise convergence only the continuity of f is required and the continuity of the wavelet ψ at a point x = x0 concerned is not required. To prove the pointwise convergence using the Hilbert space technique, we require the continuity of the function f and the wavelet ψ both at a point x = x0 concerned [4, p. 63]. It sounds quite strange but that is the way it is! Our inversion formula is valid over Rn × Rn whereas the formulas derived by Daubechies and Meyer are valid only at R × Rn. Thus our formula is more general than their inversion formula. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONTINUOUS WAVELET TRANSFORM AND WINDOW FUNCTIONS 4761 We will give a characterisation of a subclass of functions belonging to L2(Rn) ∩ L1(Rn) in the proof of the Corollary to Theorem 3.3, which are wavelets. This makes the construction of wavelets useful to applied scientists much easier. 2. Definitions and preliminaries Definition 2.1. A function f ∈ L2(Rn) is called a window function if it satisfies the following conditions: 2 n (1) x1f,x2f,...,xnf all belong to L (R ). 2 n (2) xixj f ∈ L (R ) for all i, j =1, 2,...,n, i= j. 2 n (3) xixj xkf ∈ L (R ) for all i, j, k =1, 2,...,n, i = j = k = i.Notethat i = j = k = i implies that i, j, k are all different, i = j = k may imply that i and k could be equal. Finally we have 2 n (n) x1x2 ...xn f ∈ L (R ). Here, the lower suffixes in a term are all different. Let us illustrate this definition with reference to n =2.Soiff ∈ L2(R2)wemusthave 2 2 x1f,x2f ∈ L (R ), 2 2 x1x2f ∈ L (R ). Example 2.2. Define ⎧ ⎨ ≤ ≤ 1 1, 0 x 2 , ψ(x)= −1, − 1 <x≤ 0, ⎩ 2 0, elsewhere. Then clearly ψ is a window function in L2(R), and is a continuous wavelet. −(x2+x2+···+x2 ) 2 n Example 2.3. ψ(x)=(x1,x2...xn)e 1 2 n is a window function in L (R ), and is a wavelet. The wavelet ψ that we have chosen satisfies the following conditions: (1) ψ ∈ L2(Rn). (2) ψ is a window function as defined above. ∞ ∀ (3) −∞ ψ(x)dxi =0, i =1, 2,...,n. We will show that under the set of conditions (1), (2) and (3) the wavelet kernel ψ satisfies the admissibility condition | ˆ |2 ψ(Λ) ∞ | | dΛ < , Λn Λ which enables us to prove the aforesaid wavelet inversion formula. 3. Orthants and pseudo-orthants Let Rn stand for the n-dimensional Euclidean space. Then Rn = R×R×···×R, n times (Cartesian product): n Rn R ∪ R (3.1) = |xi|≥a |xi|≤a ,a>0, i=1 R ∈ Rn | |≥ |xi|≥a = x : xi a, x =(x1,x2,...,xn) , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4762 J. N. PANDEY AND S. K. UPADHYAY where i =1, 2,...,n. | |≥ | |≥ | |≥ | |≤ | |≤ | | Let k = x : xj1 a, xj2 a,..., xjk a; xjk+1 a, xjk+2 a,..., xjn ≤ a ,wherej1,j2,...,jn are permutations of 1, 2, 3,...,n; k is known as a pseudo- Rn orthant and when a =0, k is an orthant in . Rn n n ∈ Now = k=0 k . Then the inner union has k terms for a fixed k n n ··· (0,1, 2,...,n). Varying k from 0, 1, 2,...,n, we see that there are 0 + 1 + + n n Rn n =2 pseudo-orthants. The total number of pseudo-orthants in can be easily figured out from the representation (3.1). There are n factors in the representation (3.1) of Rn and there are two elements in each of the factors so there will be 2n pseudo-orthants in Rn. From now on we will choose a = 1 as there is no loss of generality in doing so. Theorem 3.1. Let f ∈ L2(Rn) be a window function on Rn.Thenf ∈ L1(Rn). Proof. By using Holder’s inequality one can see that |f(x)|dx k | ··· | 1 = xj1 xj2 xjk f(x) dx |x x ···x | k j1 j2 jk 1 1 1 ≤ | ··· |2 2 2 xj1 xj2 xjk f(x) dx 2 dx |x x ···x | k k j1 j2 jk 1 1 1 ≤ | ··· |2 2 |2 2 xj1 xj2 xjk f(x) dx dx Rn |x x ···x k j1 j2 jk 1 1 ≤ ··· 2 xj1 xj2 xjk f(x) 2 2 2 2 dx x x ···x k j1 j2 jk −1 +∞ 1 1 ≤ ··· 2 xj1 xj2 xjk f 2 + 2 dxj1 −∞ 1 x j1 −1 +∞ −1 +∞ 1 1 1 1 × + dx 2 ··· + dx 2 2 j2 2 jk −∞ 1 x −∞ 1 x j2 jk 1 1 1 1 × ··· 2 dxjk+1 dxjk+2 dxjk+n−k −1 −1 −1 k n−k ≤ ··· 2 · 2 xj1 xj2 xjk f 22 2 n ≤ 2 ··· 2 xj1 xj2 xjk f 2, and n | | ≤ 2 ··· ··· f(x) dx 2 xj1 xj2 xji xjk f 2, k n where the number of terms in the R.H.S.
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