---
title: "Cauchy wavelet"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Cauchy_wavelet"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T07:23:27.637906+00:00"
instance: "kb-cron"
---

  
    
      
        f
        (
        t
        )
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        
          L
          
            n
          
        
        (
        f
        )
        
          e
          
            j
            n
            t
          
        
      
    
    {\displaystyle f(t)=\sum _{n=-\infty }^{\infty }L_{n}(f)e^{jnt}}
  
. 
and in fact we have Parseval's identity

  
    
      
        
          |
        
        
          |
        
        f
        
          |
        
        
          
            |
          
          
            2
          
        
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        
          |
        
        
          L
          
            n
          
        
        (
        f
        )
        
          
            |
          
          
            2
          
        
      
    
    {\displaystyle ||f||^{2}=\sum _{n=-\infty }^{\infty }|L_{n}(f)|^{2}}
  
. 
where 
  
    
      
        
          |
        
        
          |
        
        f
        
          |
        
        
          
            |
          
          
            2
          
        
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            −
            π
          
          
            π
          
        
        
          |
        
        f
        (
        t
        )
        
          
            |
          
          
            2
          
        
        
        d
        t
      
    
    {\displaystyle ||f||^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(t)|^{2}\,dt}
  
 i.e. the norm defined in 
  
    
      
        
          L
          
            2
          
        
        (
        [
        −
        π
        ,
        π
        ]
        )
      
    
    {\displaystyle L^{2}([-\pi ,\pi ])}
  
. 
Hence, in this example, the index set 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is the integer 
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  
, the vector space 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is 
  
    
      
        
          L
          
            2
          
        
        (
        [
        −
        π
        ,
        π
        ]
        )
      
    
    {\displaystyle L^{2}([-\pi ,\pi ])}
  
 and the linear form 
  
    
      
        
          L
          
            n
          
        
      
    
    {\displaystyle L_{n}}
  
 is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients 
  
    
      
        {
        
          |
        
        
          L
          
            n
          
        
        (
        f
        )
        
          |
        
        
          }
          
            n
            ∈
            
              Z
            
          
        
      
    
    {\displaystyle \{|L_{n}(f)|\}_{n\in \mathbb {Z} }}
  
 can only determine the norm of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 defined in 
  
    
      
        
          L
          
            2
          
        
        (
        [
        −
        π
        ,
        π
        ]
        )
      
    
    {\displaystyle L^{2}([-\pi ,\pi ])}
  
.

=== Unicity Theorem of the reconstruction ===
Firstly, we define the Cauchy wavelet transform as:

  
    
      
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        x
        (
        t
        )
        ]
        (
        a
        ,
        b
        )
        =
        
          
            1
            b
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        x
        (
        t
        )
        
          
            
              
                ψ
                
                  p
                
              
              (
              
                
                  
                    t
                    −
                    a
                  
                  b
                
              
              )
            
            ¯
          
        
        
        d
        t
      
    
    {\displaystyle W_{\psi _{p}}[x(t)](a,b)={\frac {1}{b}}\int _{-\infty }^{\infty }x(t){\overline {\psi _{p}({\frac {t-a}{b}})}}\,dt}
  
. 
Then, the theorem is:

Theorem. For a fixed 
  
    
      
        p
        >
        0
      
    
    {\displaystyle p>0}
  
, if exist two different numbers 
  
    
      
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        >
        0
      
    
    {\displaystyle b_{1},b_{2}>0}
  
 and the Cauchy wavelet transform defined as above. Then, if there are two real-valued functions 
  
    
      
        f
        ,
        g
        ∈
        
          L
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle f,g\in L^{2}(\mathbb {R} )}
  
 satisfied 

  
    
      
        
          |
        
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        f
        (
        t
        )
        ]
        (
        a
        ,
        
          b
          
            1
          
        
        )
        
          |
        
        =
        
          |
        
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        g
        (
        t
        )
        ]
        (
        a
        ,
        
          b
          
            1
          
        
        )
        
          |
        
      
    
    {\displaystyle |W_{\psi _{p}}[f(t)](a,b_{1})|=|W_{\psi _{p}}[g(t)](a,b_{1})|}
  
, 
  
    
      
        ∀
        a
        ∈
        
          R
        
      
    
    {\displaystyle \forall a\in \mathbb {R} }
  
 and 

  
    
      
        
          |
        
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        f
        (
        t
        )
        ]
        (
        a
        ,
        
          b
          
            2
          
        
        )
        
          |
        
        =
        
          |
        
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        g
        (
        t
        )
        ]
        (
        a
        ,
        
          b
          
            2
          
        
        )
        
          |
        
      
    
    {\displaystyle |W_{\psi _{p}}[f(t)](a,b_{2})|=|W_{\psi _{p}}[g(t)](a,b_{2})|}
  
, 
  
    
      
        ∀
        a
        ∈
        
          R
        
      
    
    {\displaystyle \forall a\in \mathbb {R} }
  
, 
then there is a 
  
    
      
        α
        ∈
        
          R
        
      
    
    {\displaystyle \alpha \in \mathbb {R} }
  
 such that 
  
    
      
        
          f
          
            +
          
        
        (
        t
        )
        =
        
          e
          
            j
            α
          
        
        
          g
          
            +
          
        
        (
        t
        )
      
    
    {\displaystyle f_{+}(t)=e^{j\alpha }g_{+}(t)}
  
. 

  
    
      
        
          f
          
            +
          
        
        (
        t
        )
        =
        
          e
          
            j
            α
          
        
        
          g
          
            +
          
        
        (
        t
        )
      
    
    {\displaystyle f_{+}(t)=e^{j\alpha }g_{+}(t)}
  
 implies that 

  
    
      
        R
        e
        {
        
          f
          
            +
          
        
        (
        t
        )
        }
        =
        R
        e
        {
        
          e
          
            j
            α
          
        
        
          g
          
            +
          
        
        (
        t
        )
        }
        
        ⟹
        
        f
        (
        t
        )
        =
        cos
        ⁡
        
          α
        
        g
        (
        t
        )
        −
        sin
        ⁡
        
          α
        
        
          g
          
            H
          
        
        (
        t
        )
      
    
    {\displaystyle Re\{f_{+}(t)\}=Re\{e^{j\alpha }g_{+}(t)\}\implies f(t)=\cos {\alpha }g(t)-\sin {\alpha }g_{H}(t)}
  
 and 

  
    
      
        I
        m
        {
        
          f
          
            +
          
        
        (
        t
        )
        }
        =
        I
        m
        {
        
          e
          
            j
            α
          
        
        
          g
          
            +
          
        
        (
        t
        )
        }
        
        ⟹
        
        
          f
          
            H
          
        
        (
        t
        )
        =
        sin
        ⁡
        
          α
        
        g
        (
        t
        )
        +
        cos
        ⁡
        
          α
        
        
          g
          
            H
          
        
        (
        t
        )
      
    
    {\displaystyle Im\{f_{+}(t)\}=Im\{e^{j\alpha }g_{+}(t)\}\implies f_{H}(t)=\sin {\alpha }g(t)+\cos {\alpha }g_{H}(t)}
  
.
Hence, we get the relation 

  
    
      
        f
        (
        t
        )
        =
        (
        cos
        ⁡
        
          α
        
        −
        sin
        ⁡
        
          α
        
        tan
        ⁡
        
          α
        
        )
        g
        (
        t
        )
        −
        tan
        ⁡
        
          α
        
        
          f
          
            H
          
        
        (
        t
        )
      
    
    {\displaystyle f(t)=(\cos {\alpha }-\sin {\alpha }\tan {\alpha })g(t)-\tan {\alpha }f_{H}(t)}
  
 
and 
  
    
      
        f
        (
        t
        )
        ,
        
          g
          
            H
          
        
        (
        t
        )
        ∈
        s
        p
        a
        n
        {
        
          f
          
            H
          
        
        (
        t
        )
        ,
        g
        (
        t
        )
        }
        =
        s
        p
        a
        n
        {
        f
        (
        t
        )
        ,
        
          f
          
            H
          
        
        (
        t
        )
        }
        =
        s
        p
        a
        n
        {
        g
        (
        t
        )
        ,
        
          g
          
            H
          
        
        (
        t
        )
        }
      
    
    {\displaystyle f(t),g_{H}(t)\in span\{f_{H}(t),g(t)\}=span\{f(t),f_{H}(t)\}=span\{g(t),g_{H}(t)\}}
  
. 

Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is 
  
    
      
        
          R
        
        ×
        {
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        }
      
    
    {\displaystyle \mathbb {R} \times \{b_{1},b_{2}\}}
  
 with 
  
    
      
        
          b
          
            1
          
        
        ≠
        
          b
          
            2
          
        
      
    
    {\displaystyle b_{1}\neq b_{2}}
  
 and 
  
    
      
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        >
        0
      
    
    {\displaystyle b_{1},b_{2}>0}
  
,  the vector space 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is 
  
    
      
        
          L
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle L^{2}(\mathbb {R} )}
  
 and the linear form 
  
    
      
        
          L
          
            (
            a
            ,
            b
            )
          
        
      
    
    {\displaystyle L_{(a,b)}}
  
 is defined as 
  
    
      
        
          L
          
            (
            a
            ,
            b
            )
          
        
        (
        f
        )
        =
        
          W
          
            
              ψ
              
                p
              
            
          
        
        [
        f
        (
        t
        )
        ]
        (
        a
        ,
        b
        )
      
    
    {\displaystyle L_{(a,b)}(f)=W_{\psi _{p}}[f(t)](a,b)}
  
. Hence, 
  
    
      
        {
        
          |
        
        
          L
          
            (
            a
            ,
            b
            )
          
        
        (
        f
        )
        
          |
        
        
          }
          
            a
            ,
            b
            ∈
            
              R
            
            ×
            {
            
              b
              
                1
              
            
            ,
            
              b
              
                2
              
            
            }
          
        
      
    
    {\displaystyle \{|L_{(a,b)}(f)|\}_{a,b\in \mathbb {R} \times \{b_{1},b_{2}\}}}
  
 determines the two dimensional subspace 
  
    
      
        s
        p
        a
        n
        {
        f
        ,
        
          f
          
            H
          
        
        }
      
    
    {\displaystyle span\{f,f_{H}\}}
  
 in 
  
    
      
        
          L
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle L^{2}(\mathbb {R} )}
  
.

== References ==