Construction of Orthogonal and Biorthogonal Product Systems Bal - PowerPoint PPT Presentation

1/18 First Back Construction of Orthogonal and Biorthogonal Product Systems Bal´ azs Kir´ aly 6 th Workshop on Fourier Analysis and Related Fields, P´ ecs, Hungary 2017.

2/18 First Back Rademacher and Walsh Systems

2/18 First Back Rademacher and Walsh Systems • The Rademacher functions r n ( n ∈ N ) can be derived from the basic function r by dilation: r n ( x ) := r (2 n x ) ( x ∈ [0 , 1) , n ∈ N ) 1 , x ∈ [ k, k + 1 � 2 ) , k ∈ Z , r ( x ) := − 1 , x ∈ [ k + 1 2 , k + 1) , k ∈ Z .

2/18 First Back Rademacher and Walsh Systems • The Rademacher functions r n ( n ∈ N ) can be derived from the basic function r by dilation: r n ( x ) := r (2 n x ) ( x ∈ [0 , 1) , n ∈ N ) 1 , x ∈ [ k, k + 1 � 2 ) , k ∈ Z , r ( x ) := − 1 , x ∈ [ k + 1 2 , k + 1) , k ∈ Z . • The Walsh system is the product system of the Rademacher system i.e. ∞ � r m k w m = k , k =0 where ∞ � m k · 2 k , m = m k ∈ { 0 , 1 } . k =0

2/18 First Back Rademacher and Walsh Systems • The Rademacher functions r n ( n ∈ N ) can be derived from the basic function r by dilation: r n ( x ) := r (2 n x ) ( x ∈ [0 , 1) , n ∈ N ) 1 , x ∈ [ k, k + 1 � 2 ) , k ∈ Z , r ( x ) := − 1 , x ∈ [ k + 1 2 , k + 1) , k ∈ Z . • The Walsh system is the product system of the Rademacher system i.e. ∞ � r m k w m = k , k =0 where ∞ � m k · 2 k , m = m k ∈ { 0 , 1 } . k =0 The w m ( m ∈ N ) Walsh system is a complete orthonormal system with respect to the scalar product 1 � � f, g � = f ( t ) · g ( t ) dt. 0

3/18 First Back Haar System

3/18 First Back Haar System • The Haar-system was presented in 1910 by Alfr´ ed Haar.

3/18 First Back Haar System • The Haar-system was presented in 1910 by Alfr´ ed Haar. • The original definition � 1 x ∈ [0 , 1) h 0 ( x ) = χ [0 , 1) = 0 otherwise, n x ∈ [ k 2 n , 2 k +1  2 2 n +1 ) 2  n 2 h (2 n x − k ) = n x ∈ [ 2 k +1 2 n +1 , k +1 h m ( x ) = h n,k ( x ) = 2 − 2 2 n ) 2 0 otherwise,  where m = 2 n + k and n ∈ N , 0 ≤ k < 2 n .

3/18 First Back Haar System • The Haar-system was presented in 1910 by Alfr´ ed Haar. • The original definition � 1 x ∈ [0 , 1) h 0 ( x ) = χ [0 , 1) = 0 otherwise, n x ∈ [ k 2 n , 2 k +1  2 2 n +1 ) 2  n 2 h (2 n x − k ) = n x ∈ [ 2 k +1 2 n +1 , k +1 h m ( x ) = h n,k ( x ) = 2 − 2 2 n ) 2 0 otherwise,  where m = 2 n + k and n ∈ N , 0 ≤ k < 2 n . • The Haar system was the first and the simplest wavelet.

3/18 First Back Haar System • The Haar-system was presented in 1910 by Alfr´ ed Haar. • The original definition � 1 x ∈ [0 , 1) h 0 ( x ) = χ [0 , 1) = 0 otherwise, n x ∈ [ k 2 n , 2 k +1  2 2 n +1 ) 2  n 2 h (2 n x − k ) = n x ∈ [ 2 k +1 2 n +1 , k +1 h m ( x ) = h n,k ( x ) = 2 − 2 2 n ) 2 0 otherwise,  where m = 2 n + k and n ∈ N , 0 ≤ k < 2 n . • The Haar system was the first and the simplest wavelet. • Wavelet construction Haar-functions can be derived from the basic function  1 , (0 ≤ x < 1 / 2) ,   h ( x ) := h 1 ( x ) = − 1 , (1 / 2 ≤ x < 1) ,  0 , (1 ≤ x < ∞ ) .  by translation and dilation: h n,k ( x ) := 2 n/ 2 h (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) .

4/18 First Back Haar Scaling Functions

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions.

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions. • These functions can be derived similarly to Haar-functions starting from χ [0 , 1) so χ n,k ( x ) = χ (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) .

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions. • These functions can be derived similarly to Haar-functions starting from χ [0 , 1) so χ n,k ( x ) = χ (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) . • For the Haar functions and for the Haar scaling functions the following are true (0 ≤ k < 2 n , n ∈ N ) , χ n,k = χ n +1 , 2 k + χ n +1 , 2 k +1 , h n,k = 2 n/ 2 ( χ n +1 , 2 k − χ n +1 , 2 k +1 ) (0 ≤ k < 2 n , n ∈ N ) .

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions. • These functions can be derived similarly to Haar-functions starting from χ [0 , 1) so χ n,k ( x ) = χ (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) . • For the Haar functions and for the Haar scaling functions the following are true (0 ≤ k < 2 n , n ∈ N ) , χ n,k = χ n +1 , 2 k + χ n +1 , 2 k +1 , h n,k = 2 n/ 2 ( χ n +1 , 2 k − χ n +1 , 2 k +1 ) (0 ≤ k < 2 n , n ∈ N ) . • The Haar-Fourier analysis and synthesis are based on these equations. ( O (2 N ) operations)

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions. • These functions can be derived similarly to Haar-functions starting from χ [0 , 1) so χ n,k ( x ) = χ (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) . • For the Haar functions and for the Haar scaling functions the following are true (0 ≤ k < 2 n , n ∈ N ) , χ n,k = χ n +1 , 2 k + χ n +1 , 2 k +1 , h n,k = 2 n/ 2 ( χ n +1 , 2 k − χ n +1 , 2 k +1 ) (0 ≤ k < 2 n , n ∈ N ) . • The Haar-Fourier analysis and synthesis are based on these equations. ( O (2 N ) operations) • The 2 n -th Dirichlet kernel of the Walsh-system is of the form 2 n − 1 n − 1 � � w k ( x ) · w k ( y ) = (1 + r j ( x ) r j ( y )) ( x, y ∈ [0 , 1) , n ∈ N ) D 2 n ( x, y ) = k =0 j =0

4/18 First Back Haar Scaling Functions • The χ n,k characteristic functions of dyadic intervals are known as the Haar scaling functions. • These functions can be derived similarly to Haar-functions starting from χ [0 , 1) so χ n,k ( x ) = χ (2 n x − k ) ( x ∈ R , 0 ≤ k < 2 n , n ∈ N ) . • For the Haar functions and for the Haar scaling functions the following are true (0 ≤ k < 2 n , n ∈ N ) , χ n,k = χ n +1 , 2 k + χ n +1 , 2 k +1 , h n,k = 2 n/ 2 ( χ n +1 , 2 k − χ n +1 , 2 k +1 ) (0 ≤ k < 2 n , n ∈ N ) . • The Haar-Fourier analysis and synthesis are based on these equations. ( O (2 N ) operations) • The 2 n -th Dirichlet kernel of the Walsh-system is of the form 2 n − 1 n − 1 � � D 2 n ( x, y ) = w k ( x ) · w k ( y ) = (1 + r j ( x ) r j ( y )) ( x, y ∈ [0 , 1) , n ∈ N ) j =0 k =0 • The Haar-functions and the Haar scaling functions can be expressed as h n,k ( x ) = 2 − n/ 2 r n ( x ) D 2 n ( x, k 2 − n ) χ n,k ( x ) = 2 − n D 2 n ( x, k 2 − n ) ( x ∈ [0 , 1) , 0 ≤ k < 2 n , n ∈ N ) .

5/18 First Back Generalization of Product system • Let us fix the number p ∈ N ∗∗ := { 2 , 3 , · · · } and the non-empty set X . Let us start from the following finite collection of the systems φ n = ( ϕ ( i ) ϕ ( i ) n , 0 ≤ i < p ) , (0 ≤ n < N ≤ ∞ , n : X → C )

5/18 First Back Generalization of Product system • Let us fix the number p ∈ N ∗∗ := { 2 , 3 , · · · } and the non-empty set X . Let us start from the following finite collection of the systems φ n = ( ϕ ( i ) ϕ ( i ) n , 0 ≤ i < p ) , (0 ≤ n < N ≤ ∞ , n : X → C ) • Then N − 1 N − 1 � ϕ ( m k ) � m k · p k m k ∈ { 0 , 1 , . . . , p − 1 } Φ m = , m = k k =0 k =0 is the generalized product system of the systems φ .

5/18 First Back Generalization of Product system • Let us fix the number p ∈ N ∗∗ := { 2 , 3 , · · · } and the non-empty set X . Let us start from the following finite collection of the systems φ n = ( ϕ ( i ) ϕ ( i ) n , 0 ≤ i < p ) , (0 ≤ n < N ≤ ∞ , n : X → C ) • Then N − 1 N − 1 ϕ ( m k ) � � m k · p k Φ m = , m = m k ∈ { 0 , 1 , . . . , p − 1 } k k =0 k =0 is the generalized product system of the systems φ . • It is really the generalization of idea of the product system, because in special case when p = 2 and ϕ (0) m = 1, ϕ (1) m = r m we reobtain the Walsh system.

6/18 First Back Biorthogonality with respect to a p -fold map • Let us fix the number p ∈ N ∗∗ := { 2 , 3 , · · · } and the set X � = ∅ . The map A : X → X ′ is called p -fold map on set X if every x ∈ X ′ has exactly p preimages, i.e. the set A − 1 ( x ) = { x 0 , x 1 , . . . , x p − 1 } x ∈ X ′ . has p elements.

6/18 First Back Biorthogonality with respect to a p -fold map • Let us fix the number p ∈ N ∗∗ := { 2 , 3 , · · · } and the set X � = ∅ . The map A : X → X ′ is called p -fold map on set X if every x ∈ X ′ has exactly p preimages, i.e. the set A − 1 ( x ) = { x 0 , x 1 , . . . , x p − 1 } x ∈ X ′ . has p elements. • Let f ( j ) , g ( j ) : X → C , j = 0 , 1 , . . . , p − 1 . and ρ : X → (0 , + ∞ ) is a positive weight-function. The system F = ( f (0) , f (1) , . . . f ( p − 1) ) and the system G = ( g (0) , g (1) , . . . g ( p − 1) ) is called ( A, ρ ) -biorthogonal if for every x ∈ X ′ p − 1 � f ( i ) ( t ) · g ( j ) ( t ) ρ ( t ) = � f ( i ) ( x k ) · g ( j ) ( x k ) ρ ( x k ) = δ ij ( � ) k =0 t ∈ A − 1 ( x ) if F = G the system F is called ( A, ρ ) -orthonormal .