Structure of shift-invariant subspaces and their bases for the - PowerPoint PPT Presentation

Structure of shift-invariant subspaces and their bases for the Heisenberg group Azita Mayeli City University of New York Queensborough College Texas A & M University July 17, 2012 Concentration week - Larsonfest 2012

This is a joint work with Brad Curry and Vignon Oussa of St. Louis University

Why do we study shift-invariant subspaces? • Shift-invariant spaces are used as models for spaces of signals and images in math and engineering applications. • The scales of shift-invariant subspaces are "good" approximation spaces of any function, in particular, potential functions, such as Sobolev functions. • They are core spaces of MRAs. • These spaces are used in sampling theory.

Historical comments • In 1994, De Boor, De Vore, and Ron characterized finitely generated shift invariant subspaces of L 2 ( R n ) in terms of range functions introduced by Helson (1964). • In 2000, Bownik extended the results to countably generated shift invariant subspaces of L 2 ( R n ) , and characterized frame and Reisz families. • Kamyabi, and et al. (2008) and Cabrelli and et al. (2009) studied shift-invariant subspaces in the context of locally compact abelian groups. • Aldroubi, Cabrelli, Heil, Hernandez, Kornelson, Molter, Speegle, Sun, Weiss, Wilson, · · ·

The Heisenberg group The Heisenberg group N ≡ R 2 × R . We let the group product ( p , q , t )( p ′ , q ′ , t ′ ) = ( p + p ′ , q + q ′ , t + t ′ + pq ′ ) , ( p , q , t ) − 1 = ( − p , − q , − t + pq ) [( p , q , t ) , ( p ′ , q ′ , t ′ )] = ( 0 , 0 , pq ′ − qp ′ ) . We fix the standard Euclidean measure on R 3 .

The Heisenberg group The Heisenberg group N ≡ R 2 × R . We let the group product ( p , q , t )( p ′ , q ′ , t ′ ) = ( p + p ′ , q + q ′ , t + t ′ + pq ′ ) , ( p , q , t ) − 1 = ( − p , − q , − t + pq ) [( p , q , t ) , ( p ′ , q ′ , t ′ )] = ( 0 , 0 , pq ′ − qp ′ ) . We fix the standard Euclidean measure on R 3 . Schrödinger representation. For λ ∈ R / { 0 } π λ : N → U ( L 2 ( R )) π λ ( p , q , t ) f ( x ) = e 2 π i λ t e − 2 π iq λ x f ( x − p ) = e 2 π i λ t M q λ l p f ( x )

The Heisenberg group The Heisenberg group N ≡ R 2 × R . We let the group product ( p , q , t )( p ′ , q ′ , t ′ ) = ( p + p ′ , q + q ′ , t + t ′ + pq ′ ) , ( p , q , t ) − 1 = ( − p , − q , − t + pq ) [( p , q , t ) , ( p ′ , q ′ , t ′ )] = ( 0 , 0 , pq ′ − qp ′ ) . We fix the standard Euclidean measure on R 3 . Schrödinger representation. For λ ∈ R / { 0 } π λ : N → U ( L 2 ( R )) π λ ( p , q , t ) f ( x ) = e 2 π i λ t e − 2 π iq λ x f ( x − p ) = e 2 π i λ t M q λ l p f ( x ) Dual space. � N ≡ R / { 0 } and we simply show λ ∈ � N .

The Heisenberg group Fourier transform. For f ∈ L 1 ( N ) ∩ L 2 ( N ) and λ ∈ � N � ˆ f ( λ ) = f ( n ) π λ ( n ) dn . n ∈ N

The Heisenberg group Fourier transform. For f ∈ L 1 ( N ) ∩ L 2 ( N ) and λ ∈ � N � ˆ f ( λ ) = f ( n ) π λ ( n ) dn . n ∈ N Definition: f , g , h ∈ L 2 ( R ) ( f ⊗ g )( h ) = � f , h � g

The Heisenberg group Fourier transform. For f ∈ L 1 ( N ) ∩ L 2 ( N ) and λ ∈ � N � ˆ f ( λ ) = f ( n ) π λ ( n ) dn . n ∈ N Definition: f , g , h ∈ L 2 ( R ) ( f ⊗ g )( h ) = � f , h � g Plancherel transform. � ⊕ F : L 2 ( N ) − L 2 ( R ) ⊗ L 2 ( R ) | λ | d λ → � N → � f = { � f − f ( λ ) } � N � � ˆ f ( λ ) � 2 � f � = HS | λ | d λ. � N HS := HS ( L 2 ( R )) = L 2 ( R ) ⊗ L 2 ( R ) . Notation:

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ]

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ] Techniques: Plancherel transform and periodization.

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ] Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L 2 ( N ) . � � 1 � � f � 2 = � � f � 2 = � � � ( α + j ) 1 / 2 � f ( λ ) � 2 f ( α + j ) � 2 HS | λ | d λ = HS d α � N 0 j ∈ Z

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ] Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L 2 ( N ) . � � 1 � � f � 2 = � � f � 2 = � � � ( α + j ) 1 / 2 � f ( λ ) � 2 f ( α + j ) � 2 HS | λ | d λ = HS d α � N 0 j ∈ Z ˆ f α ( j ) := ( α + j ) 1 / 2 ˆ Let f ( α + j ) .

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ] Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L 2 ( N ) . � � 1 � � f � 2 = � � f � 2 = � � � ( α + j ) 1 / 2 � f ( λ ) � 2 f ( α + j ) � 2 HS | λ | d λ = HS d α � N 0 j ∈ Z ˆ f α ( j ) := ( α + j ) 1 / 2 ˆ Let f ( α + j ) . Define T : f → Tf Tf : ( 0 , 1 ] → l 2 ( Z , HS ); α �→ ˆ f α := { ˆ f α ( j ) } j ∈ Z .

The Heisenberg group Lemma � ⊕ L 2 ( N ) ∼ l 2 ( Z , HS ) d α. = ( 0 , 1 ] Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L 2 ( N ) . � � 1 � � f � 2 = � � f � 2 = � � � ( α + j ) 1 / 2 � f ( λ ) � 2 f ( α + j ) � 2 HS | λ | d λ = HS d α � N 0 j ∈ Z f α ( j ) := ( α + j ) 1 / 2 ˆ ˆ Let f ( α + j ) . Define T : f → Tf Tf : ( 0 , 1 ] → l 2 ( Z , HS ); α �→ ˆ f α := { ˆ f α ( j ) } j ∈ Z . We show T is isometric isomorphism and � 1 � f � 2 = � Tf � 2 = � Tf ( α ) � 2 l 2 ( Z , HS ) d α. 0

Shift-invariant spaces Recall: N = R 2 × R . Fix Γ = ( a Z × b Z ) × Z , a , b > 0 .

Shift-invariant spaces Recall: N = R 2 × R . Fix Γ = ( a Z × b Z ) × Z , a , b > 0 . Shift-invariant space. We say a closed subspace V ≤ L 2 ( N ) is shift-invariant if l γ f = f ( γ − 1 · ) ∈ V ∀ f ∈ V , ∀ γ ∈ Γ

Shift-invariant spaces Recall: N = R 2 × R . Fix Γ = ( a Z × b Z ) × Z , a , b > 0 . Shift-invariant space. We say a closed subspace V ≤ L 2 ( N ) is shift-invariant if l γ f = f ( γ − 1 · ) ∈ V ∀ f ∈ V , ∀ γ ∈ Γ A version of range function . Given a Hilbert space H , a measurable map J J : α ∈ ( 0 , 1 ] → the family of closed subspaces of H J : α �→ J ( α ) ≤ H Fiber. We say { J ( α ) } α is fiber of the range function J .

Shift-invariant spaces Recall: N = R 2 × R . Fix Γ = ( a Z × b Z ) × Z , a , b > 0 . Shift-invariant space. We say a closed subspace V ≤ L 2 ( N ) is shift-invariant if l γ f = f ( γ − 1 · ) ∈ V ∀ f ∈ V , ∀ γ ∈ Γ A version of range function . Given a Hilbert space H , a measurable map J J : α ∈ ( 0 , 1 ] → the family of closed subspaces of H J : α �→ J ( α ) ≤ H Fiber. We say { J ( α ) } α is fiber of the range function J . Lemma The map is unitary. π λ : N → U ( l 2 ( Z , HS )) , � ( � π λ ( n ) h ) ( j ) = π λ + j ( n ) ◦ h ( j )

Structure of shift-invariant spaces Theorem Given V ≤ L 2 ( N ) and Γ = ( a Z × b Z ) × Z , the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := J V such that J ( α ) ≤ l 2 ( Z , HS ) , and � π α (Γ 1 )( J ( α )) ⊆ J ( α ) , a . e . α ∈ ( 0 , 1 ] �� ⊕ � and V = T − 1 J ( α ) d α . ( 0 , 1 ]

Structure of shift-invariant spaces Theorem Given V ≤ L 2 ( N ) and Γ = ( a Z × b Z ) × Z , the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := J V such that J ( α ) ≤ l 2 ( Z , HS ) , and � π α (Γ 1 )( J ( α )) ⊆ J ( α ) , a . e . α ∈ ( 0 , 1 ] �� ⊕ � and V = T − 1 J ( α ) d α . ( 0 , 1 ] Sketch of proof. “( ii ) ⇒ ( i )” We need to show that if φ ∈ V , then l γ 1 γ 0 φ ∈ V for any γ 1 γ 0 ∈ Γ .

Structure of shift-invariant spaces Theorem Given V ≤ L 2 ( N ) and Γ = ( a Z × b Z ) × Z , the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := J V such that J ( α ) ≤ l 2 ( Z , HS ) , and � π α (Γ 1 )( J ( α )) ⊆ J ( α ) , a . e . α ∈ ( 0 , 1 ] �� ⊕ � and V = T − 1 J ( α ) d α . ( 0 , 1 ] Sketch of proof. “( ii ) ⇒ ( i )” We need to show that if φ ∈ V , then l γ 1 γ 0 φ ∈ V for any γ 1 γ 0 ∈ Γ . This follows by T ( l γ 1 γ 0 φ )( α ) = e 2 π i αγ 0 � π α ( γ 1 )( T φ ( α )) .

Structure of shift-invariant spaces Theorem Given V ≤ L 2 ( N ) and Γ = ( a Z × b Z ) × Z , the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := J V such that J ( α ) ≤ l 2 ( Z , HS ) , and � π α (Γ 1 )( J ( α )) ⊆ J ( α ) , a . e . α ∈ ( 0 , 1 ] and �� ⊕ � V = T − 1 J ( α ) d α . ( 0 , 1 ] � 1 Sketch of proof. “( i ) ⇒ ( ii )” Recall that L 2 ( N ) ≡ 0 l 2 ( Z , HS ) d α .

Structure of shift-invariant spaces Theorem Given V ≤ L 2 ( N ) and Γ = ( a Z × b Z ) × Z , the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := J V such that J ( α ) ≤ l 2 ( Z , HS ) , and � π α (Γ 1 )( J ( α )) ⊆ J ( α ) , a . e . α ∈ ( 0 , 1 ] and �� ⊕ � V = T − 1 J ( α ) d α . ( 0 , 1 ] � 1 Sketch of proof. “( i ) ⇒ ( ii )” Recall that L 2 ( N ) ≡ 0 l 2 ( Z , HS ) d α . Take { e m } m ONB for l 2 ( Z , HS ) .