Characterizations of some types of linear independence of integer - PowerPoint PPT Presentation

Characterizations of some types of linear independence of integer translates Sandra Saliani Department of Mathematics and Computer Science University of Basilicata Italy February Fourier Talks 2012 February 16 - 17, 2012 S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 1

Setting Systems of integer translates of a ψ ∈ L 2 ( R ) B = { ψ k , k ∈ Z } , ψ k ( x ) = ψ ( x − k ) , occur in approximation theory, frame theory, and wavelet analysis. Specific subset of the affine system generate by ψ { 2 j / 2 ψ ( 2 j x − k ) , j , k ∈ Z } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 2

Periodization function Definition Let ψ ∈ L 2 ( R ) , the periodization function is ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z p ψ ∈ L 1 ( T ) Many properties of B = { ψ k , k ∈ Z } can be completely described in terms of the periodization function p ψ . [Hernández, Šiki´ c, Weiss, and Wilson (2010)]. S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 3

Periodization function Definition Let ψ ∈ L 2 ( R ) , the periodization function is ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z p ψ ∈ L 1 ( T ) Many properties of B = { ψ k , k ∈ Z } can be completely described in terms of the periodization function p ψ . [Hernández, Šiki´ c, Weiss, and Wilson (2010)]. S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 3

Periodization function Definition Let ψ ∈ L 2 ( R ) , the periodization function is ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z p ψ ∈ L 1 ( T ) Many properties of B = { ψ k , k ∈ Z } can be completely described in terms of the periodization function p ψ . [Hernández, Šiki´ c, Weiss, and Wilson (2010)]. S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 3

Definition A sequence ( e n ) n ∈ N in a Hilbert space H is a Riesz basis if it is complete in H and there exist constants A , B > 0 such that, for any ( c n ) n ∈ N ∈ ℓ 2 ( N ) + ∞ + ∞ + ∞ | c n | 2 ≤ � c n e n � 2 ≤ B | c n | 2 . � � � A n = 0 n = 0 n = 0 Definition A sequence ( e n ) n ∈ N in a Hilbert space H is a Frame if There exist two real constants A , B > 0 such that, for any x ∈ H + ∞ A � x � 2 ≤ | < x , e n > | 2 ≤ B � x � 2 . � n = 0 S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 4

Definition A sequence ( e n ) n ∈ N in a Hilbert space H is a Riesz basis if it is complete in H and there exist constants A , B > 0 such that, for any ( c n ) n ∈ N ∈ ℓ 2 ( N ) + ∞ + ∞ + ∞ | c n | 2 ≤ � c n e n � 2 ≤ B | c n | 2 . � � � A n = 0 n = 0 n = 0 Definition A sequence ( e n ) n ∈ N in a Hilbert space H is a Frame if There exist two real constants A , B > 0 such that, for any x ∈ H + ∞ A � x � 2 ≤ | < x , e n > | 2 ≤ B � x � 2 . � n = 0 S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 4

Periodization function Let B = { ψ k , k ∈ Z } � ψ � = span ( B ) ⊂ L 2 ( R ) denotes the shift invariant space generated by ψ. ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z Theorem 1) B is Bessel for � ψ � with bound B ⇔ p ψ ( ξ ) ≤ B a . e . 2) B is an Orthonormal basis for � ψ � ⇔ p ψ ( ξ ) = 1 a . e . � B is a Riesz basis for � ψ � 3) ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . with bounds A , B � B is a Frame for � ψ � ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . 4) ξ ∈ T / { p ψ = 0 } with bounds A , B S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 5

Periodization function Let B = { ψ k , k ∈ Z } � ψ � = span ( B ) ⊂ L 2 ( R ) denotes the shift invariant space generated by ψ. ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z Theorem 1) B is Bessel for � ψ � with bound B ⇔ p ψ ( ξ ) ≤ B a . e . 2) B is an Orthonormal basis for � ψ � ⇔ p ψ ( ξ ) = 1 a . e . � B is a Riesz basis for � ψ � 3) ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . with bounds A , B � B is a Frame for � ψ � ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . 4) ξ ∈ T / { p ψ = 0 } with bounds A , B S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 5

Periodization function Let B = { ψ k , k ∈ Z } � ψ � = span ( B ) ⊂ L 2 ( R ) denotes the shift invariant space generated by ψ. ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z Theorem 1) B is Bessel for � ψ � with bound B ⇔ p ψ ( ξ ) ≤ B a . e . 2) B is an Orthonormal basis for � ψ � ⇔ p ψ ( ξ ) = 1 a . e . � B is a Riesz basis for � ψ � 3) ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . with bounds A , B � B is a Frame for � ψ � ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . 4) ξ ∈ T / { p ψ = 0 } with bounds A , B S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 5

Periodization function Let B = { ψ k , k ∈ Z } � ψ � = span ( B ) ⊂ L 2 ( R ) denotes the shift invariant space generated by ψ. ψ ( ξ + k ) | 2 � | ˆ p ψ ( ξ ) = k ∈ Z Theorem 1) B is Bessel for � ψ � with bound B ⇔ p ψ ( ξ ) ≤ B a . e . 2) B is an Orthonormal basis for � ψ � ⇔ p ψ ( ξ ) = 1 a . e . � B is a Riesz basis for � ψ � 3) ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . with bounds A , B � B is a Frame for � ψ � ⇔ A ≤ p ψ ( ξ ) ≤ B a . e . 4) ξ ∈ T / { p ψ = 0 } with bounds A , B S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 5

Example Let ψ ( x ) = ( 1 − | x | ) χ [ − 1 , 1 ] ( x ) be a linear Spline. Then ψ ( ξ + k ) | 2 = 1 3 ( 1 + 2 cos 2 ( πξ )) . � | ˆ p ψ ( ξ ) = k ∈ Z B is a Riesz basis for the space { f ∈ L 2 ( R ) ∩ C ( R ) , f linear in intervals [ k , k + 1 ) , k ∈ Z } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 6

Different concepts of linear independence Definition We say that a sequence ( e n ) n ∈ N in a Hilbert space H is (i) Linearly independent if each finite subsequence is linearly independent. + ∞ (ii) ℓ 2 - Linearly independent if whenever the series � c n e n is n = 0 convergent and equal to zero for some coefficients ( c n ) n ∈ N ∈ ℓ 2 ( N ) , then necessarily c n = 0 for all n ∈ N . + ∞ � (iii) ω -Independent if whenever the series c n e n is convergent and n = 0 equal to zero for some scalar coefficients ( c n ) n ∈ N , then necessarily c n = 0 for all n ∈ N . (iv) Minimal if for all k ∈ N , e k / ∈ span { e n , n � = k } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 7

Different concepts of linear independence Definition We say that a sequence ( e n ) n ∈ N in a Hilbert space H is (i) Linearly independent if each finite subsequence is linearly independent. + ∞ (ii) ℓ 2 - Linearly independent if whenever the series � c n e n is n = 0 convergent and equal to zero for some coefficients ( c n ) n ∈ N ∈ ℓ 2 ( N ) , then necessarily c n = 0 for all n ∈ N . + ∞ � (iii) ω -Independent if whenever the series c n e n is convergent and n = 0 equal to zero for some scalar coefficients ( c n ) n ∈ N , then necessarily c n = 0 for all n ∈ N . (iv) Minimal if for all k ∈ N , e k / ∈ span { e n , n � = k } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 7

Different concepts of linear independence Definition We say that a sequence ( e n ) n ∈ N in a Hilbert space H is (i) Linearly independent if each finite subsequence is linearly independent. + ∞ (ii) ℓ 2 - Linearly independent if whenever the series � c n e n is n = 0 convergent and equal to zero for some coefficients ( c n ) n ∈ N ∈ ℓ 2 ( N ) , then necessarily c n = 0 for all n ∈ N . + ∞ � (iii) ω -Independent if whenever the series c n e n is convergent and n = 0 equal to zero for some scalar coefficients ( c n ) n ∈ N , then necessarily c n = 0 for all n ∈ N . (iv) Minimal if for all k ∈ N , e k / ∈ span { e n , n � = k } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 7

Different concepts of linear independence Definition We say that a sequence ( e n ) n ∈ N in a Hilbert space H is (i) Linearly independent if each finite subsequence is linearly independent. + ∞ (ii) ℓ 2 - Linearly independent if whenever the series � c n e n is n = 0 convergent and equal to zero for some coefficients ( c n ) n ∈ N ∈ ℓ 2 ( N ) , then necessarily c n = 0 for all n ∈ N . + ∞ � (iii) ω -Independent if whenever the series c n e n is convergent and n = 0 equal to zero for some scalar coefficients ( c n ) n ∈ N , then necessarily c n = 0 for all n ∈ N . (iv) Minimal if for all k ∈ N , e k / ∈ span { e n , n � = k } . S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 7

Periodization function and linear independence ψ ( ξ + k ) | 2 . | ˆ B = { ψ k , k ∈ Z } , � p ψ ( ξ ) = k ∈ Z Fact 1 ∈ L 1 ( T ) B is minimal ⇐ ⇒ p ψ ⇓ B is ω -independent ⇓ B is ℓ 2 - linearly independent p ψ ( ξ ) > 0 a.e. ⇐ = ⇓ B is linearly independent Always true S. Saliani - Università della Basilicata Linear independence of integer translates FFT 2012 8