Characterization of all dual frames Result by Shidong Li, 1991: Theorem: Let { f k } ∞ k = 1 be a frame for H . The dual frames of { f k } ∞ k = 1 are precisely the families ∞ ∞ � { g k } ∞ S − 1 f k + h k − � S − 1 f k , f j � h j k = 1 = , (1) j = 1 k = 1 where { h k } ∞ k = 1 is a Bessel sequence in H . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 16 / 85
Characterization of all dual frames Result by Shidong Li, 1991: Theorem: Let { f k } ∞ k = 1 be a frame for H . The dual frames of { f k } ∞ k = 1 are precisely the families ∞ ∞ � { g k } ∞ S − 1 f k + h k − � S − 1 f k , f j � h j k = 1 = , (1) j = 1 k = 1 where { h k } ∞ k = 1 is a Bessel sequence in H . Allows us to optimize the duals: • Which dual has the best approximation theoretic properties? • Which dual has the smallest support? • Which dual has the most convenient expression? • Can we find a dual that is easy to calculate? (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 16 / 85
An example: Sigma-Delta quantization Work by Lammers, Powell, and Yilmaz: Consider a frame { f k } N k = 1 for R d . Letting { g k } N k = 1 denote a dual frame, each f ∈ R d can be written � N f = � f , g k � f k . k = 1 (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 17 / 85
An example: Sigma-Delta quantization Work by Lammers, Powell, and Yilmaz: Consider a frame { f k } N k = 1 for R d . Letting { g k } N k = 1 denote a dual frame, each f ∈ R d can be written � N f = � f , g k � f k . k = 1 In practice: the coefficients � f , g k � must be quantized, i.e., replaced by some coefficients d k from a discrete set such that d k ≈ � f , g k � , which leads to � N f ≈ d k f k . k = 1 Note: increased redundancy (large N ) increases the chance of a good approximation. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 17 / 85
An example: Sigma-Delta quantization • For each r ∈ N there is a procedure ( r th order sigma-delta quantization) to find appropriate coefficients d k . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 18 / 85
An example: Sigma-Delta quantization • For each r ∈ N there is a procedure ( r th order sigma-delta quantization) to find appropriate coefficients d k . • r the order sigma-delta quantization with the canonical dual frame does not provide approximation order N − r . • Approximation order N − r can be obtained using other dual frames. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 18 / 85
Tight frames versus dual pairs • For some years: focus on construction of tight frame. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 19 / 85
Tight frames versus dual pairs • For some years: focus on construction of tight frame. • Do not forget the extra flexibility offered by convenient dual frame pairs! (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 19 / 85
Tight frames versus dual pairs • For some years: focus on construction of tight frame. • Do not forget the extra flexibility offered by convenient dual frame pairs! Theorem: For each Bessel sequence { f k } ∞ k = 1 in a Hilbert space H , there exist a family of vectors { g k } ∞ k = 1 such that { f k } ∞ k = 1 ∪ { g k } ∞ k = 1 is a tight frame for H . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 19 / 85
Tight frames versus dual pairs j = 1 be an orthonormal basis for C 10 and consider the frame Example Let { e j } 10 { f j } 10 j = 1 := { 2 e 1 } ∪ { e j } 10 j = 2 . There exist 9 vectors { h j } 9 j = 1 such that { f j } 10 j = 1 ∪ { h j } 9 j = 1 is a tight frame for C 10 - and 9 is the minimal number to add. A pair of dual frames can be obtained by adding just one element: { f j } 10 j = 1 ∪ {− 3 e 1 } and { f j } 10 j = 1 ∪ { e 1 } form dual frames in C 10 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 20 / 85
Tight frames versus dual pairs Theorem (Casazza and Fickus): Given a sequence of positive numbers j = 1 for R N with a 1 ≥ a 2 ≥ · · · ≥ a M , there exists a tight frame { f j } M || f j || = a j , j = 1 , . . . , M , if and only if M � 1 ≤ 1 a 2 a 2 j . (2) N j = 1 (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 21 / 85
Tight frames versus dual pairs Theorem (Casazza and Fickus): Given a sequence of positive numbers j = 1 for R N with a 1 ≥ a 2 ≥ · · · ≥ a M , there exists a tight frame { f j } M || f j || = a j , j = 1 , . . . , M , if and only if M � 1 ≤ 1 a 2 a 2 j . (2) N j = 1 Theorem (C., Powell, Xiao, 2010): Given any sequence { α j } M j = 1 of real numbers, and assume that M > N . Then the following are equivalent: j = 1 and { � j = 1 for R N such that (i) There exist a pair of dual frames { f j } M f j } M α j = � f j , � f j � for all j = 1 , . . . , M . (ii) N = � M j = 1 α j . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 21 / 85
Part II: Gabor frames and wavelet frames in L 2 ( R ) (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 22 / 85
Operators on L 2 ( R ) T a : L 2 ( R ) → L 2 ( R ) , ( T a f )( x ) = f ( x − a ) . Translation by a ∈ R : Modulation by b ∈ R : E b : L 2 ( R ) → L 2 ( R ) , ( E b f )( x ) = e 2 π ibx f ( x ) . Dilation by a > 0 : D a : L 2 ( R ) → L 2 ( R ) , ( D a f )( x ) = √ a f ( x 1 a ) . D : L 2 ( R ) → L 2 ( R ) , ( Df )( x ) = 2 1 / 2 f ( 2 x ) . Dyadic scaling: All these operators are unitary on L 2 ( R ) . Important commutator relations: T a E b = e − 2 π iba E b T a , T b D a = D a T b / a , D a E b = E b / a D a (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 23 / 85
The Fourier transform For f ∈ L 1 ( R ) , the Fourier transform is defined by � ∞ f ( x ) e − 2 π ix γ dx , γ ∈ R . F f ( γ ) = ˆ f ( γ ) := −∞ The Fourier transform can be extended to a unitary operator on L 2 ( R ) . Plancherel’s equation: � ˆ g � = � f , g � , ∀ f , g ∈ L 2 ( R ) , and || ˆ f , ˆ f || = || f || . Important commutator relations: F T a = E − a F , F E a = T a F , D 1 / a F , F D = D − 1 F . F D a = (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 24 / 85
Splines The B-splines B N , N ∈ N , are given by B 1 = χ [ 0 , 1 ] , B N + 1 = B N ∗ B 1 . Theorem: Given N ∈ N , the B-spline B N has the following properties: (i) supp B N = [ 0 , N ] and B N > 0 on ] 0 , N [ . � ∞ (ii) −∞ B N ( x ) dx = 1. (iii) � k ∈ Z B N ( x − k ) = 1 (iv) For any N ∈ N , � 1 − e − 2 π i γ � N � B N ( γ ) = . 2 π i γ (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 25 / 85
Splines 1 1 0 0 2 1 1 2 3 4 2 1 1 2 3 4 K K K K Figure: The B-splines B 2 and B 3 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 26 / 85
Classical wavelet theory • Given a function ψ ∈ L 2 ( R ) and j , k ∈ Z , let ψ j , k ( x ) := 2 j / 2 ψ ( 2 j x − k ) , x ∈ R . • In terms of the operators T k f ( x ) = f ( x − k ) and Df ( x ) = 2 1 / 2 f ( 2 x ) , ψ j , k = D j T k ψ, j , k ∈ Z . • If { ψ j , k } j , k ∈ Z is an orthonormal basis for L 2 ( R ) , the function ψ is called a wavelet. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 27 / 85
Multiresolution analysis - a tool to construct a wavelet Definition: A multiresolution analysis for L 2 ( R ) consists of a sequence of closed subspaces { V j } j ∈ Z of L 2 ( R ) and a function φ ∈ V 0 , such that the following conditions hold: (i) · · · V − 1 ⊂ V 0 ⊂ V 1 · · · . (ii) ∪ j V j = L 2 ( R ) and ∩ j V j = { 0 } . (iii) f ∈ V j ⇔ [ x → f ( 2 x )] ∈ V j + 1 . (iv) f ∈ V 0 ⇒ T k f ∈ V 0 , ∀ k ∈ Z . (v) { T k φ } k ∈ Z is an orthonormal basis for V 0 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 28 / 85
Construction of wavelet ONB • The function φ in a multiresolution analysis satisfies a scaling equation, φ ( 2 γ ) = H 0 ( γ )ˆ ˆ φ ( γ ) , a . e . γ ∈ R , for some 1-periodic function H 0 ∈ L 2 ( 0 , 1 ) . • Let H 1 ( γ ) = H 0 ( γ + 1 2 ) e − 2 π i γ . • Then the function ψ defined via ψ ( 2 γ ) = H 1 ( γ )ˆ ˆ φ ( γ ) generates a wavelet orthonormal basis { D j T k ψ } j , k ∈ Z . • Explicitly: if H 1 ( γ ) = � k ∈ Z d k e 2 π ik γ , then ψ ( x ) = 2 � k ∈ Z d k φ ( 2 x + k ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 29 / 85
Construction of wavelet ONB Theorem: Let φ ∈ L 2 ( R ) , and let V j := span { D j T k φ } k ∈ Z . Assume that the following conditions hold: (i) inf γ ∈ ] − ǫ,ǫ [ | ˆ φ ( γ ) | > 0 for some ǫ > 0 ; (ii) The scaling equation φ ( 2 γ ) = H 0 ( γ )ˆ ˆ φ ( γ ) , is satisfied for a bounded 1-periodic function H 0 ; (iii) { T k φ } k ∈ Z is an orthonormal system. Then φ generates a multiresolution analysis, and there exists a wavelet ψ of the form � ψ ( x ) = 2 d k φ ( 2 x + k ) . k ∈ Z (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 30 / 85
Spline wavelets • The B-splines B N , N ∈ N , are given by B 1 = χ [ 0 , 1 ] , B N + 1 = B N ∗ B 1 . • One can consider any order splines B N and define associated multiresolution analyses, which leads to wavelets of the type � ψ ( x ) = c k B N ( 2 x + k ) . k ∈ Z • These wavelets are called Battle–Lemari´ e wavelets. • Only shortcoming: except for the case N = 1, all coefficients c k are non-zero, which implies that the wavelet ψ has support equal to R . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 31 / 85
Spline wavelets - can we do better for N > 1? (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 32 / 85
Spline wavelets - can we do better for N > 1? Can show: • There does not exists an ONB { D j T k ψ } j , k ∈ Z for L 2 ( R ) generated by a finite linear combination � ψ ( x ) = c k B N ( 2 x + k ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 32 / 85
Spline wavelets - can we do better for N > 1? Can show: • There does not exists an ONB { D j T k ψ } j , k ∈ Z for L 2 ( R ) generated by a finite linear combination � ψ ( x ) = c k B N ( 2 x + k ) . • There does not exists a tight frame { D j T k ψ } j , k ∈ Z for L 2 ( R ) generated by a finite linear combination � ψ ( x ) = c k B N ( 2 x + k ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 32 / 85
Spline wavelets - can we do better for N > 1? Can show: • There does not exists an ONB { D j T k ψ } j , k ∈ Z for L 2 ( R ) generated by a finite linear combination � ψ ( x ) = c k B N ( 2 x + k ) . • There does not exists a tight frame { D j T k ψ } j , k ∈ Z for L 2 ( R ) generated by a finite linear combination � ψ ( x ) = c k B N ( 2 x + k ) . • There does not exists a pairs of dual wavelet frames { D j T k ψ } j , k ∈ Z and { D j T k ˜ ψ } j , k ∈ Z for which ψ and ˜ ψ are finite linear combinations of functions DT k B N , j , k ∈ Z . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 32 / 85
Spline wavelets Solution: consider systems of the wavelet-type, but generated by more than one function. Setup for construction of tight wavelet frames by Ron and Shen: Let ψ 0 ∈ L 2 ( R ) and assume that (i) There exists a function H 0 ∈ L ∞ ( T ) such that ψ 0 ( 2 γ ) = H 0 ( γ ) � � ψ 0 ( γ ) . (ii) lim γ → 0 � ψ 0 ( γ ) = 1 . Further, let H 1 , . . . , H n ∈ L ∞ ( T ) , and define ψ 1 , . . . , ψ n ∈ L 2 ( R ) by � ψ ℓ ( 2 γ ) = H ℓ ( γ ) � ψ 0 ( γ ) , ℓ = 1 , . . . , n . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 33 / 85
The unitary extension principle • We want to find conditions on the functions H 1 , . . . , H n such that ψ 1 , . . . , ψ n generate a multiwavelet frame for L 2 ( R ) . • Let H denote the ( n + 1 ) × 2 matrix-valued function defined by H 0 ( γ ) T 1 / 2 H 0 ( γ ) H 1 ( γ ) T 1 / 2 H 1 ( γ ) H ( γ ) = · · , γ ∈ R . · · H n ( γ ) T 1 / 2 H n ( γ ) (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 34 / 85
The unitary extension principle Theorem (Ron and Shen, 1997): Let { ψ ℓ , H ℓ } n ℓ = 0 be as in the general setup, and assume that H ( γ ) ∗ H ( γ ) = I for a.e. γ ∈ T . Then the multiwavelet system { D j T k ψ ℓ } j , k ∈ Z ,ℓ = 1 ,..., n constitutes a tight frame for L 2 ( R ) with frame bound equal to 1 . Can be applied to any order B-spline! 1 1 0 0 K 2 K 1 1 2 K 2 K 1 1 2 K 1 K 1 Figure: The two wavelet frame generators ψ 1 and ψ 2 associated with ψ 0 = B 2 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 35 / 85
Gabor systems Gabor systems: have the form { e 2 π imbx g ( x − na ) } m , n ∈ Z for some g ∈ L 2 ( R ) , a , b > 0. Short notation: { E mb T na g } m , n ∈ Z = { e 2 π imbx g ( x − na ) } (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 36 / 85
Gabor systems Gabor systems: have the form { e 2 π imbx g ( x − na ) } m , n ∈ Z for some g ∈ L 2 ( R ) , a , b > 0. Short notation: { E mb T na g } m , n ∈ Z = { e 2 π imbx g ( x − na ) } Example: • { e 2 π imx χ [ 0 , 1 ] ( x ) } m ∈ Z is an ONB for L 2 ( 0 , 1 ) • For n ∈ Z , { e 2 π im ( x − n ) χ [ 0 , 1 ] ( x − n ) } m ∈ Z = { e 2 π imx χ [ 0 , 1 ] ( x − n ) } m ∈ Z is an ONB for L 2 ( n , n + 1 ) • { e 2 π imx χ [ 0 , 1 ] ( x − n ) } m , n ∈ Z is an ONB for L 2 ( R ) (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 36 / 85
Gabor frames and Riesz bases (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 37 / 85
Gabor frames and Riesz bases • If { E mb T na g } m , n ∈ Z is a frame, then ab ≤ 1 ; (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 37 / 85
Gabor frames and Riesz bases • If { E mb T na g } m , n ∈ Z is a frame, then ab ≤ 1 ; • If { E mb T na g } m , n ∈ Z is a frame, then { f k } ∞ k = 1 is a Riesz basis ⇔ ab = 1 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 37 / 85
Gabor frames and Riesz bases • If { E mb T na g } m , n ∈ Z is a frame, then ab ≤ 1 ; • If { E mb T na g } m , n ∈ Z is a frame, then { f k } ∞ k = 1 is a Riesz basis ⇔ ab = 1 . For the sake of time-frequency analysis: we want the Gabor frame { E mb T na g } m , n ∈ Z to be generated by a continuous function g with compact support. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 37 / 85
Gabor systems Lemma: If g is be a continuous function with compact support, then • { E mb T na g } m , n ∈ Z can not be an ONB. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 38 / 85
Gabor systems Lemma: If g is be a continuous function with compact support, then • { E mb T na g } m , n ∈ Z can not be an ONB. • { E mb T na g } m , n ∈ Z can not be a Riesz basis. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 38 / 85
Gabor systems Lemma: If g is be a continuous function with compact support, then • { E mb T na g } m , n ∈ Z can not be an ONB. • { E mb T na g } m , n ∈ Z can not be a Riesz basis. • { E mb T na g } m , n ∈ Z can be a frame if 0 < ab < 1 ; Thus, it is necessary to consider frames if we want Gabor systems with good properties. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 38 / 85
Pairs of dual Gabor frames Two Bessel sequences { E mb T na g } m , n ∈ Z and { E mb T na h } m , n ∈ Z form dual frames if � � f , E mb T na h � E mb T na g , ∀ f ∈ L 2 ( R ) . f = m , n ∈ Z Ron & Shen, A.J.E.M. Janssen (1998): Theorem: Two Bessel sequences { E mb T na g } m , n ∈ Z and { E mb T na h } m , n ∈ Z form dual frames if and only if � g ( x − n / b − ka ) h ( x − ka ) = b δ n , 0 , a . e . x ∈ [ 0 , a ] . k ∈ Z (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 39 / 85
History • B¨ olsckei, Janssen, 1998-2000: For b rational, characterization of Gabor frames with compactly supported window having a compactly supported dual window; • Feichtinger, Gr¨ ochenig (1997): window in S 0 implies that the canonical dual window is in S 0 ; • Krishtal, Okoudjou, 2007: window in W ( L ∞ , ℓ 1 ) implies that the canonical dual window is in W ( L ∞ , ℓ 1 ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 40 / 85
Duality principle The duality principle (Janssen, Daubechies, Landau, and Landau, and Ron and Shen: concerns the relationship between frame properties for a function g with respect to the lattice { ( na , mb ) } m , n ∈ Z and with respect to the so-called dual lattice { ( n / b , m / a ) } m , n ∈ Z : Theorem: Let g ∈ L 2 ( R ) and a , b > 0 be given. Then the following are equivalent: (i) { E mb T na g } m , n ∈ Z is a frame for L 2 ( R ) with bounds A , B ; 1 (ii) { √ ab E m / a T n / b g } m , n ∈ Z is a Riesz sequence with bounds A , B. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 41 / 85
Duality principle The duality principle (Janssen, Daubechies, Landau, and Landau, and Ron and Shen: concerns the relationship between frame properties for a function g with respect to the lattice { ( na , mb ) } m , n ∈ Z and with respect to the so-called dual lattice { ( n / b , m / a ) } m , n ∈ Z : Theorem: Let g ∈ L 2 ( R ) and a , b > 0 be given. Then the following are equivalent: (i) { E mb T na g } m , n ∈ Z is a frame for L 2 ( R ) with bounds A , B ; 1 (ii) { √ ab E m / a T n / b g } m , n ∈ Z is a Riesz sequence with bounds A , B. Intuition: If { E mb T na g } m , n ∈ Z is a frame for L 2 ( R ) , then ab ≤ 1 , i.e., the sampling points { ( na , mb ) } m , n ∈ Z are “sufficiently dense.” Therefore the 1 points { ( n / b , m / a ) } m , n ∈ Z are “sparse,” and therefore { ab E m / a T n / b g } m , n ∈ Z √ are linearly independent and only span a subspace of L 2 ( R ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 41 / 85
Wexler-Raz’ Theorem Wexler-Raz’ Theorem: If the Gabor systems { E mb T na g } m , n ∈ Z and { E mb T na h } m , n ∈ Z are dual frames, then the Gabor systems 1 1 { ab E m / a T n / b g } m , n ∈ Z and { ab E m / a T n / b h } m , n ∈ Z are biorthogonal, i.e., √ √ � 1 1 √ E m / a T n / b g , √ E m ′ / a T n ′ / b h � = δ m , m ′ δ n , n ′ . ab ab (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 42 / 85
Explicit construction of dual pairs of Gabor frames k = 1 to be useful, we need a dual frame { g k } ∞ In order for a frame { f k } ∞ k = 1 , i.e., a frame such that ∞ � f = � f , g k � f k , ∀ f ∈ H . k = 1 How can we construct convenient dual frames? Ansatz/suggestion: Given a window function g ∈ L 2 ( R ) generating a frame { E mb T na g } m , n ∈ Z , look for a dual window of the form K � h ( x ) = c k g ( x + k ) . k = − K The structure of h makes it easy to derive properties of h based on properties of g (regularity, size of support, membership un various vector spaces,....) (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 43 / 85
Explicit construction of dual pairs of Gabor frames Theorem:(C., 2006) Let N ∈ N . Let g ∈ L 2 ( R ) be a real-valued bounded function for which • supp g ⊆ [ 0 , N ] , • � n ∈ Z g ( x − n ) = 1 . 1 Let b ∈ ] 0 , 2 N − 1 ] . Then the function g and the function h defined by N − 1 � h ( x ) = bg ( x ) + 2 b g ( x + n ) n = 1 generate dual frames { E mb T n g } m , n ∈ Z and { E mb T n h } m , n ∈ Z for L 2 ( R ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 44 / 85
Candidates for g - the B-splines For the B-spline case: • The functions B N and the dual window N − 1 � h ( x ) = bB N ( x ) + 2 b B N ( x + n ) n = 1 are splines; • B N and h have compact support, i.e., perfect time–localization; • By choosing N sufficiently large, polynomial decay of � B N and h of any desired order can be obtained. Note: � sin πγ � N � e − π iN γ . B N ( γ ) = πγ (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 45 / 85
Example For the B-spline x ∈ [ 0 , 1 [ , x B 2 ( x ) = 2 − x x ∈ [ 1 , 2 [ , 0 x / ∈ [ 0 , 2 [ , we can use the result for b ∈ ] 0 , 1 / 3 ] . For b = 1 / 3 we obtain the dual generator 1 3 B 2 ( x ) + 2 h ( x ) = 3 B 2 ( x + 1 ) 2 3 ( x + 1 ) x ∈ [ − 1 , 0 [ , 1 = 3 ( 2 − x ) x ∈ [ 0 , 2 [ , x / ∈ [ − 1 , 2 [ . 0 (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 46 / 85
Example 1 1 0 0 3 2 1 1 2 3 4 3 2 1 1 2 3 4 K K K K K K Figure: The B-spline N 2 and the dual generator h for b = 1 / 3; and the B-spline N 3 and the dual generator h with b = 1 / 5. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 47 / 85
Other choices? Yes! Theorem: (C., Kim, 2007) Let N ∈ N . Let g ∈ L 2 ( R ) be a real-valued bounded function for which • supp g ⊆ [ 0 , N ] , • � n ∈ Z g ( x − n ) = 1 . 1 2 N − 1 ] . Define h ∈ L 2 ( R ) by Let b ∈ ] 0 , N − 1 � h ( x ) = a n g ( x + n ) , n = − N + 1 where a 0 = b , a n + a − n = 2 b , n = 1 , 2 , · · · , N − 1 . Then g and h generate dual frames { E mb T n g } m , n ∈ Z and { E mb T n h } m , n ∈ Z for L 2 ( R ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 48 / 85
Example: B-splines revisited 1) Take a 0 = b , a n = 0 for n = − N + 1 , . . . , − 1 , a n = 2 b , n = 1 , . . . N − 1 . This is the previous Theorem. This choice gives the shortest support. 2) Take a − N + 1 = a − N + 2 = · · · = a N − 1 = b : if g is symmetric, this leads to a symmetric dual generator � N − 1 h ( x ) = b g ( x + n ) . n = − N + 1 Note: h ( x ) = b on supp g . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 49 / 85
B-splines revisited 1 0.7 0.8 0.6 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0.1 0 0 -2 -1 0 1 2 3 4 -2 0 2 4 6 x x Figure: The generators B 2 and B 3 and their dual generators via 2). (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 50 / 85
Explicit constructions Another class of examples: Exponential B-splines: Given a sequence of scalars β 1 , β 2 , . . . , β N ∈ R , let E N := e β 1 ( · ) χ [ 0 , 1 ] ( · ) ∗ e β 2 ( · ) χ [ 0 , 1 ] ( · ) ∗ · · · ∗ e β N ( · ) χ [ 0 , 1 ] ( · ) . • The function E N is supported on [ 0 , N ] . • If β k = 0 for at least one k = 1 , . . . , N , then E N satisfies the partition of unity condition (up to a constant). • If β j � = β k for j � = k , an explicit expression for E N is known (C., Peter Massopust, 2010). (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 51 / 85
Explicit constructions Assume that β k = ( k − 1 ) β, k = 1 , . . . , N . Then E N ( x ) = N − 1 � 1 e β kx , 1 x ∈ [ 0 , 1 ]; β N − 1 � N k = 0 ( k + 1 − j ) j = 1 j � = k + 1 � [ e β j 1 + · · · + e β j ℓ − 1 ] 0 ≤ j 1 < ··· < j ℓ − 1 ≤ N − 1 � N − 1 x ∈ [ ℓ − 1 , ℓ ] ( − 1 ) ℓ − 1 j 1 ,..., j ℓ − 1 � = k − 1 e β k ( x − ℓ + 1 ) , ℓ = 2 , . . . , N . β N − 1 � N k = 0 ( k + 1 − j ) j = 1 j � = k + 1 (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 52 / 85
Explicit constructions � � N − 1 � e β m − 1 � m = 1 E N ( x − k ) = β N − 1 ( N − 1 )! . k ∈ Z Via the Theorem: construction of dual Gabor frames with generators N − 1 � E N , h N = c k E N ( x + K ) . k = − N + 1 (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 53 / 85
From Gabor frames to wavelet frames - duality conditions: Theorem: Two Bessel sequences { E mb T na g } m , n ∈ Z and { E mb T na h } m , n ∈ Z form dual frames if and only if (i) � k ∈ Z g ( x − ka ) h ( x − ka ) = b , a . e . x ∈ [ 0 , a ] . (ii) � k ∈ Z g ( x − n / b − ka ) h ( x − ka ) = 0 , a . e . x ∈ [ 0 , a ] , n ∈ Z \ { 0 } . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 54 / 85
From Gabor frames to wavelet frames - duality conditions: Theorem: Two Bessel sequences { E mb T na g } m , n ∈ Z and { E mb T na h } m , n ∈ Z form dual frames if and only if (i) � k ∈ Z g ( x − ka ) h ( x − ka ) = b , a . e . x ∈ [ 0 , a ] . (ii) � k ∈ Z g ( x − n / b − ka ) h ( x − ka ) = 0 , a . e . x ∈ [ 0 , a ] , n ∈ Z \ { 0 } . Theorem: Given a > 1 , b > 0 , two Bessel sequences { D a j T kb ψ } j , k ∈ Z and { D a j T kb � ψ } j , k ∈ Z , where ψ, � ψ ∈ L 2 ( R ) , form dual wavelet frames for L 2 ( R ) if and only if the following two conditions are satisfied: (i) � ψ ( a j γ ) � j ∈ Z � � ψ ( a j γ ) = b for a.e. γ ∈ R . (ii) For any number α � = 0 of the form α = m / a j , m , j ∈ Z , � ψ ( a j γ ) � � � ψ ( a j γ + m / b ) = 0 , a . e . γ ∈ R . { ( j , m ) ∈ Z 2 | α = m / a j } (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 54 / 85
From Gabor frames to wavelet frames C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet frames based on dual pairs of Gabor frames. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 55 / 85
From Gabor frames to wavelet frames C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet frames based on dual pairs of Gabor frames. Let θ > 1 be given. Associated with a function g ∈ L 2 ( R ) with the property that g ( log θ | · | ) ∈ L 2 ( R ) we define a function ψ ∈ L 2 ( R ) by � ψ ( γ ) = g ( log θ ( | γ | )) . Then � ψ ( a j γ ) = g ( j log θ ( a ) + log θ ( | γ | )) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 55 / 85
From Gabor frames to wavelet frames C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet frames based on dual pairs of Gabor frames. Let θ > 1 be given. Associated with a function g ∈ L 2 ( R ) with the property that g ( log θ | · | ) ∈ L 2 ( R ) we define a function ψ ∈ L 2 ( R ) by � ψ ( γ ) = g ( log θ ( | γ | )) . Then � ψ ( a j γ ) = g ( j log θ ( a ) + log θ ( | γ | )) . When applied to exponential B-splines: Construction of dual pairs of wavelet frames with generators ψ and � ψ, for which ψ and � • � � ψ are compactly supported splines with geometrically distributed knot sequences. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 55 / 85
Example The exponential B-spline E 2 with β 1 = 0 , β 2 = 1 : 0 , x / ∈ [ 0 , 2 ] , e x − 1 , E 2 ( x ) = x ∈ [ 0 , 1 ] , e − e − 1 e x , x ∈ [ 1 , 2 ] . Then � E 2 ( x − k ) = e − 1 , x ∈ R , k ∈ Z so we consider the function g ( x ) := ( e − 1 ) − 1 E 2 ( x ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 56 / 85
Example • Let ψ be defined via ∈ [ 1 , e 2 ] , 0 , | γ | / � | γ |− 1 ψ ( γ ) = e − 1 , | γ | ∈ [ 1 , e ] , e − e − 1 | γ | | γ | ∈ [ e , e 2 ] . , e − 1 • � ψ is a geometric spline with knots at the points ± 1 , ± e , ± e 2 . • Let b = 15 − 1 . Then the function � ψ defined by 1 � ψ ( γ ) = 1 � � � ψ ( | e n γ | ) 15 n = − 1 is a dual generator. • � � ψ is a geometric spline with knots at ± e − 1 , ± 1 , ± e 2 , ± e 3 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 57 / 85
Example ψ and � Figure: Plots of the geometric splines � � ψ . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 58 / 85
From to wavelet frames to Gabor frames The Meyer wavelet is the function ψ ∈ L 2 ( R ) defined via e i πγ sin ( π 2 ( ν ( 3 | γ | − 1 ))) , if 1 / 3 ≤ | γ | ≤ 2 / 3, � e i πγ cos ( π ψ ( γ ) = 2 ( ν ( 3 | γ | / 2 − 1 ))) , if 2 / 3 ≤ | γ | ≤ 4 / 3, 0 , if | γ | / ∈ [ 1 / 3 , 4 / 3 ] , where ν : R → R is any continuous function for which � 0 , if x ≤ 0 , ν ( x ) = 1 , if x ≥ 1 , and ν ( x ) + ν ( 1 − x ) = 1 , x ∈ R . Known: { D 2 j T k ψ } j , k ∈ Z is an orthonormal basis for L 2 ( R ) , and supp � ψ = [ − 4 / 3 , − 1 / 3 ] ∪ [ 1 / 3 , 4 / 3 ] . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 59 / 85
From to wavelet frames to Gabor frames Let � − { exp [ x / ( 1 − x )] − 1 } − 1 � , if 0 < x < 1, exp ν 0 ( x ) = 0 , if x ≤ 0, 1 , if x ≥ 1, ν ( x ) := 1 2 ( ν 0 ( x ) − ν 0 ( 1 − x ) + 1 ) , x ∈ R . 2 ( ν ( 3 · 2 x − 1 ))) , 2 sin ( π 1 if − ln 3 ln 2 ≤ x ≤ 1 − ln 3 √ ln 2 , 2 · 2 x − 1 ))) , 1 2 cos ( π 2 ( ν ( 3 if 1 − ln 3 ln 2 ≤ x ≤ 2 − ln 3 τ ( x ) := ln 2 , √ ∈ [ − ln 3 ln 2 , 2 − ln 3 0 , if x / ln 2 ] , Then τ is real-valued, compactly supported, belongs to C ∞ ( R ) , and { E m / 2 T n τ } m , n ∈ Z is a tight frame with bound A = 1 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 60 / 85
From to wavelet frames to Gabor frames Figure: The function τ, which is C ∞ and has compact support. The Gabor system { E m / 2 T n τ } m , n ∈ Z is a tight frame with bound A = 1 . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 61 / 85
Part III: Research problems related to frames and operator theory (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 62 / 85
Open problems • An extension problem for wavelet frames • The duality principle in general Hilbert spaces (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 63 / 85
An extension problem for wavelet frames (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 64 / 85
Tight frames versus dual pairs Theorem: For each Bessel sequence { f k } ∞ k = 1 in a Hilbert space H , there exist a family of vectors { g k } ∞ k = 1 such that k = 1 ∪ { g k } ∞ { f k } ∞ k = 1 is a tight frame for H . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 65 / 85
Tight frames versus dual pairs Theorem: For each Bessel sequence { f k } ∞ k = 1 in a Hilbert space H , there exist a family of vectors { g k } ∞ k = 1 such that k = 1 ∪ { g k } ∞ { f k } ∞ k = 1 is a tight frame for H . Theorem (D. Li and W.Sun, 2009): Let { E mb T na g 1 } m , n ∈ Z be Bessel sequences in L 2 ( R ) , and assume that ab ≤ 1 . Then the following hold: • There exists a Gabor systems { E mb T na g 2 } m , n ∈ Z such that { E mb T na g 1 } m , n ∈ Z ∪ { E mb T na g 2 } m , n ∈ Z is a tight frame for L 2 ( R ) . (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 65 / 85
Tight frames versus dual pairs Theorem: For each Bessel sequence { f k } ∞ k = 1 in a Hilbert space H , there exist a family of vectors { g k } ∞ k = 1 such that k = 1 ∪ { g k } ∞ { f k } ∞ k = 1 is a tight frame for H . Theorem (D. Li and W.Sun, 2009): Let { E mb T na g 1 } m , n ∈ Z be Bessel sequences in L 2 ( R ) , and assume that ab ≤ 1 . Then the following hold: • There exists a Gabor systems { E mb T na g 2 } m , n ∈ Z such that { E mb T na g 1 } m , n ∈ Z ∪ { E mb T na g 2 } m , n ∈ Z is a tight frame for L 2 ( R ) . • If g 1 has compact support and | supp g 1 | ≤ b − 1 , then g 2 can be chosen to have compact support. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 65 / 85
Tight frames versus dual pairs Theorem (C., Kim, Kim, 2011): Let { E mb T na g 1 } m , n ∈ Z and { E mb T na h 1 } m , n ∈ Z be Bessel sequences in L 2 ( R ) , and assume that ab ≤ 1 . Then the following hold: • There exist Gabor systems { E mb T na g 2 } m , n ∈ Z and { E mb T na h 2 } m , n ∈ Z in L 2 ( R ) such that { E mb T na g 1 } m , n ∈ Z ∪ { E mb T na g 2 } m , n ∈ Z and { E mb T na h 1 } m , n ∈ Z ∪ { E mb T na h 2 } m form a pair of dual frames for L 2 ( R ) . • If g 1 and h 1 have compact support, the functions g 2 and h 2 can be chosen to have compact support. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 66 / 85
Tight frames versus dual pairs Theorem (C., Kim, Kim, 2011): Let { E mb T na g 1 } m , n ∈ Z and { E mb T na h 1 } m , n ∈ Z be Bessel sequences in L 2 ( R ) , and assume that ab ≤ 1 . Then the following hold: • There exist Gabor systems { E mb T na g 2 } m , n ∈ Z and { E mb T na h 2 } m , n ∈ Z in L 2 ( R ) such that { E mb T na g 1 } m , n ∈ Z ∪ { E mb T na g 2 } m , n ∈ Z and { E mb T na h 1 } m , n ∈ Z ∪ { E mb T na h 2 } m form a pair of dual frames for L 2 ( R ) . • If g 1 and h 1 have compact support, the functions g 2 and h 2 can be chosen to have compact support. Note: closely related to work by Han (2009), where it is assumed that { E mb T na g 1 } m , n ∈ Z and { E mb T na h 1 } m , n ∈ Z are dual frames for a subspace. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 66 / 85
The wavelet case Theorem (C., Kim, Kim, 2011): Let { D j T k ψ 1 } j , k ∈ Z and { D j T k � ψ 1 } j , k ∈ Z be Bessel sequences in L 2 ( R ) . Assume that the Fourier transform of � ψ 1 satisfies supp � � ψ 1 ⊆ [ − 1 , 1 ] . Then there exist wavelet systems { D j T k ψ 2 } j , k ∈ Z and { D j T k � ψ 2 } j , k ∈ Z such that { D j T k ψ 1 } j , k ∈ Z ∪ { D j T k ψ 2 } j , k ∈ Z and { D j T k � ψ 1 } j , k ∈ Z ∪ { D j T k � ψ 2 } j , k ∈ Z form dual frames for L 2 ( R ) . Corollary: (C., Kim, Kim, 2011): In the above setup, assume that � ψ 1 is compactly supported and that supp � � ψ 1 ⊆ [ − 1 , 1 ] \ [ − ǫ, ǫ ] for some ǫ > 0. Then the functions ψ 2 and � ψ 2 can be chosen to have compactly supported Fourier transforms as well. (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 67 / 85
The wavelet case - Open problems: (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 68 / 85
The wavelet case - Open problems: • Let { D j T k ψ 1 } j , k ∈ Z and { D j T k � ψ 1 } j , k ∈ Z be Bessel sequences in L 2 ( R ) . Assume that supp � � ψ 1 is NOT contained in [ − 1 , 1 ] . Does there exist wavelet systems { D j T k ψ 2 } j , k ∈ Z and { D j T k � ψ 2 } j , k ∈ Z such that { D j T k ψ 1 } j , k ∈ Z ∪ { D j T k ψ 2 } j , k ∈ Z and { D j T k � ψ 1 } j , k ∈ Z ∪ { D j T k � ψ 2 } j , k ∈ Z form dual frames for L 2 ( R )? (DTU Mathematics) Talk, Concentration Week, Texas A & M July 16, 2012 68 / 85
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