Frames and operator representations of frames Ole Christensen Joint work with Marzieh Hasannasab HATA – DTU DTU Compute, Technical University of Denmark HATA: Harmonic Analysis - Theory and Applications https://hata.compute.dtu.dk/ Ole Christensen Jakob Lemvig Mads Sielemann Jakobsen Marzieh Hasannasab Kamilla Haahr Nielsen Ehsan Rashidi Jordy van Velthoven August 17, 2017 (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 1 / 27
Frames and overview of the talk • If a sequence { f k } ∞ k = 1 in a Hilbert spaces H is a frame, there exists another frame { g k } ∞ k = 1 such that ∞ � f = � f , g k � f k , ∀ f ∈ H . k = 1 Similar to the decomposition in terms of an orthonormal basis, but MUCH MORE flexible. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 2 / 27
Frames and overview of the talk • If a sequence { f k } ∞ k = 1 in a Hilbert spaces H is a frame, there exists another frame { g k } ∞ k = 1 such that ∞ � f = � f , g k � f k , ∀ f ∈ H . k = 1 Similar to the decomposition in terms of an orthonormal basis, but MUCH MORE flexible. • We will consider representations of frames on the form { f k } ∞ k = 1 = { T n f 1 } ∞ n = 0 = { f 1 , Tf 1 , T 2 f 1 , · · · } , where T : H → H is a linear operator, possibly bounded. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 2 / 27
Frames and overview of the talk • If a sequence { f k } ∞ k = 1 in a Hilbert spaces H is a frame, there exists another frame { g k } ∞ k = 1 such that ∞ � f = � f , g k � f k , ∀ f ∈ H . k = 1 Similar to the decomposition in terms of an orthonormal basis, but MUCH MORE flexible. • We will consider representations of frames on the form { f k } ∞ k = 1 = { T n f 1 } ∞ n = 0 = { f 1 , Tf 1 , T 2 f 1 , · · · } , where T : H → H is a linear operator, possibly bounded. Main conclusion: Frame theory is operator theory, with several interesting and challenging open problems! (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 2 / 27
Bessel sequences Definition A sequence { f k } ∞ k = 1 in H is called a Bessel sequence if there exists a constant B > 0 such that ∞ |� f , f k �| 2 ≤ B || f || 2 , ∀ f ∈ H . � k = 1 (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 3 / 27
Bessel sequences Definition A sequence { f k } ∞ k = 1 in H is called a Bessel sequence if there exists a constant B > 0 such that ∞ |� f , f k �| 2 ≤ B || f || 2 , ∀ f ∈ H . � k = 1 Theorem Let { f k } ∞ k = 1 be a sequence in H , and B > 0 be given. Then { f k } ∞ k = 1 is a Bessel sequence with Bessel bound B if and only if ∞ T : { c k } ∞ � k = 1 → c k f k k = 1 √ defines a bounded operator from ℓ 2 ( N ) into H and || T || ≤ B . (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 3 / 27
Bessel sequences Pre-frame operator or synthesis operator associated to a Bessel sequence: ∞ T : ℓ 2 ( N ) → H , T { c k } ∞ � k = 1 = c k f k k = 1 The adjoint operator - the analysis operator: T ∗ : H → ℓ 2 ( N ) , T ∗ f = {� f , f k �} ∞ k = 1 . The frame operator: ∞ S : H → H , Sf = TT ∗ f = � � f , f k � f k . k = 1 The series defining S converges unconditionally for all f ∈ H . (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 4 / 27
Frames Definition: A sequence { f k } ∞ k = 1 in H is a frame if there exist constants A , B > 0 such that ∞ A || f || 2 ≤ |� f , f k �| 2 ≤ B || f || 2 , ∀ f ∈ H . � k = 1 A and B are called frame bounds. Note: • Any orthonormal basis is a frame; • Example of a frame which is not a basis: { e 1 , e 1 , e 2 , e 3 , . . . } , where { e k } ∞ k = 1 is an ONB. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 5 / 27
The frame decomposition If { f k } ∞ k = 1 is a frame, the frame operator � S : H → H , Sf = � f , f k � f k is well-defined, bounded, invertible, and self-adjoint. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 6 / 27
The frame decomposition If { f k } ∞ k = 1 is a frame, the frame operator � S : H → H , Sf = � f , f k � f k is well-defined, bounded, invertible, and self-adjoint. Theorem - the frame decomposition Let { f k } ∞ k = 1 be a frame with frame operator S . Then ∞ � � f , S − 1 f k � f k , ∀ f ∈ H . f = k = 1 It might be difficult to compute S − 1 ! (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 6 / 27
The frame decomposition If { f k } ∞ k = 1 is a frame, the frame operator � S : H → H , Sf = � f , f k � f k is well-defined, bounded, invertible, and self-adjoint. Theorem - the frame decomposition Let { f k } ∞ k = 1 be a frame with frame operator S . Then ∞ � � f , S − 1 f k � f k , ∀ f ∈ H . f = k = 1 It might be difficult to compute S − 1 ! Important special case: If the frame { f k } ∞ k = 1 is tight, A = B , then S = A I and ∞ f = 1 � � f , f k � f k , ∀ f ∈ H . A k = 1 (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 6 / 27
General dual frames A frame which is not a basis is said to be overcomplete . Theorem: Assume that { f k } ∞ k = 1 is an overcomplete frame. Then there exist frames { g k } ∞ k = 1 � = { S − 1 f k } ∞ k = 1 for which ∞ ∞ � � � f , S − 1 f k � f k , ∀ f ∈ H . f = � f , g k � f k = k = 1 k = 1 • { g k } ∞ k = 1 is called a dual frame of { f k } ∞ k = 1 . • The excess of a frame is the maximal number of elements that can be removed such that the remaining set is still a frame. The excess equals dim N ( T ) - the dimension of the kernel of the synthesis operator. • When the excess is large, the set of dual frames is large. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 7 / 27
General dual frames Note: Let { f k } ∞ k = 1 be a Bessel sequence with pre-frame operator ∞ T : H → ℓ 2 ( N ) , T { c k } ∞ � [ T ∗ f = {� f , f k �} ∞ k = 1 = k = 1 ] c k f k k = 1 and { g k } ∞ k = 1 be a Bessel sequence with pre-frame operator ∞ � U : H → ℓ 2 ( N ) , U { c k } ∞ [ U ∗ f = {� f , g k �} ∞ k = 1 = c k g k k = 1 ] k = 1 k = 1 and { g k } ∞ Then { f k } ∞ k = 1 are dual frames if and only if ∞ � f = � f , g k � f k , ∀ f ∈ H , k = 1 i.e., if and only if TU ∗ = I . (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 8 / 27
Key tracks in frame theory: • Frames in finite-dimensional spaces; • Frames in general separable Hilbert spaces • Concrete frames in concrete Hilbert spaces: • Gabor frames in L 2 ( R ) , L 2 ( R d ) ; • Wavelet frames; • Shift-invariant systems, generalized shift-invariant (GSI) systems; • Shearlets, etc. • Frames in Banach spaces; • (GSI) Frames on LCA groups • Frames via integrable group representations, coorbit theory. (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 9 / 27
Key tracks in frame theory: • Frames in finite-dimensional spaces; • Frames in general separable Hilbert spaces • Concrete frames in concrete Hilbert spaces: • Gabor frames in L 2 ( R ) , L 2 ( R d ) ; • Wavelet frames; • Shift-invariant systems, generalized shift-invariant (GSI) systems; • Shearlets, etc. • Frames in Banach spaces; • (GSI) Frames on LCA groups • Frames via integrable group representations, coorbit theory. Research Group HATA DTU (Harmonic Analysis - Theory and Applications , Technical University of Denmark), https://hata.compute.dtu.dk/ (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 9 / 27
Key tracks in frame theory: • Frames in finite-dimensional spaces; • Frames in general separable Hilbert spaces • Concrete frames in concrete Hilbert spaces: • Gabor frames in L 2 ( R ) , L 2 ( R d ) ; • Wavelet frames; • Shift-invariant systems, generalized shift-invariant (GSI) systems; • Shearlets, etc. • Frames in Banach spaces; • (GSI) Frames on LCA groups • Frames via integrable group representations, coorbit theory. Research Group HATA DTU (Harmonic Analysis - Theory and Applications , Technical University of Denmark), https://hata.compute.dtu.dk/ An Introduction to frames and Riesz bases, 2.edition, Birkh¨ auser 2016 (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 9 / 27
Towards concrete frames - 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 ) . All these operators are unitary on L 2 ( R ) . (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 10 / 27
Towards concrete frames - 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 ) . All these operators are unitary on L 2 ( R ) . Gabor systems in L 2 ( R ) : 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 ) } m , n ∈ Z (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 10 / 27
Gabor systems in L 2 ( R ) It is known how to construct frames and dual pairs of frames with the Gabor structure { E mb T na g } m , n ∈ Z = { e 2 π imbx g ( x − na ) } m , n ∈ Z • Typical choices of g : B-splines or the Gaussian. • { E mb T na g } m , n ∈ Z can only be a frame if ab ≤ 1 . • If { E mb T na g } m , n ∈ Z is a frame, then it is a basis if and only if ab = 1 . • Gabor frames { E mb T na g } m , n ∈ Z are always linearly independent, and they have infinite excess if ab < 1 . (DTU ) IWOTA, Chemnitz 2017 August 17, 2017 11 / 27
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