Frames and operator representations of frames

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1 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 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, / 27

2 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, / 27

3 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 = {Tn 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, / 27

4 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 = {Tn 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, / 27

5 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, / 27

6 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 inh, 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, / 27

7 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, / 27

8 Frames Definition: A sequence {f k } k=1 inhis 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: where {e k } k=1 is an ONB. {e 1, e 1, e 2, e 3,...}, (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

9 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, / 27

10 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 = f, S 1 f k f k, f H. k=1 It might be difficult to compute S 1! (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

11 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 = f, S 1 f k f k, f H. 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, / 27

12 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 = f, g k f k = k=1 f, S 1 f k f k, f H. 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, / 27

13 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 } k=1 = c k f k [T f = { f, f k } k=1 ] k=1 and {g k } k=1 be a Bessel sequence with pre-frame operator U : H l 2 (N), U{c k } k=1 = c k g k [U f = { f, g k } k=1 ] k=1 Then{f k } k=1 and {g 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, / 27

14 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, / 27

15 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), (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

16 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), An Introduction to frames and Riesz bases, 2.edition, Birkhäuser 2016 (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

17 Towards concrete frames - operators on L 2 (R) Translation by a R: T a : L 2 (R) L 2 (R), (T a f)(x) = f(x a). Modulation by b R : E b : L 2 (R) L 2 (R), (E b f)(x) = e 2πibx f(x). All these operators are unitary on L 2 (R). (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

18 Towards concrete frames - operators on L 2 (R) Translation by a R: T a : L 2 (R) L 2 (R), (T a f)(x) = f(x a). Modulation by b R : E b : L 2 (R) L 2 (R), (E b f)(x) = e 2πibx f(x). All these operators are unitary on L 2 (R). Gabor systems in L 2 (R): have the form {e 2πimbx 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 2πimbx g(x na)} m,n Z (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

19 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 2πimbx 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, / 27

20 Dynamical Sampling Introduced in papers by Aldroubi, Davis & Krishtal, and Aldroubi, Cabrelli, Molter & Tang. Further developed in papers by Aceska, Aldroubi, Cabrelli, Çakmak, Kim, Molter, Paternostro, Petrosyan, Philipp. Let H denote a Hilbert space, and A a class of operators T : H H. For T A and ϕ H, consider the iterated system {T n ϕ} n=0 = {ϕ, Tϕ, T 2 ϕ }. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

21 Dynamical Sampling Introduced in papers by Aldroubi, Davis & Krishtal, and Aldroubi, Cabrelli, Molter & Tang. Further developed in papers by Aceska, Aldroubi, Cabrelli, Çakmak, Kim, Molter, Paternostro, Petrosyan, Philipp. Let H denote a Hilbert space, and A a class of operators T : H H. For T A and ϕ H, consider the iterated system Key questions: {T n ϕ} n=0 = {ϕ, Tϕ, T 2 ϕ }. Can{T n ϕ} n=0 be a basis for H for some T A,ϕ H? Can{T n ϕ} n=0 be a frame for H for some T A,ϕ H? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

22 Dynamical Sampling and standard operator theory Consider a bounded operator T : H H. Recall: A vector ϕ H is cyclic if span{t n ϕ} n=0 = H. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

23 Dynamical Sampling and standard operator theory Consider a bounded operator T : H H. Recall: A vector ϕ H is cyclic if span{t n ϕ} n=0 = H. This is much weaker than the condition that {T n ϕ} n=0 is a frame. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

24 Dynamical Sampling and standard operator theory Consider a bounded operator T : H H. Recall: A vector ϕ H is cyclic if span{t n ϕ} n=0 = H. This is much weaker than the condition that {T n ϕ} n=0 is a frame. A vector ϕ H is hypercyclic if {T n ϕ} n=0 is dense inh. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

25 Dynamical Sampling and standard operator theory Consider a bounded operator T : H H. Recall: A vector ϕ H is cyclic if span{t n ϕ} n=0 = H. This is much weaker than the condition that {T n ϕ} n=0 is a frame. A vector ϕ H is hypercyclic if {T n ϕ} n=0 is dense inh. This is too strong in the frame context - it excludes that {T n ϕ} n=0 is a Bessel sequence. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

26 Dynamical Sampling Recall the key questions in dynamical sampling: Can{T n ϕ} n=0 be a basis for H for some T A,ϕ H? Can{T n ϕ} n=0 be a frame for H for some T A,ϕ H? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

27 Dynamical Sampling Recall the key questions in dynamical sampling: Can{T n ϕ} n=0 be a basis for H for some T A,ϕ H? Can{T n ϕ} n=0 be a frame for H for some T A,ϕ H? Dual approach by C. & Marzieh Hasannasab: When does a given frame {f k } k=1 has a representation {f k } k=1 = {Tn ϕ} n=0 for some operator T : H H? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

28 Dynamical Sampling Recall the key questions in dynamical sampling: Can{T n ϕ} n=0 be a basis for H for some T A,ϕ H? Can{T n ϕ} n=0 be a frame for H for some T A,ϕ H? Dual approach by C. & Marzieh Hasannasab: When does a given frame {f k } k=1 has a representation for some operator T : H H? {f k } k=1 = {Tn ϕ} n=0 Under what conditions is such a representation possible with a bounded operator T? What are the properties of such frames? What are the properties of the relevant operators T? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

29 Dynamical sampling Plan for the rest of the talk: A sample of results from the literature A characterization of the frames that have a representation {f k } k=1 = {Tn ϕ} n=0 for some operator T : span{f k} k=1 H. Characterizations of the case where T can be chosen to be bounded. Properties of {f k } k=1 and properties of T. Open problems. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

30 Results from the literature If T is normal, then {T n ϕ} n=0 is not a basis (Aldroubi, Cabrelli, Çakmak, Molter, Petrosyan). If T is unitary, then {T n ϕ} n=0 is not a frame (Aldroubi, Petrosyan). If T is compact, then {T n ϕ} n=0 is not a frame (C., Hasannasab, Rashidi). (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

31 A positive result from the literature Example (Aldroubi, Cabrelli, Molter, Tang) Consider an operator T of the form T = k=1 λ kp k, where P k, k N, are rank 1 orthogonal projections such that P j P k = 0, j k, k=1 P k = I, and λ k < 1 for all k N. There exists an ONB{e k } k=1 such that Tf = λ k f, e k e k, f H. k=1 Assume that {λ k } k=1 satisfies the Carleson condition, i.e., λ j λ k inf k 1 λ j k j λ k > 0. Lettingϕ := k=1 1 λk 2 e k, the family{t n ϕ} n=0 is a frame for H. Concrete case: λ k = 1 α k for someα > 1. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

32 The key question, formulated inl 2 (N) Motivated by all the talks on Hilbert/Hankel/Helson/Toeplitz matrices: Key question: Identify a class of infinite non-diagonalizable matrices A = (a m,n ) m,n 1 and some vector ϕ l 2 (N) such that the collection of vectors form a frame for l 2 (N). {A n ϕ} n=0 = {ϕ, Aϕ, A 2 ϕ, } (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

33 Existence of the representation{f k } k=1 = {Tn f 1 } n=0 Proposition (Hasannasab & C., 2016): Consider a frame {f k } k=1 for an infinite-dimensional Hilbert space H. Then the following are equivalent: There exists a linear operator T : span{f k } k=1 H such that {f k } k=1 = {Tn f 1 } n=0, The sequence {f k } k=1 is linearly independent. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

34 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 If {f k } k=1 is a Bessel sequence, the synthesis operator U : l 2 (N) H, U{c k } k=1 := c k f k k=1 is well-defined and bounded. The kernel of U is { N(U) = {c k } k=1 l2 (N) } c k f k = 0. k=1 Consider the right-shift operator T on l 2 (N), defined by T (c 1, c 2, ) = (0, c 1, c 2, ). (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

35 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Theorem (Hasannasab & C., 2017): Consider a frame {f k } k=1. Then the following are equivalent: (i) The frame has a representation {f k } k=1 = {Tn f 1 } n=0 for some bounded operator T : H H. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

36 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Theorem (Hasannasab & C., 2017): Consider a frame {f k } k=1. Then the following are equivalent: (i) The frame has a representation {f k } k=1 = {Tn f 1 } n=0 for some bounded operator T : H H. (ii) {f k } k=1 is linearly independent and the kernel N(U) of the synthesis operator U is invariant under the right-shift operator T. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

37 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Theorem (Hasannasab & C., 2017): Consider a frame {f k } k=1. Then the following are equivalent: (i) The frame has a representation {f k } k=1 = {Tn f 1 } n=0 for some bounded operator T : H H. (ii) {f k } k=1 is linearly independent and the kernel N(U) of the synthesis operator U is invariant under the right-shift operator T. (iii) For some dual frame {g k } k=1 (and hence all), f j+1 = f j, g k f k+1, j N. k=1 (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

38 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Theorem (Hasannasab & C., 2017): Consider a frame {f k } k=1. Then the following are equivalent: (i) The frame has a representation {f k } k=1 = {Tn f 1 } n=0 for some bounded operator T : H H. (ii) {f k } k=1 is linearly independent and the kernel N(U) of the synthesis operator U is invariant under the right-shift operator T. (iii) For some dual frame {g k } k=1 (and hence all), f j+1 = f j, g k f k+1, j N. k=1 In the affirmative case, letting {g k } k=1 denote an arbitrary dual frame of {f k } k=1, the operator T has the form Tf = f, g k f k+1, f H. k=1 (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

39 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Corollary: Any Riesz basis {f k } k=1 has the form {f k} k=1 = {Tn f 1 } n=0 for some bounded operator T : H H. Surprisingly, the availability of such a representation fails for frames with finite excess: Proposition: Assume that {f k } k=1 is a frame and {f k} k=1 = {Tn f 1 } n=0 for a linear operator T. If {f k } k=1 has finite and strictly positive excess, then T is unbounded. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

40 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Note: The properties of a frame with a representation {f k } k=1 = {Tn f 1 } n=0 (linear independence and infinite excess) match precisely the properties of Gabor frames {E mb T na g} m,n Z for ab < 1! (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

41 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Note: The properties of a frame with a representation {f k } k=1 = {Tn f 1 } n=0 (linear independence and infinite excess) match precisely the properties of Gabor frames {E mb T na g} m,n Z for ab < 1! Can a Gabor frame {E mb T na g} m,n Z with ab < 1 be represented on this form with a bounded operator T? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

42 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Note: The properties of a frame with a representation {f k } k=1 = {Tn f 1 } n=0 (linear independence and infinite excess) match precisely the properties of Gabor frames {E mb T na g} m,n Z for ab < 1! Can a Gabor frame {E mb T na g} m,n Z with ab < 1 be represented on this form with a bounded operator T? No! (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

43 Boundedness of T in the representation{f k } k=1 = {Tn f 1 } n=0 Note: The properties of a frame with a representation {f k } k=1 = {Tn f 1 } n=0 (linear independence and infinite excess) match precisely the properties of Gabor frames {E mb T na g} m,n Z for ab < 1! Can a Gabor frame {E mb T na g} m,n Z with ab < 1 be represented on this form with a bounded operator T? No! Question: Can a Gabor frame {E mb T na g} m,n Z with ab < 1 be represented on the {E mb T na g} m,n Z = {a n T n g} n=0 for some scalars a n > 0 and a bounded operator T : L 2 (R) L 2 (R)? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

44 A comment on the indexing Let us index the frames byzinstead of N 0 : Frames with a representation {f k } k=1 = {Tk f 0 } k Z have a similar characterization as for the index set N 0. The conditions for boundedness of T are similar. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

45 A comment on the indexing Let us index the frames byzinstead of N 0 : Frames with a representation {f k } k=1 = {Tk f 0 } k Z have a similar characterization as for the index set N 0. The conditions for boundedness of T are similar. The standard operators in applied harmonic analysis easily leads to such frames represented by bounded operators: Translation: {T k ϕ} k Z = {(T 1 ) k ϕ} k Z ; Modulation: {E mb ϕ} m Z = {(E b ) m ϕ} m Z ; Scaling: {D a jϕ} j Z = {(D a ) j ϕ} j Z ; (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

46 A comment on the indexing Let us index the frames byzinstead of N 0 : Frames with a representation {f k } k=1 = {Tk f 0 } k Z have a similar characterization as for the index set N 0. The conditions for boundedness of T are similar. The standard operators in applied harmonic analysis easily leads to such frames represented by bounded operators: Translation: {T k ϕ} k Z = {(T 1 ) k ϕ} k Z ; Modulation: {E mb ϕ} m Z = {(E b ) m ϕ} m Z ; Scaling: {D a jϕ} j Z = {(D a ) j ϕ} j Z ; Question: Consider a Gabor frame{e mb T na g} m,n Z for L 2 (R) with ab < 1. Does the frame {E mb T na g} m,n Z have a representation {E mb T na g} m,n Z = {T n ϕ} n= for a bounded operator T : L 2 (R) L 2 (R) and someϕ L 2 (R)? (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

47 Frames of the form{f k } k=1 = {Tn f 1 } n=0 are special! Theorem (C., Hasannasab, Rashidi, 2017) Assume that {T n ϕ} n=0 is an overcomplete frame for some ϕ H and some bounded operator T : H H. Then the following hold: (i) The image chain for the operator T has finite length q(t). (ii) If N N, then T N ϕ span{t n ϕ} n=n+1 N q(t). For any N q(t), let V := span{t n ϕ} n=n. Then the following hold: (iii) The space V is independent of N and has finite codimension. (iv) The sequence {T n ϕ} n=n+l is a frame for V for all l N 0. (v) V is invariant under T, and T : V V is surjective. (vi) If the null chain of T has finite length then T : V V is injective; in particular this is the case if T is normal. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

48 A way to achieve boundedness - multiple operators Theorem (Hasannasab & C., 2016) Consider a frame {f k } k=1 which is norm-bounded below. Then there is a finite collection of vectors from {f k } k=1, to be called ϕ 1,...,ϕ J, and corresponding bounded operators T j : H H, such that J {f k } k=1 = {Tj n ϕ j} n=0. j=1 Remark: The assumption that {f k } k=1 is norm-bounded below can not be removed. (DTU ) IWOTA, Chemnitz 2017 August 17, / 27

49 Christensen, O., and Hasannasab, M.: An open problem concerning operator representations of frames. Preprint, 2017 (DTU ) IWOTA, Chemnitz 2017 August 17, / 27 References Aceska, R., and Yeon Hyang Kim, Y. H.: Scalability of frames generated by dynamical operators. arxiv preprint arxiv: (2016). Aldroubi, A., Cabrelli, C., Molter, U., and Tang, S.: Dynamical sampling. Appl. Harm. Anal. Appl. 42 no. 3 (2017), Aldroubi, A., Cabrelli, C., Çakmak, A. F., Molter, U., and Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272 no. 3 (2017), Cabrelli, C., Molter, U, Paternostro, V., and Philipp, F.: Dynamical sampling on finite index sets. Preprint, Christensen, O., and Hasannasab, M.: Frame properties of systems arising via iterative actions of operators Preprint, Christensen, O., and Hasannasab, M.: Operator representations of frames: boundedness, duality, and stability. Int. Eq. & Op. Theory, to appear.

Operator representations of frames: boundedness, duality, and stability.

arxiv:1704.08918v1 [math.fa] 28 Apr 2017 Operator representations of frames: boundedness, duality, and stability. Ole Christensen, Marzieh Hasannasab May 1, 2017 Abstract The purpose of the paper is to