The Duality Principle in General Hilbert Spaces

The duality principle in general Hilbert spaces

Ole Christensen Joint work with Diana Stoeva

Department of Applied Mathematics and Computer Science Technical University of Denmark Denmark

June 2, 2014

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 1/22 Outline of the talk

Gabor analysis, the duality principle • The R-dual of a frame (Casazza, Kutyniok, Lammers), a generalization • of the duality principle for some (all?) Gabor frames R-duals ( of type III), a generalization of the duality principle for all • Gabor frames.

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 2/22 Gabor systems

Gabor systems: have the form

2πimbx e g(x na) m,n Z { − } ∈ for some g L2(R), a, b > 0. Short notation: ∈ 2πimbx EmbTnag m,n Z = e g(x na) { } ∈ { − }

If EmbTnag m,n Z is a frame, then ab 1. • { } ∈ ≤ If EmbTnag m,n Z is a frame, then • { } ∈

EmbTnag m,n Z is a Riesz basis ab = 1 { } ∈ ⇔ 1 EmbTnag m,n Z is integer-oversampled if N 1 . • { } ∈ ab ∈ \ { }

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 3/22 Duality principle

A sequence fi i I in a Hilbert space is a frame if • { } ∈ 2 2 2 A, B > 0 : A f f , fi B f , f . ∃ || || ≤ X |h i| ≤ || || ∀ ∈H i I ∈ fi i I is a Riesz sequence if fi i I is a Riesz basis for span fi i I, i.e., • { } ∈ { } ∈ { } ∈ 2 2 2 A, B > 0 : A ci cifi B ci ∃ X | | ≤ || X || ≤ X | | i I ∈ for all ﬁnite sequences ci . { } The duality principle: Theorem: (Ron & Shen, Daubechies & Landau & Landau, Janssen, 1994) Let g L2(R) and a, b > 0 be given. Then the following are equivalent: ∈ 2 (i) EmbTnag m,n Z is a frame for L (R) with bounds A, B; { 1 } ∈ (ii) Em/aTn/bg m,n Z is a Riesz sequence with bounds A, B. { √ab } ∈ (DTU Mathematics) Talk, Strobl, 2014 June2,2014 4/22 Wexler-Raz’ Theorem

Wexler-Raz’ Theorem: (1994) If the Gabor systems EmbTnag m,n Z and { } ∈ EmbTnah m,n Z are Bessel sequences, then the following are equivalent: { } ∈ (i) The Gabor systems EmbTnag m,n Z and EmbTnah m,n Z are dual frames; { } ∈ { } ∈ 1 1 (ii) The Gabor systems Em/aTn/bg m,n Z and Em/aTn/bh m,n Z { √ab } ∈ { √ab } ∈ are biorthogonal, i.e., 1 1 Em/aTn/bg, Em′/aTn′/bh = δm,m′ δn,n′ . h√ab √ab i

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 5/22 Abstract duality in a Hilbert space H

Can the duality principle in Gabor analysis be recast as a special case of a general theory, valid for general frames?

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 6/22 Abstract duality in a Hilbert space H

Can the duality principle in Gabor analysis be recast as a special case of a general theory, valid for general frames?

R-dual of a sequence fi i I in a Hilbert space , introduced by Casazza, Kutyniok, and Lammers{ } in∈ 2004: H

Deﬁnition: Let ei i I and hi i I denote orthonormal bases for , and let { } ∈ { } ∈ 2 H fi i I be any sequence in for which i I fi, ej < for all j I. The { } ∈ H P ∈ |h i| ∞ ∈ R-dual (of type I) of fi i I with respect to the orthonormal bases ei i I and { } ∈ { } ∈ hi i I is the sequence ωj j I given by { } ∈ { } ∈

ωj = fi, ej hi, i I. (1) Xh i ∈ i I ∈

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 6/22 Abstract duality - results by Casazza, Kutyniok, Lammers:

Theorem: Consider the R-dual ωj j I of a sequence fi i I, i.e., { } ∈ { } ∈ ωj = fi, ej hi, i I. Xh i ∈ i I ∈ (i) fi i I is the R-dual of ωj j I w.r.t. the ONB’s hi i I and ei i I, i.e., { } ∈ { } ∈ { } ∈ { } ∈ fi = ωj, hi ej. Xh i j I ∈ (ii) fi i I is a Bessel sequence if and only ωi i I is a Bessel sequence. { } ∈ { } ∈ (iii) fi i I satisﬁes the lower frame condition with bound A if and only if { } ∈ ωj j I satisﬁes the lower Riesz sequence condition with bound A. { } ∈ (iv) fi i I is a frame for with bounds A, B if and only if ωj j I is a Riesz { } ∈ H { } ∈ sequence in with bounds A, B. H (v) Two Bessel sequences fi i I and gi i I in are dual frames if and { } ∈ { } ∈ H only if the associated R-dual sequences ωj j I and γj j I satisfy that { } ∈ { } ∈ ωj, γk = δj k, j, k I. h i , ∈ (DTU Mathematics) Talk, Strobl, 2014 June2,2014 7/22 The duality principle in Gabor analysis

The duality principle:

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 8/22 The duality principle in Gabor analysis

The duality principle:

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 8/22 The duality principle in Gabor analysis

The duality principle:

Theorem: Let g L2(R) and a, b > 0 be given. Then the Gabor system ∈ 2 EmbTnag m,n Z is a frame for L (R) with bounds A, B if and only if { 1 } ∈ Em/aTn/bg m,n Z is a Riesz sequence with bounds A, B. { √ab } ∈ Can this result be derived as a consequence of the abstract duality • concept? 1 That is, can Em/aTn/bg m,n Z be realized as the R-dual of • { √ab } ∈ EmbTnag m,n Z w.r.t. certain choices of orthonormal bases em,n m,n Z { } ∈ { } ∈ and hm,n m,n Z? { } ∈ That is, can we ﬁnd ONB’s em,n m,n Z, hm,n m,n Z such that • { } ∈ { } ∈ 1 ′ ′ ′ ′ Z Em/aTn/bg = Em bTn ag, em,n hm ,n , m, n ? √ab ′X′ h i ∀ ∈ m ,n Z ∈

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 8/22 Abstract duality

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 9/22 Abstract duality

Results by Casazza, Kutyniok, Lammers (2004): 1 If EmbTnag m,n Z is a frame and ab = 1, then Em/aTn/bg m,n Z • { } ∈ { √ab } ∈ can be realized as the R-dual of EmbTnag m,n Z w.r.t. certain choices of { } ∈ orthonormal bases ei i I and hi i I. ∈ ∈ { } { } 1 If EmbTnag m,n Z is a tight frame, then Em/aTn/bg m,n Z can be • { } ∈ { √ab } ∈ realized as the R-dual of EmbTnag m,n Z w.r.t. certain choices of { } ∈ orthonormal bases ei i I and hi i I. { } ∈ { } ∈ The general case is still open!

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 9/22 1a) Daubechies, I., Landau, H. J., and Landau, Z.: Gabor time-frequency lattices and the Wexler-Raz identity. J. Fourier Anal. Appl. 1 (1995), 437–478. 1b) Janssen, A. J. E. M.: Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1 no. 4 (1995), 403–436. 1c) Ron, A. and Shen, Z. Weyl-Heisenberg systems and Riesz bases in L2(Rd). Duke Math. J. 89 (1997), 237–282. 2) Casazza, P., Kutyniok, G., and Lammers, M.: Duality principles in abstract frame theory. J. Fourier Anal. Appl. 10 4, 2004, 383–408. 3a) Christensen, O., Kim, H.O., and Kim, R.Y.: On the duality principle by Casazza, Kutyniok, and Lammers. J. Fourier Anal. Appl. 17 (2011), 640–655. 3b) Fan, Z., and Shen, Z.: Dual Gramian analysis: duality principle and unitary extension principle. Preprint, 2013. 4a) Dutkay, D., Han, D., and Larson, D.: A duality principle for groups. J. Funct. Anal. 257 (2009), 1133–1143. 4b) Xiao, X. M. and Zhu, Y. C.: Duality principles of frames in Banach spaces. Acta. Math. Sci. Ser. A. Chin. 29 (2009), 94–102 4c) Christensen, O., Xiao, X. C., and Zhu, Y. C.: Characterizing R-duality in Banach spaces. Acta Mathematica Sinica, Eng. Series 29 no.1 (2013), 75–84. 5) Stoeva, D., and Christensen, O.: On R-duals and the duality principle in Gabor analysis. Preprint, 2014 (ArXive)

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 10/22 Towards generalized R-duals

Deﬁnition Let ei i I and hi i I denote orthonormal bases for , and let { } ∈ { } ∈ 2 H fi i I be any sequence in for which i I fi, ej < for all j I. The { } ∈ H P ∈ |h i| ∞ ∈ R-dual of type IV of fi i I with respect to the Riesz bases ei i I and hi i I { } ∈ { } ∈ { } ∈ is the sequence ωj j I given by { } ∈

ωj = fi, ej hi, i I. Xh i ∈ i I ∈

Can show: This deﬁnition is too broad - the relations between fi i I and { } ∈ ωj j I do not necessarily correspond to the properties of the R-duals, or the duality{ } ∈ principle.

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 11/22 Towards generalized R-duals

Proposition (Stoeva, C., 2014) If EmbTnag m,n Z is an integer-oversampled { } ∈ or tight frame, there exist Riesz bases xm,n m,n Z, ym,n m,n Z such that { } ∈ { } ∈ 1 Em/aTn/bg = Em′bTn′ag, xm,n ym′,n′ . √ab ′X′ h i m ,n Z ∈ 1/2 1/2 In fact, xm n = S em n, ym n = S hm n for some ONB’s em n , hm n , − , , , { , } { , } Proof in the case 1/(ab) N : ∈ S 1/2 commutes with the operators E T for all m, n Z. • − mb na 1/2 2 ∈ EmbTnaS− g m,n Z is a tight frame for L (R) with frame bound 1. • { } ∈ By Casazza & al. there exist ONB’s em,n m,n Z, hm,n m,n Z such that • { } ∈ { } ∈ 1 1/2 1/2 Em/aTn/bS− g = Em′bTn′aS− g, em,n hm′,n′ . √ab ′X′ h i m ,n Z ∈

Since 1/(ab)= K N, E T form a subclass of EmbTna, m, n Z • ∈ m/a n/b ∈ (DTU Mathematics) Talk, Strobl, 2014 June2,2014 12/22 Towards generalized R-duals

Deﬁnition Let fi i I be a frame for with frame operator S. Let ei i I and { } ∈ H { } ∈ hi i I denote orthonormal bases for . { } ∈ H (i) The R-dual of type II of fi i I w.r.t. ei i I and hi i I is the sequence { } ∈ { } ∈ { } ∈ 1/2 1/2 ωj = fi, S− ej S hi, j I. Xh i ∈ i I ∈ (ii) Let Q : be a bounded bijective operator with Q S and 1 H→H 1 k k≤ p|| || Q− S− . The R-dual of type III of fi i I with respect to the k k≤ p|| || { } ∈ triplet ( ei i I , hi i I , Q), is the sequence ωj j I deﬁned by { } ∈ { } ∈ { } ∈

∞ 1/2 ωj := S− fi, ej Qhi. Xh i i=1 Note: R-duals of type III also exists if fi i I is just a frame sequence. • 1 { } ∈ Qhi i I has the bounds S−1 , S , the optimal bounds for fi i I. • { } ∈ || || || || { } ∈ (DTU Mathematics) Talk, Strobl, 2014 June2,2014 13/22 Towards generalized R-duals

I II III IV

Proposition (Stoeva, C., 2014) The class of type I R-duals of f is contained in the class of type III • i i I R-duals; and type II is contained{ } in∈ type III. Assume that f is a tight frame for . Then the classes of R-duals of • i i I type I, II and{ III} coincide.∈ H

This is good news - for tight frames we know that R-duals of type I are enough for the duality principle! (DTU Mathematics) Talk, Strobl, 2014 June2,2014 14/22 R-duals of type II

Proposition (Stoeva, C., 2014) Let fi i I be a frame for with frame { } ∈ H operator S and let ωj j I be the R-dual of type II of fi i I with respect to { } ∈ { } ∈ some orthonormal bases ei i I and hi i I, i.e., { } ∈ { } ∈ 1/2 1/2 ωj = fi, S− ej S hi, j I. Xh i ∈ i I ∈ The following statements hold. 1/2 1/2 (i) fi = j I ωj, S− hi S ej, i I. (almost)P ∈ Likeh R-dualsi of type∀ I. ∈ (ii) Let γj j I denote the R-dual of type II of the canonical dual frame {1 } ∈ S− fi i I with respect to ei i I and hi i I . Then ωj j I and γj j I are{ biorthogonal.} ∈ { } ∈ { } ∈ { } ∈ { } ∈ Like R-duals of type I - but only for the canonical dual

Question: R-duals of type II generalize the duality principle for integer- oversampled and tight Gabor frames - what about general Gabor frames? (DTU Mathematics) Talk, Strobl, 2014 June2,2014 15/22 R-duals of type II/III

Proposition (Stoeva, C., 2014) Let fi i I be a frame sequence and ωi i I an { } ∈ { } ∈ R-dual of fi i I of type III. Then the following hold. { } ∈ (i) fi i I is a frame if and only if ωj j I is a Riesz sequence; in the { } ∈ { } ∈ afﬁrmative case the bounds for fi i I are also bounds for ωj j I. Almost like for R-duals of type{ I } ∈ { } ∈

(ii) fi i I is a Riesz sequence if and only if ωj j I is a frame; in the { } ∈ { } ∈ afﬁrmative case the bounds for fi i I are also bounds for ωj j I. Almost like for R-duals of type{ I } ∈ { } ∈

(iii) ωj j I is a Riesz basis if and only if fi i I is a Riesz basis. Like{ } ∈ for R-duals of type I { } ∈

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 16/22 Towards generalized R-duals

Recall: If fi i I is a Bessel sequence, the preframe operator is { } ∈ 2 T : ℓ (N) , T ci i I = cifi →H { } ∈ X i I ∈ Lemma (Casazza, Kutyniok, Lammers, 2004) Asume that

fi i I is a frame with synthesis operator T; • { } ∈ ωj j I is an R-dual of type I of fi i I , • { } ∈ { } ∈ Then

dim(ker T)= dim(span ωj j⊥ I.) { } ∈

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 17/22 R-duals of type II and III

Characterization of R-duals of type II and III: Theorem (Stoeva, C., 2014) Assume that

fi i I is a frame for , with synthesis operator T. • { } ∈ H ωj j I is a Riesz sequence in ; • { } ∈ H The bounds of fi i I are also bounds for ωj j I . • { } ∈ { } ∈ Then the following hold. 1/2 (i) ωj j I is an R-dual of type II of fi i I if and only if S− ωj j I is an orthonormal{ } ∈ system and { } ∈ { } ∈

dim(ker T)= dim(span ωj j⊥ I ) (2) { } ∈

(ii) ωj j I is an R-dual of type III of fi i I if and only if (2) holds. { } ∈ { } ∈

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 18/22 R-duals of type III and Gabor systems

2 Corollary (Stoeva, C., 2014) Let EmbTnag m,n Z be a Gabor frame for L (R). 1 { } ∈ Then Em/aTn/bg m,n Z can be realized as an R-dual of type III of { √ab } ∈ EmbTnag m,n Z. { } ∈ Proof: Letting T be the preframe operator for EmbTnag m,n Z, we must show that { } ∈

dim(ker T)= dim(span Em/aTn/bg m⊥,n I ) { } ∈ This holds if ab = 1, so assume that ab < 1. Then, using results about deﬁcit/excess by Balan, Casazza, and Heil:

EmbTnag m,n Z has infinite excess, so dim(ker T)= ; • ∈ {1 } ∞ > 1, so Em/aTn/bg m,n I has infinite deficit, i.e., • ab { } ∈

dim(span Em/aTn/bg m⊥,n I )= { } ∈ ∞

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 19/22 1 Canonical R-dual of type III of S− fi i I. { } ∈ Let ωj j I be an R-dual of type III of a frame fi i I with respect to • { } ∈ { } ∈ ONB’s ei i I , hi i I and a bounded bijective operator Q, i.e., { } ∈ { } ∈ ∞ 1/2 ωj := S− fi, ej Qhi. Xh i i=1 1 1 The frame operator associated with S− fi i I is S− , and the bijective • 1 { } ∈ operator Q := (Q∗)− satisﬁes that e 1 1 1 1 1 Q = Q− S , and Q− = Q (S ) . || || || || ≤ q|| − || || || || || ≤ q|| − − || e 1 e (Q∗)− hi i I is the canonical dual Riesz basis of hi i I . • { } ∈ 1 { } ∈ The canonical R-dual of type III of S− fi i I : the R-dual of type III of • 1 { } ∈ 1 S− fi i I w.r.t. the ONB’s ei i I , hi i I and the operator Q = (Q∗)− { } ∈ { } ∈ { } ∈ Speciﬁcally, it is the sequence γj j I , where e { } ∈ 1 1/2 1 1/2 1 γj = S− fi, S ej (Q∗)− hi = S− fi, ej (Q∗)− hi. Xh i Xh i i I i I ∈ ∈ (DTU Mathematics) Talk, Strobl, 2014 June2,2014 20/22 Towards generalized R-duals

Theorem (Stoeva, C., 2014) Let fi i I be a frame and ωi i I an R-dual of { } ∈ { } ∈ fi i I of type III, w.r.t. orthonormal bases ei i I , hi i I and a bounded { } ∈ { } ∈ { } ∈ bijective operator Q, i.e.,

∞ 1/2 ωj := S− fi, ej Qhi. Xh i i=1

Denote the frame operator of fi i I by S. Then the following hold: { } ∈ 1 1/2 (i) fi = j I ωj, (Q∗)− hi S ej, i I. AlmostP ∈ likeh R-duals ofi type I. ∀ ∈

(ii) The R-dual of type III of fi i I with respect to some orthonormal bases { } ∈ ei i I , hi i I, and an operator Q, is biorthogonal to the canonical { } ∈ { } ∈ 1 R-dual of type III of S− fi i I . Similar to results for{ the R-duals} ∈ of type I, but only for the canonical R-dual of type III.

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 21/22 Conclusion

R-duals of type III generalize the duality principle for all Gabor frames, • and share most of the properties of the R-duals of type I (by Casazza et. al); It is unknown whether R-duals of type I and II generalize the duality • principle for all Gabor frames.

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 22/22 Balan, R., Casazza, P., and Heil, C.: Deﬁcits and excesses of frames. Adv. Comp. Math. 18 (2003), 93–116. Casazza, P., Kutyniok, G., and Lammers, M.: Duality principles in abstract frame theory. J. Fourier Anal. Appl. 10 4, 2004, 383–408. Christensen, O.: Frames and bases in mathematics and engineering. An introductory course. Birkh¨auser 2007. Christensen, O., Kim, H.O., and Kim, R.Y.: On the duality principle by Casazza, Kutyniok, and Lammers. J. Fourier Anal. Appl. 17 (2011), 640–655. Christensen, O., Xiao, X. C., and Zhu, Y. C.: Characterizing R-duality in Banach spaces. Acta Mathematica Sinica, Eng. Series 29 no.1 (2013), 75–84. Daubechies, I., Landau, H. J., and Landau, Z.: Gabor time-frequency lattices and the Wexler-Raz identity. J. Fourier Anal. Appl. 1 (1995), 437–478.

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 22/22 Dutkay, D., Han, D., and Larson, D.: A duality principle for groups. J. Funct. Anal. 257 (2009), 1133–1143. Fan, Z., and Shen, Z.: Dual Gramian analysis: duality principle and unitary extension principle. Preprint, 2013. Janssen, A. J. E. M.: Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1 no. 4 (1995), 403–436. Ron, A. and Shen, Z.: Weyl-Heisenberg systems and Riesz bases in L2(Rd). Duke Math. J. 89 (1997), 237–282. Xiao, X. M. and Zhu, Y. C.: Duality principles of frames in Banach spaces. Acta. Math. Sci. Ser. A. Chin. 29 (2009), 94–102

(DTU Mathematics) Talk, Strobl, 2014 June2,2014 22/22