Sampling and Interpolation on Some Nilpotent Lie Groups

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1 Sampling and Interpolation on Some Nilpotent Lie Groups SEAM 013 Vignon Oussa Bridgewater State University March 013 ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

2 Background A Prominent Example (Whittaker, Shannon, Kotel nikov) Let G = R and Γ = Z. The Paley Wiener space PW = f L (R) : support b 1 f, 1 1 The map f 7! (f (k)) kz is an isometry Any function in PW is completely determined by its values on Z. The series converges unif. in L (R). 3 fsinc (x k) : k Zg is an ONB. f (t) = f (k) sinc (x k) kz 4 PW is a sampling space with interpolation property. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 / 8

3 A Starting Point Background Let G be a locally compact group, Γ G. Let H L (G) be a left-invariant closed subspace of L (G) consisting of continuous functions. H is a sampling space with respect to Γ if 1 f 7! (f (γ)) γγ is an isometry: jf (γ)j = kfk. γγ There exists S H such that for all f H, f (x) = f (γ) S γ 1 x. γγ and S is called a sinc-type function. 3 A sampling space has the interpolation property if f 7! (f (γ)) γγ is unitary. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

4 The Heisenberg group Background B. Currey and A. Mayeli proved in 01 that the Heisenberg group admits sampling spaces with interpolation property with respect to Γ < 1 x z z = H = y 5 : 4 y 5 R 3 : ; x < 1 m k k = Γ = l 5 : 4 l 5 Z 3 : ; m The interpolation property was a surprising fact!!! Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

5 Open Questions Background Let N be a locally compact group. We consider L (N) with left regular representation L. L (y) f (x) = f y 1 x 1 How do we generalize the concept of bandlimitation? How do we pick a subset Γ N such that f 7! (f (γ)) γγ is an isometry? 3 What is the structure of Γ? 4 Do sampling subspaces of L (N) with respect to Γ exist? If yes do they have the interpolation property? 5 How do we construct sampling subspaces of L (N)? Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

6 Background Sampling on Non-abelian groups 1 J. Christensen, Sampling in Reproducing Kernel Banach Spaces on Lie Groups, Journal of Approximation Theory, Vol. 164 Issue 1 (01), B. Currey, A. Mayeli, A Density Condition for Interpolation on the Heisenberg Group, Rocky Mountain Journal of Mathematics 3 A. Dooley, A Nonabelian Version of the Shannon Sampling Theorem. Siam. J. Math. Anal. 0, (1989). 4 H. Führ, K. Gröchenig, Sampling Theorems on Locally Compact Groups from Oscillation Estimates, Math. Z. 55 (007), I. Pesenson, Sampling of Paley-Wiener functions on strati ed groups, J. Fourier Anal. Appl. 4 (1998), V. Oussa, Bandlimited Spaces on Some -step Nilpotent Lie Groups With One Parseval Frame Generator, to appear in Rocky Mountain Journal of Mathematics, (01) 7 V. Oussa, Sampling and Interpolation Conditions on Some Two-Step Nilpotent Lie Groups, preprint Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

7 Nilpotent Lie Groups Content of the talk 1 Consider a class of groups of non-commutative unipotent matrices Develop the concept of bandlimitation 3 Show existence of sampling spaces 4 Give conditions for interpolation Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

8 Nilpotent Lie Groups A Class of Nilpotent Lie Groups Let N be a non-commutative simply connected, connected two step nilpotent Lie group with Lie algebra n 1 n = z b a, z is the center of n. dm [a, b] z where b = RY k, a = 3 a, b are commutative algebras. 4 dm RX k [X 1, Y 1 ] [X 1, Y d ] B C. 5A [X d, Y 1 ] [X d, Y d ] is a non-vanishing polynomial over [n, n] Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

9 Some Facts Nilpotent Lie Groups Theorem If N satis es the previous conditions then N = R n d o R d and there is nite dimensional faithful matrix representation of N in GL (n + 1, R). ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

10 Example 1 Nilpotent Lie Groups The Heisenberg Lie group with Lie algebra spanned by fz, Y, Xg with Lie brackets [X, Y] = Z 8 N < = 4 : 1 x z 0 1 y : 4 z y x 3 5 R 3 9 = ; Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

11 Example Nilpotent Lie Groups Let N = exp (RZ 1 RZ ) exp (RY 1 RY ) exp (RX 1 RX ) with Lie brackets [X 1, Y 1 ] = Z 1, [X, Y 1 ] = Z, [X 1, Y ] = Z, [X, Y ] = Z 1. Here is a matrix representation of N x 1 x y 1 y z x x 1 y y 1 z >< N y 1 = y : >: z 1 z y 1 y x 1 x 3 9 >= R >; Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

12 Frames Nilpotent Lie Groups Given a countable sequence ff i g ii of functions in a Hilbert space H, we say ff i g ii forms a frame i there exist strictly positive real numbers A, B s.t. A kf k jhf, f i ij B kf k ii 1 If A = B, ff i g ii is a tight frame. If A = B = 1, ff i g ii is a Parseval frame. 3 If ff i g ii is a Parseval frame such that for all i I, kf i k = 1 then ff i g ii is an orthonormal basis for H. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

13 Nilpotent Lie Groups Gabor System Let Λ = AZ d BZ d be a full rank separable lattice in R d and g L R d. The system G g, AZ d BZ d = ne πihk,xi g (x n) : k BZ d, n AZ do is called a Gabor system. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

14 Nilpotent Lie Groups Density Condition Given a separable full rank lattice Λ = AZ d BZ d in R d. The following facts are equivalent 1 There exits φ L (R d ) such that G (φ, Λ) is a Parseval frame in L R d. vol (Λ) = jdet A det Bj 1. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

15 Nilpotent Lie Groups Harmonic Analysis on N The Fourier transform is de ned on L (N) \ L 1 (N) by where for B (λ) = Z P (f ) (λ) = λ X i, Y j Σ 1i,jd π λ (n) f (n) dn, 1 Σ is a Zariski open subset of the linear dual of the z Σ = fλ z : det (B (λ)) 6= 0g bn = fπ λ : λ Σg where π λ (exp a exp b) f = ne πihb(λ)y,ti f (t x) : y R d, x R do π λ (exp z) f = ne πiλ(z) f : z R n do and f L R d Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

16 Nilpotent Lie Groups Harmonic Analysis on N 1 P L (N) = R Σ L R d L R d dµ (λ) P L P 1 = R Σ π λ 1 L (R d ) dµ (λ) 3 The Plancherel measure: dµ (λ) = jdet (B (λ))j dλ where 3 λ [X 1, Y 1 ] λ [X 1, Y d ] 6 B (λ) = λ [X d, Y 1 ] λ [X d, Y d ] 4 Plancherel Theorem kf k L (N) = R Σ kp (f ) (λ)k HS dµ (λ). Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

17 Nilpotent Lie Groups Example N = exp (RZ 1 RZ ) exp (RY 1 RY ) exp (RX 1 RX ) with [X 1, Y 1 ] = Z 1, [X 1, Y ] = Z, [X, Y 1 ] = Z, [X, Y ] = Z 1. Then 1 Σ = (λ 1, λ, 0, 0, 0, 0) n : λ 1 + λ 1 6= 0 The Plancherel measure is λ λ dλ1 dλ Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

18 Nilpotent Lie Groups Theorem ( Oussa, 01) For a xed λ Σ, let L a be a lattice in a = R d and let L b be a lattice in b = R d. is a Gabor system. π λ (exp L a exp L b ) f ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

19 Results and Examples Concept of Bandlimitation 1 We say a Hilbert space H is a multiplicity-free left-invariant closed subspace of L (N) if and only if bh = Z Σ L R d f λ dµ (λ) where ff λ : λ Σ, kf λ k = 1g We say a function φ L (N) is bandlimited if its Plancherel transform is supported on a bounded measurable subset of Σ. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

20 Lattice in N Results and Examples Lemma If n has a rational structure and Γ = n d exp (ZZ k ) d exp (ZY k ) d exp (ZX k ) then there is a choice of Lie algebra for which the group generated by Γ is a lattice subgroup of N ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

21 Results and Examples Theorem ( Oussa) De ne E = fλ Σ : jdet B (λ)j 1g and bh K\E = Z K\E Assume that K is compact s.t L R d f λ jdet B (λ)j dλ. L (K,dλ). Then there exists φ H K\E such that bφ (λ) = n o e πihk,λi χ K (λ) is a Parseval frame for 1 p jdet B (λ)j (g λ f λ ) and the system G g λ, Z d B (λ) Z d is a Gabor Parseval frame in L R d for all λ K \ E, and L (Γ) φ is a Parseval frame in H K\E ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

22 Results and Examples Existence of Sampling Spaces with Sinc-type Functions. De ne W φ : H K\E! L (N) where W φ f (n) = hf,l (n) φi = fφ (n) and φ (x) = φ (x 1 ) Recall Γ = n d exp (ZZ k ) d exp (ZY k ) d exp (ZX k ). Theorem (Oussa, 01) The Hilbert space W φ (H K\E ) is a Γ-sampling space with sinc-type function φ φ ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 / 8

23 An Example Results and Examples 1 N = Γ = exp (RZ k ) exp (ZZ k ) exp (RY k ) exp (ZY k ) exp (RX k ) exp (ZX k ) 3 [X 1, Y 1 ] = Z 1, [X, Y ] = Z 1, [X, Y 1 ] = Z, [X 1, Y ] = Z. Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

24 Example (cont) Results and Examples 1 A matrix realization of the group 8 >< N = 6 4 >: 1 0 x 1 x y 1 y z x x 1 y y 1 z y y : E = (λ 1, λ, 0, 0, 0, 0) : λ 1 λ 1, and λ 1 λ 6= 0 z 1 z y 1 y x 1 x 9 3 >= 7 5 >; Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

25 Results and Examples 1 There is a restriction on how to pick K. The choice of K depends on the structure constants of n 1 De ne bφ (λ) = q jλ 1 λ j g λ f λ G g λ, Z λ1 λ Z λ λ 1 is a Parseval Gabor frame 3 W φ (H E\K ) is a sampling space with Sinc-type function φ φ Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

26 An Example (Part I) Results and Examples 1 N = Γ = exp (RZ k ) exp (ZZ k ) 3 3 exp (RY k ) exp (ZY k ) 3 3 exp (RX k ) exp (ZX k ) [X 1, Y 1 ] = Z 1, [X, Y 1 ] = Z, [X 3, Y 1 ] = Z [X 1, Y ] = Z, [X, Y ] = Z, [X 3, Y ] = Z 1 [X 1, Y 3 ] = Z, [X, Y 3 ] = Z 1, [X 3, Y 3 ] = Z. 3 (λ1, λ E =, 0, 0, 0, 0) : λ λ 1λ λ 3 1, λ λ 1λ λ 3 6= 0 Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

27 Results and Examples Acceptable Bandlimitation Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8

28 Results and Examples A Necessary Condition for Interpolation. Theorem (Oussa, 01) Let H I be a bandlimited multiplicity-free space such that bh I = Z I L R d f λ jdet B (λ)j dλ. If H I is a sampling space with the interpolation property then 1 there exists φ H I such that L (Γ) φ is a Parseval frame R jdet B (λ)j dλ = 1. I ignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March / 8