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 013 1 / 8
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
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 013 3 / 8
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 8 3 3 9 < 1 x z z = H = 4 0 1 y 5 : 4 y 5 R 3 : ; 0 0 1 x 8 3 3 9 < 1 m k k = Γ = 4 0 1 l 5 : 4 l 5 Z 3 : ; 0 0 1 m The interpolation property was a surprising fact!!! Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 4 / 8
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 013 5 / 8
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), 179-03 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, 64-633 (1989). 4 H. Führ, K. Gröchenig, Sampling Theorems on Locally Compact Groups from Oscillation Estimates, Math. Z. 55 (007), 177 194. 5 I. Pesenson, Sampling of Paley-Wiener functions on strati ed groups, J. Fourier Anal. Appl. 4 (1998), 71-81. 6 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 013 6 / 8
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 013 7 / 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. 0 31 [X 1, Y 1 ] [X 1, Y d ] B6 det @ 4.... 7C. 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 013 8 / 8
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 013 9 / 8
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 0 0 1 3 5 : 4 z y x 3 5 R 3 9 = ; Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 10 / 8
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. 8 3 1 0 x 1 x y 1 y z 1 0 1 x x 1 y y 1 z >< N 0 0 1 0 0 0 y 1 = 0 0 0 1 0 0 y : 6 0 0 0 0 1 0 0 6 7 4 4 0 0 0 0 0 1 0 5 >: 0 0 0 0 0 0 1 z 1 z y 1 y x 1 x 3 9 >= R 6. 7 5 >; Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 11 / 8
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 013 1 / 8
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 013 13 / 8
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 013 14 / 8
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 013 15 / 8
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) = 4.... 7. 5 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 013 16 / 8
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 1 + 1 λ dλ1 dλ Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 17 / 8
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 013 18 / 8
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 013 19 / 8
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 013 0 / 8
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 013 1 / 8
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
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 013 3 / 8
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 1 0 1 x x 1 y y 1 z 0 0 1 0 0 0 y 1 0 0 0 1 0 0 y 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 3 : 6 7 4 5 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 013 4 / 8
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 013 5 / 8
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 3 1 + 3λ 1λ λ 3 1, λ 3 1 + 3λ 1λ λ 3 6= 0 Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 6 / 8
Results and Examples Acceptable Bandlimitation Vignon Oussa (Bridgewater State University)Sampling and Interpolation on Some Nilpotent Lie Groups March 013 7 / 8
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 013 8 / 8