A simple proof of the existence of sampling spaces with the interpolation property on the Heisenberg group

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1 A simple proof of the existence of sampling spaces with the interpolation property on the Heisenberg group Vignon Oussa Abstract A surprisingly short geometric proof of the existence of sampling spaces with the interpolation property on the Heisenberg Lie group is given. This result was originally proved by of B. Currey and A. Mayeli [3]. In the loving memory of my father Dr. Germain Oussa. Let N be a locally compact group, and let Γ be a discrete subset of N. Let H be a leftinvariant closed subspace of L 2 N consisting of continuous functions. We call H a sampling space Section 2.6 [5] with respect to Γ if the restriction mapping R Γ : H l 2 Γ, R Γ f = f γ γ Γ is an isometry, and there exists a vector s H such that for any vector f H, f x = γ Γ f γ s γ 1 x with convergence in the norm of H. If R Γ is onto, then R Γ is a unitary map and we say that the sampling space H has the interpolation property. For example the vector space of square-integrable continuous functions on the real line whose Fourier transforms are supported on the interval [ 1, 1 2 2] is a sampling space with interpolation property with respect to the lattice subgroup Z. This example is provided by the well-known Whittaker, Shannon, Kotel nikov Theorem see xample 2.52 [5]. As far as I know, the first example of a sampling space with interpolation property on a non-commutative nilpotent Lie group using the Plancherel transform was defined over the three-dimensional Heisenberg Lie group. This remarkable example is due to Currey and Mayeli [3]. In [6], I gave sufficient conditions for the existence of sampling spaces with the interpolation property on a class of non-commutative nilpotent Lie groups. The work presented in [6, 3] suggests that in general the investigation of sampling spaces with the interpolation property over non-commutative groups is by no mean an easy task. The main objective of the present work is to offer a surprisingly simpler and shorter proof than the one given in [3] of the fact that there exist sampling subspaces defined over the Heisenberg group which also enjoy the interpolation property with respect to a discrete uniform subgroup. I would like to point out that the proof given here is not a substitute for the work of Currey and Mayeli in [3]. In fact, the authors of [3] prove more than the mere existence of a sampling space with the interpolation property over the Heisenberg group. They also gave an explicit construction of such a space. 1

2 Let n be a three-dimensional nilpotent Lie algebra which is spanned by X, Y, Z with nontrivial Lie brackets [X, Y ] = Z. Put N = exp RZ exp RY exp RX. Clearly, N is a connected and simply connected nilpotent Lie group with Lie algebra n. Moreover, it often convenient to realize N as a matrix group via the isomorphism 1 x z exp zz exp yy exp xx 0 1 y As is well-known see Section 7.6, [4] the unitary irreducible representations of N are up the a null-set with respect to the Lebesgue measure on the real line parametrized by the punctured real line. More precisely, for every λ R there is a corresponding unitary irreducible representation π λ which is realized as acting in L 2 R as follows: π λ exp zz exp yy exp xx f t = e 2πiλz e 2πiλyt f t x. Let P be the Plancherel transform which is defined on L 2 N. Then P is a unitary operator P : L 2 N k=1 L 2 R λ dλ R yielding a decomposition of the left regular representation into a direct integral of representations of the Heisenberg group. More precisely, the Plancherel transform P defined on L 2 N intertwines the left regular representation L N of N with R k=1π λ λ dλ. This decomposition is the central decomposition of the left regular representation of the Heisenberg group see Page 234 [4] or Theorem 3.48 [5]. Characteristics of left-invariant subspaces of L 2 N are easily derived from Theorem 3.48, [5]. Indeed, let H be a Hilbert subspace of L 2 N which is left-invariant. That is, H is stable under the action of the left regular representation. Up to a set of Lebesgue measure zero, there exist a unique measurable set R and a multiplicity function n : N { } defined on such that P H = nλ k=1 L2 R λ dλ and the restriction of L N to the Hilbert space H is unitarily equivalent to nλ k=1 π λ dλ. We remark that since the measure λ dλ is absolutely continuous with respect to the Lebesgue dλ defined on the real line, the decomposition of the left regular representation of the Heisenberg group provided above makes sense. Now, let Γ = exp ZZ exp ZY exp ZX be a discrete uniform subgroup of N. Put Γ 1 = exp ZZ exp ZY. Clearly Γ 1 is a maximal normal abelian subgroup of Γ. 2

3 Lemma 1 Let λ 0. Then π λ Γ R Z χλ,tλ dt and χ λ,tλ is a unitary character of the abelian group Γ 1 for each λ, t R R Z. A proof of the lemma above can be found in Theorem 1.3, [1]. For measurable sets and F contained in R, we define S,F = {λ, tλ : λ, t F }. We observe that if F R is a measurable fundamental domain for R, the direct integral Z π λ Γ dλ is unitarily equivalent to F χλ,tλ dtdλ S,F χ s ds. Lemma 2 Let L Γ be the left regular representation of Γ. Then L Γ χ s ds and χ s is a unitary character of Γ 1 for each s R2. Proof. Let e be the identity element in Γ. First, we observe that L Γ Ind Γ {e} 1. Inducing in stages, we obtain L Γ Ind Γ {e} 1 Ind Γ 1 {e} 1. Next, Ind Γ 1 {e} 1 is the left regular representation of Γ 1 which is an abelian discrete group. Therefore, L Γ χ s ds χ s ds. This completes the proof. Define β : R R R R such that β λ, t = λ, tλ. It is easy to check that β is a diffeomorphism. Lemma 3 There exists a subrepresentation of the left regular representation of N whose restriction to Γ is unitarily equivalent to the left regular representation of Γ. Proof. Put = [ 1, 1] R and F = [0, 1. First, we claim that the image of the set F via the map β is up to a null set a fundamental domain for R2 which is contained in. First, {λ, tλ : λ, t F } is equal to {λ, tλ : λ, t [ 1, 0 F } {λ, tλ : λ, t 0, 1] F }. 3

4 Put A = {λ, tλ : λ, t [ 1, 0 F } and B = {λ, tλ : λ, t 0, 1] F }. A B Then A + 1, 1 B is up to a null set equal to [0, 1 2 which is clearly a -tiling set in. A + 1, 1 B Therefore, π λ Γ dλ [0,1 2 χ s ds L Γ. Clearly, π λdλ is equivalent to a subrepresentation of the left regular representation of the Heisenberg group. Next, the restriction of this direct integral representation to the discrete group Γ is equivalent to the representations described above. This completes the proof of the lemma. Finally, appealing to Lemma 3 and to Proposition 2.61, [5] which states that there exists a sampling space with interpolation property with respect to Γ if and only if there exists a representation of N whose restriction to Γ is unitarily equivalent to the left regular representation of Γ, the result below is immediate. Theorem 4 Let N be the Heisenberg group. There exists a Hilbert subspace of L 2 N which is a sampling space with respect to Γ with the interpolation property. 4

5 References [1] M. Bekka, P. Driutti, Restrictions of irreducible unitary representations of nilpotent Lie groups to lattices. J. Funct. Anal , no. 2, [2] L. Corwin, F. Greenleaf, Representations of Nilpotent Lie Groups and their Applications. Part I. Basic Theory and xamples, Cambridge Studies in Advanced Mathematics, 18. Cambridge University Press, Cambridge, 1990 [3] B. Currey, A. Mayeli, A Density Condition for Interpolation on the Heisenberg Group, Rocky Mountain J. Math. Volume 42, Number , [4] G. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995 [5] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer Lecture Notes in Math. 1863, [6] V. Oussa, Sampling and interpolation on some nilpotent Lie groups, Forum Mathematicum. ISSN Online , ISSN Print , DOI: /forum , September