Poisson summation formulas over homogeneous spaces of compact groups

Abstract This paper presents the abstract notion of Poisson summation formulas for homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G , and μ be the normalized G -invariant measure over the left coset space G / H associated to the Weil’s formula. We prove that the abstract Fourier transform over G / H satisﬁes a generalized version of the Poisson summation formula.


Introduction
The classical Poisson summation formula is a spectacular result with many important applications in physics and engineering [1][2][3]. Various connections of the Poisson formula with group representations have been discovered for a some time already. In the most natural domain of commutative groups this can be already found in [4,14]. Inspiring connections with the Heisenberg group have been discussed in [1].
The abstract aspect of harmonic analysis over homogeneous spaces of compact non-Abelian groups, more precisely, left (resp. right) coset spaces of non-normal sub-B Arash Ghaani Farashahi arash.ghaani.farashahi@univie.ac.at; ghaanifarashahi@hotmail.com (2.1) If π(x) is represented by the matrix (π i j (x)) ∈ C d π ×d π . Then f (π ) ∈ C d π ×d π is the matrix with entries given by f (π ) i j = d −1 π c π ji ( f ) which satisfies where c π i, j ( f ) = d π f, π i j L 2 (G) . Then as a consequence of Peter-Weyl Theorem we get and Let μ be a Radon measure on G/H and x ∈ G. The translation μ x of μ is defined by The homogeneous space G/H has a normalized G-invariant measure μ, which satisfies the following Weil's formula [4,21] G/H and also the following norm-decreasing formula , for all f ∈ L 1 (G).

Abstract harmonic analysis over homogeneous spaces of compact groups
Throughout this paper we assume that G is a compact group with the probability Haar measure dx, H is a closed subgroup of G with the probability Haar measure dh, and μ is the normalized G-invariant measure on the compact homogeneous space G/H associated to the Weil's formula (2.5) with respect to the probability measures of G and H . From now on, we may say μ is the normalized G-invariant measure over the compact homogeneous space G/H , at times. This section is devoted to present a list of useful results concerning the classical study for abstract harmonic analysis of Banach function spaces over homogeneous spaces of compact groups, for proofs we refer the readers to see [5,7,8].
The following proposition shows that the linear map T H : C(G) → C(G/H ) is uniformly continuous. Next theorem proves that the linear map T H is norm-decreasing in other L p -spaces, when p > 1.

Theorem 3.2 Let H be a closed subgroup of a compact group G, μ be the normalized G-invariant measure on G/H , and p ≥ 1. The linear map T H : C(G) → C(G/H ) has an extension to a bounded linear map from L p (G) onto L p (G/H, μ).
As an immediate consequence of Theorem 3.2 we deduce the following corollary.

Proposition 3.4 Let H be a closed subgroup of a compact group G and μ be the normalized G-invariant measure on G/H . Then T H : L
Remark 3.7 Invoking Corollary 3.6, one can regard the Hilbert space is an orthogonal projection onto L 2 (G/H, μ).

Abstract duality over homogeneous spaces of compact groups
In this section, we present the abstract notion of dual homogeneous spaces associated to homogeneous spaces of compact groups [8].
Then by definition we have If G is Abelian, each closed subgroup H of G is normal and the compact group G/H is Abelian and so G/H is precisely the set of all characters (one dimensional irreducible representations) of G which are constant on H , that is precisely H ⊥ . If G is a non-Abelian group and H is a closed normal subgroup of G, then the dual space G/H which is the set of all unitary equivalence classes of unitary representations of the quotient group G/H , has meaning and it is well-defined. Indeed, G/H is a non-Abelian group. In this case, the map : [4,19]. Thus if H is normal, H ⊥ coincides with the classic definitions of the dual space either when G is Abelian or non-Abelian. For a closed subgroup H of G and a continuous unitary representation (π, H π ) of G, define where the operator valued integral (4.3) is considered in the weak sense.
In other words, The function h → π(h)ζ, ξ is bounded and continuous on H . Since H is compact, the right integral is the ordinary integral of a function in L 1 (H ). Therefore, T π H is a bounded linear operator on H π with T π H ≤ 1.
Evidently, any closed subgroup H of G satisfies Let (π, H π ) be a continuous unitary representation of G such that T π H = 0. Then the functions π H ζ,ξ : for ξ, ζ ∈ H π , are called H -matrix elements of (π, H π ). For x H ∈ G/H and ζ, ξ ∈ H π , we have Also, we can write Invoking definition of the linear map T H and T π H , we have which implies T H (π ζ,ξ ) = π H ζ,ξ . (4.9) Next we show that the reverse inclusion of (4.6) holds, if H is a normal subgroup of G.

Theorem 4.3 Let H be a closed normal subgroup of a compact group G. Then
Proof Let H be a closed normal subgroup of a compact group G. It is sufficient to Therefore, π(x)T π H = T π H π(x) for x ∈ G, which implies T π H ∈ C(π ). Irreducibility of π guarantees that T π H = α I for some non-zero α ∈ C with |α| ≤ 1. Thus, for t ∈ H , we can write

Abstract Poisson summation formulas over homogeneous spaces of compact groups
Throughout this section, we present the abstract notion of operator-valued Fourier transforms and Poisson summation formulas over homogeneous spaces of compact groups. It is still assumed that H is a closed subgroup of a compact group G and μ is the normalized G-invariant measure on the compact homogeneous space G/H .

Let ϕ ∈ L 1 (G/H, μ) and [π ] ∈ G/H . The Fourier transform of ϕ at [π ] is defined as the operator
Thus, we deduce that the abstract Fourier transform defined by (5.1) coincides with the classical Fourier transform over the compact quotient group G/H if H is normal in G.
The operator-valued integral (5.1) is also considered in the weak sense. That is for all ζ, ξ ∈ H π . In other words, for [π ] ∈ G/H and ζ, ξ ∈ H π , we have Indeed, we can write Thus, for ζ, ξ ∈ H π , we get Therefore, ϕ(π) is a bounded linear operator on H π satisfying From now on, we may use ϕ(π) or F G/H (ϕ)(π ) at times. The following proposition gives us the connection of the Fourier transform over the homogeneous space G/H with the Fourier transform on the group G.

Proposition 5.2 Let H be a closed subgroup of a compact group G and μ be the normalized G-invariant measure on G/H . Let ϕ ∈ L 1 (G/H, μ) and [π ] ∈ G/H. Then
Proof Using (5.4) and also the Weil's formula, for ζ, ξ ∈ H π , we can write which implies (5.6).
The following proposition states generic properties of the Fourier transform (5.1).

Theorem 5.4 Let H be a closed subgroup of a compact group G and μ be the normalized G-invariant measure on G/H. Each ϕ ∈ L 2 (G/H, μ) satisfies the Plancherel formula
Proof See Theorem 4.6 of [8] The following corollary is a consequence of Theorem 5.4.

Corollary 5.5 Let H be a closed subgroup of a compact group G and μ be the normalized G-invariant measure on G/H . Then
(1) For ϕ, ψ ∈ L 2 (G/H, μ), we have Proof (1) Using the polarization identity and Theorem 5.4, we get (5.8).
Next theorem presents an explicit L 2 -inversion formula for the Fourier transform given in (5.1). Proof Let ϕ ∈ L 2 (G/H ) and x ∈ G. Then, ϕ q ∈ L 2 (G) and hence the series converges and also y → [π ]∈ G d π tr[ ϕ q (π )π(y)] defines a function in L 2 (G). Thus, we deduce that the series is converges, because Remark 5.14 It should be mentioned that in [13] author presented an extension for Poisson summation formulas over homogeneous spaces of compact groups in a different direction. In this framework, Corollary 5.13 coincides with Proposition 4.6 of [13].