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Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group

This work addresses an extension of Fourier-Stieltjes transform of a vector measure defined on compact groups to locally compact groups, by using a group representation induced by a representation of one of its compact subgroups.


Introduction
The Fourier transform is one of the most important and widely used computational tasks. It is nowadays an essential tool in many technological innovations.
A foundational tool commonly used to analyze the spectral representation of signals. Its applications include audio/video processing, radar and GPS systems, wireless communications, medical imaging and spectroscopy, the processing of seismic data, and many other tasks.
Among the Fourier transforms, we have the Fourier transform of integrable functions and that of measures.
During the last decades, countless studies on vector measures have been made [5], [9] (and references therein), and also [1] for some applications on compact groups.
The project tackled in this paper is a generalization of the work of Assiamoua [1,2] on the Fourier-Stieltjes transformation of a vector measure and that of an absolutely continuous integrable function with respect to a Haar measure. This is a continuation of recent work by Akakpo and al [4] in which they set up the foundations of the Fourier-Stieltjes transform of a vector measure and that of an integrable function of a locally compact group G.
The paper is organized as follows. In Section 1, we give some preliminaries useful in the sequel. In Section 2, we construct some spaces called S p spaces, very useful in our theory, and give some of their main properties, show that the Fourier transform define in [4] verify the main fundamental theorems in analysis.

Preliminaries and notations
In this section, for the clarity of the development, we briefly recall main useful known definitions and results, and set our notations.
We consider a locally compact space G, the Banach spaces A and F over the field K, (K = R or C), and denote by K(G, A) the vector space of all continuous functions f : G −→ A with a compact support, and by C(G, A) the space of continuous functions f : G −→ A. As a matter of notation simplification, we write K(G) instead of K(G, R) or K(G, C). For each subset K of G, denote by K K (G, A) the space of functions with support contained in K.
For every function f ∈ K(G, A), we define as usual The mapping f → f is a norm on each space K K (G, A), and defines the topology of uniform convergence on G over K(G).
On K(G, A), the topology of the compact convergence is the locally convex topology defined by the family of seminorms where K runs over the set of compact subsets of G.
for the topology of the compact convergence. A vector measure on G with respect to two spaces A and F , or an (A, F )-measure on G, is any linear mapping m : K(G, A) −→ F having the property that, for each compact set K ⊂ G, the restriction m to the subspace K K (G, A) is continuous for the topology of uniform convergence, i.e for each compact set, there exists a number a K > 0 such that The value m(f ) of m for a function f ∈ K(G, A) is called the integral of f with respect to m, also denoted by G f dm or G f (t)dm(t). A vector measure is said to be dominated if there exists a positive measure µ such that If m is dominated, then there exists a smallest positive measure |m|, called the modulus or the variation of m, that dominates it. A positive measure is said to be bounded if it is continous in the uniform norm topology of K(G). A vector measure is said to be bounded if it is dominated by a bounded positive measure. If m is bounded, then |m| is also bounded. Denoting by M 1 (G, A) the Banach algebra of bounded vector measures on G, the mapping m → m = G χ G d|m| is a norm on M 1 (G, A), where χ G denotes the characteristic function of G.
In the sequel, K will designate a compact subgroup of G, ν and λ left Haar measures respectively on K and G.

G-invariant measure.
Definition 1.5. Let µ be a Radon measure on G/K, the homogenous space of left K-cosets and g an element of G. Define µ g by µ g (E) = µ(gE) for Borel subsets E of G/K. The measure µ is called G−invariant measure if µ g = µ, for g ∈ G. See [3], [8] for more details.

Group representation.
Definition 1.7. A unitary representation of G is a homomorphism L from G into the group U (H) of the unitary linear operators on some nonzero Hilbert space H, which is continuous with respect to the strong operator topology satisfying: for g 1 , g 2 ∈ G, L (g1,g2) = L g1 L g2 and L e = Id H .
H is called the representation space of L, and its dimension is called the dimension or degree of L.
Consider now the subgroup K, Σ the coset space (called the dual object of K), σ ∈ Σ, L σ a representative of σ, H σ a representation space of L σ , and d σ its dimension.
Proposition 1.11. The mapping: is an inner product on H L σ 0 . Proposition 1.12. G acts on H L σ 0 by left translation, u −→ L t u, so we obtain a unitary representation of G with respect to this inner product on H L σ 0 . The inner product is preserved by left translations, since µ is invariant. Hence, if we denote by H L σ the Hilbert space completion of H L σ 0 , the translation operators L t extend to unitary operators on H L σ . Then the map t −→ L t u is continuous from G to H L σ for each u ∈ H L σ 0 , and then, since the operators L t are uniformly bounded, they are strongly continuous on H L σ . Hence they define a unitary representation of G, called the representation induced by L σ , denoted by U L σ : 1.5. Fourier-Stieltjes transform defined by induced representation on locally compact groups. [4] We have K compact, H σ is finite and separable; then H L σ is also separable. Each of these spaces admits a Hilbertian basis according to Gram-Schmidt process.
In this work, we suppose that K is chosen such that L σ and U L σ are both irreducible. Theorem 1.14. We have where In the case of a particular orthonormal basis, the orthogonality relation reduces to the following: In the sequel, (θ i ) ∞ i=1 will designate an orthonormal basis of Definition 1.16. Assume m ∈ M 1 (G, A). The Fourier-Stieltjes transform of an arbitrary measure m is defined as the family (m(σ)) σ∈Σ of sesquilinear mappings on H L σ × H L σ into A, given by the relation (1.12) Definition 1.17. For any f ∈ L 1 (G, A, λ), where λ denotes a Haar measure on G, the Fourier transform of f is a family (f (σ)) σ∈Σ of sesquilinear mappings of H L σ × H L σ into A, given by the relation is a vector space for addition and multiplication by a scalar of mappings.
Proposition 1.18. Let us consider Φ in S(Σ, A). There is the matrix (a σ ij ) 1≤i,j≤∞ , a σ ij ∈ A such that Corollary 1.19. For any f ∈ L 1 (G, A, λ), there is a matrix (a ij ) 1≤i,j≤∞ , a ij ∈ A such that ) is linear, injective, and continuous. (1.14) (1.15) 1.6. Tensor Products of Vector Spaces. [11] We recall that a mapping D from the cartesian product E × F of vector spaces into the vector space S is bilinear if it is linear in each variable, that is, D(α 1 x 1 + α 2 x 2 , y) = α 1 D(x 1 , y) + α 2 D(x 2 , y) and D(x, β 1 y 1 + β 2 y 2 ) = β 1 D(x, y 1 ) + β 2 D(x, y 2 ) for all x i , x ∈ E, y i , y ∈ F and all scalars α i , β i , i = 1, 2. We write B(E × F, S) for the vector space of bilinear mappings from the product E × F into S; when S is the scalar field we denote the corresponding space of bilinear forms simply by B(E × F ).
Definition 1.24. The tensor product, E ⊗ F , of the vector spaces E, F can be constructed as a space of linear functionals on B(E × F ), in the following way: for x ∈ E, y ∈ F , we denote by x ⊗ y the functional given by evaluation at the point (x, y). In other words, for each bilinear form D of E ×F . The tensor product E ⊗F is the subspace of the dual B(E ×F ) * spanned by these elements. Thus, a typical tensor in E ⊗ F has the form where n is a natural number, λ i ∈ K, x i ∈ E and y i ∈ F .
Remark 1.25. It is important to realize that the representation of u is not unique -in general, there will be many different ways to write a given tensor in the above form.
is a norm in E ⊗ F known as the projective norm.
We shall denote by E ⊗ π F tensor product E ⊗ F endowed with the projective norm, π. Unless the spaces E and F are finite dimensional, this space is not complete. We denote its completion by E⊗ π F . The Banach space E⊗ π F will be referred to as the projective tensor product of the Banach of E and F . Proposition 1.27. Let S and A be Banach spaces and µ any measure on S. L 1 (S, µ, K)⊗ π = L 1 (S, µ, A).
Proposition 1.28. Let E and F be Banach spaces. The mapping ε defined by We shall denote by E ⊗ ε F tensor product E ⊗ F endowed with the injective norm, π. Unless the spaces E and F are finite dimensional, this space is not complete. We denote its completion by E⊗ ε F . The Banach space E⊗ ε F will be referred to as the injective tensor product of the Banach of E and F .   Let Ω be a locally compact space, which is non-compact, let C K 0 (Ω) be the algebra of continuous K-valued functions on Ω that vanish at infinity, as above, equipped with the supremum norm and let V ⊂ C K 0 (Ω) with the following separation properties: (i) for any two points ω 1 , ω 2 ∈ Ω with ω 1 = ω 2 there exists f ∈ V such that f (ω 1 ) = f (ω 2 ) (ii) for any ω ∈ Ω there exists f ∈ V with f (ω) = 0.

Main results
2.1. S p spaces. In this section we define some particular spaces of Fourier transform and give some properties of them.

Proof
(i) Let us first show that for 1 ≤ p ≤ q < ∞, S p (Σ, A) ⊂ S q (Σ, A). For this, we show that The summation in the ̺ q expression can permit to retrieve the term d σ b q σ . Then, we and we have: Hence Φ q ≤ Φ p .
There exists also a finite set I σ Hence L(G) ⊗ A = S 00 (Σ, A).

Proof
We identify L(G, A) with L(G, C)⊗ ε A the injective tensor product of L(G, C) and A. According to theorem 1.30 L(G, C) is dense in C 0 (G, C) then L(G, A) is dense in C 0 (G, A) because C 0 (G, A) is isomorphic to C 0 (G, C)⊗ ε A as a subspace of C(G, A) = C(G, C)⊗ ε A according to proposition1.29.
Proof L(G, A) is dense in C 0 (G, A) for the topology of uniform convergence. Then L(G, A) is dense in K(G, A) with the same norm. K(G, A) is also dense in L 1 (G, A) for . 1 .
Let f be an element of L 1 (G, A), f is a limit of a sequence in L(G, A) thenf belongs to L(G, A) = Morever {σ ∈ Σ : f (σ) = 0} is countable because for every n ∈ N * the set {σ ∈ Σ: f (σ) > 1 n } is finite.

Plancherel theorem.
Proposition 2.7. Suppose that A is a Hilbert space then the mapping is a scalar product in S 2 (Σ, A).
Then the mapping is well defined. The sequel of the proof is trivial, it depends to the fact that , A is a scalar product in A.
The following theorem is the Plancherel theorem in this context.
Theorem 2.8. Let A be a Hilbert space then the mapping f −→f is an isometry of L 2 (G, A, λ) into S 2 (Σ, A) and then S 2 (Σ, A) is a Hilbert space.
Proof Since A is a Hilbert space then so is L 2 (G, A, λ). Let f be an element of L 2 (G, A, λ), put a σ ij =f (σ)(θ i , θ j )

Concluding remarks
The aspects approached in this paper are consequences of the work on the Fourier-Stieltjes transform on locally compact group recently elaborated in [4].
We have through this work given meaning to the theory developed in [4]. We have shown that the Fourier-Stieltjes transform defined in a locally compact group satisfies the main and classic theorems in analysis. We have among other established the theorems of Lebesgue and that of Plancherel then the identity of Parseval relating to our concept.
We intend in the future to study, by way of application, this theory in certain much more particular groups such as the linear groups GL(n, C) and some of its subgroups.