Fourier transforms in the “classical sense”, Schur spaces and a new formula for the Fourier transforms of slowly increasing, O(p,q)O(p,q)-invariant functions

The calculation of Fourier transforms FTFT of integrable distributions T∈DL1′ gives rise to the question of its “pointwise” calculation, i.e., the question if the relation limj→∞〈φ,Tj〉=〈φ,T〉limj→∞〈φ,Tj〉=〈φ,T〉 for each φ∈DL∞φ∈DL∞, is sufficient to prove the convergence of the sequence (Tj)j∈N,Tj∈DL1′, to the limit TT. Since DL1′ is a Schur space, pointwise convergence suffices (for sequences). The fact that DL1′ is a Schur space can be derived from the isomorphism DL1′≃ℓ1⊗ˆs′. A generalization of this reasoning is given in Chapter 2.In Chapter 3, a representation of the Fourier transform F(f([x,x]))F(f([x,x])) is given, [,][,] denoting the quadratic form x12+⋯+xp2−xp+12−⋯−xn2 and ff a slowly increasing C∞C∞–function.The representation is a vector-valued integral 〈1σ,Ff(σ)K(σ,ξ)〉〈1σ,Ff(σ)K(σ,ξ)〉 with Ff(σ)K(σ,ξ)∈DL1,σ′(OC,ξ′) and with an explicitly given kernel K(σ,ξ)K(σ,ξ).