# Fourier Transforms in the Complex Domain by Raymond E. A. C. Paley and Norbert Wiener

By Raymond E. A. C. Paley and Norbert Wiener

Via Fourier-Mellin transforms as a device in research, the authors have been in a position to assault such assorted analytic questions as these of quasi-analytic capabilities, Mercer's theorem on summability, Milne's indispensable equation of radiative equilibrium, the theorems of Munz and Szasz about the closure of units of powers of an issue, Titchmarsh's thought of whole capabilities of semi-exponential style with genuine unfavorable zeros, trigonometric interpolation and advancements in polynomials of the shape $\sum^N_1A_ne^{i\lambda_nx}$, lacunary sequence, generalized harmonic research within the advanced area, the zeros of random features, etc.

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Xl−1 , xl , x1 , . . , xl−1 , −xl ) . It follows that ν(αi ) = αi , i = 1, . . , l − 2, ν(αl−1 ) = αl . 3) g = E6 . We refer to [1], Ch. VI, §4, for a proof of the relation −e ∈ / W which implies ν = id. (ii) Any Weyl chamber for g has the form D = D1 × . . × Ds , where Di is a Weyl chamber for gi , and w0 = (w0 )1 × . . × (w0 )s , where (w0 )i is the element of the Weyl group of gi mapping Di onto −Di . Thus, ν = ν1 × . . × νs , which implies our assertion. (iii) For any s ∈ Aut Π, we have ϕ˜ = sˆϕˆ s−1 ∈ Int g, where ϕ ∈ N satisﬁes ˜ = ϕ|t(R) ˜ = (s )−1 w0 s ∈ W .

26 §3. Real forms and involutive automorphisms Finally, let us consider another compact real structure τ1 in g and the corresponding bijection between conjugacy classes of antiinvolutions and involutions. We claim that it coincides with the original bijection determined by τ . In fact, by Corollary of Proposition 6, we have τ1 = ατ α−1 , where α ∈ Int g. Let σ → σ → θ = στ , where σ is conjugate to σ and commutes with τ , be the mapping giving the original bijection. Then, clearly, the new bijection is determined by the mapping σ → ασ α−1 → θ1 = (ασ α−1 )τ1 = αθα−1 .

We are now going to extend this semidirect decomposition to the entire group Aut g. Theorem 1. Let g be a complex semisimple Lie algebra. Then Aut g = Int g Ψ(Aut Π) , (3) and the corresponding projection Aut g → Aut Π coincides with Φ on Aut(g, t, Π). Proof. First we prove that Aut g = (Int g) Aut(g, t, Π). 1), there exists ϕ ∈ Int g such that ϕθ(t) = t. 12), ϕθ permutes the Weyl chambers in t(R). Let D be the Weyl chamber corresponding to Π. 15), we have w(ϕθ(D)) = D for an element w ∈ W . 13), w = ψ|t(R) for an automorphism ψ ∈ Int g.