Exponential Polynomials

An exponential polynomial is an entire function f of the form $$ f(z) = \sum\limits_{{1 \leqslant j \leqslant m}} {{{P}_{j}}} (z){{e}^{{ajz}}}, $$ where α _j ∈ ℂ, P _j ∈ ℂ[z]. We assume that the α _j are distinct and the polynomials P _j not zero. The P _j are called the coefficients of f and α _j the frequencies. (Sometimes the α _j are called the exponents especially in the Russian literature. In some contexts α _j = iλ _j ,λ _j ∈ℝ , and the λ _j are called the frequencies and τ _j = 2π/λ _j (when λ _j ≠ 0) the periods; clearly eiλjz periodic of period τ _j .) It is immediate that there is a unique analytic functional T whose Fourier—Borel transform F (T) coincides with f, i.e.,F(T) (z) = 〈T _ζ , ezζ〉 = f (z). Namely, T is representable as a distribution in ℂ of the form $$T = \sum\limits_{j,v} {{a_{j,v}}\delta _{\alpha j}^{(v)}} $$ where δα _j is the Dirac measure at the point α _j , δ _αj (υ) is a “holomorphic” derivative of order υ, δ _αj (υ) = (∂/∂z)υδα _j , and the a_j,υ are complex constants. In the case where the frequencies α _j are purely imaginary, i.e., α _j = iλj, λ _{j ∈ ℝ}then f is the Fourier transform of a distribution μ∈έ(ℝ). That is, we let $$\mu :\sum\limits_{j,v} {{a_{j,v}}{i^v}\frac{{{d^v}}}{{d{x^v}}}{\delta _{\lambda j}}} $$ with δ_λj the Dirac mass at the point λ _j ∈ℝ (acting on C∞ functions in ℝ and $$f(z) = < {\mu _x},{e^{ - ixz}} >$$