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) υ δα j , and the aj,υ are complex constants. In the case where the frequencies α j are purely imaginary, i.e., α j = iλj, λ j ∈ ℝthen 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 δλj the Dirac mass at the point λ j ∈ℝ (acting on C∞ functions in ℝ and $$f(z) = $$