A uniqueness theorem for the exponential series of Herglotz

Extract

If {λ;_n}, {b_n} are sequences of complex numbers, and we consider the series ∑b_n exp (−λ_nx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing

(if λ_n = 0 the corresponding term is ½b_nx²) where the series is supposed is to be uniformly convergent in (0, 1), we have

for 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φ_v(x)} of integrable functions with the property that

then, on multiplying (1) by φ_v(x) and integrating over (0,1), we obtain a formula for b_v in terms of F(x). On integrating by parts twice, b_v will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑b_n exp (−λ_nx).