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

(if λn = 0 the corresponding term is ½bnx2) 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 {φv(x)} of integrable functions with the property that

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