Tauberian theorems for integral transforms

Remarks

This is the Hardy–Littlewood Tauberian theorem for the Laplace transform; the condition $f \ge 0$ is the Tauberian condition. For $c > 0$ it is the case $\rho = 1$ of Karamata's Tauberian theorem, applied to $U(x) = \int_0^x f(t)\, dt$. The proof follows Karamata. One shows $s\int_0^\infty f(t) e^{-st} P(e^{-st})\, dt \to c \int_0^1 P(x)\, dx$ for polynomials $P$. Then one approximates an indicator function by polynomials from above and below, using $f \ge 0$. The Tauberian condition cannot be dropped. Let $a_n = 1 + (-1)^n n$ and let $f = a_n$ on $[n, n + 1)$; then $sF(s) = (1 - x)\sum_n a_n x^n$ with $x = e^{-s}$, which tends to $1$ as $s \to 0^+$, as in the Hardy–Littlewood Tauberian theorem for series. But $\frac{1}{N} \int_0^N f = \frac{1}{N} \sum_{k < N} a_k$ is close to $3/2$ for odd $N$ and to $1/2$ for even $N$, so it has no limit. Converse direction: the Abelian theorem gives $sF(s) \to c$ from the convergence of the averages, with no sign condition (see Abelian theorems for integral transforms).

Going deeper

For a student

Tauberian theorems are partial converses of Abelian theorems, and each needs a side condition such as positivity, monotonicity or a growth bound. Karamata's theorem extends the statement to regularly varying growth, with a factor $\Gamma(1 + \rho)$. Wiener's Tauberian theorem treats general convolution kernels. The Wiener–Ikehara theorem, a Tauberian theorem for the Laplace transform with a condition on the boundary line, gives a short proof of the prime number theorem. In probability, this theorem relates the behaviour of a Laplace transform near $0$ to the tail of a distribution.