Talk:Laplace Transform of Integral

The theorem's proof assumes $f(t)$ continuous, which is missing from the hypothesis of the theorem. The assumption on $f(t)$ in classical Laplace theory is not continuity but piecewise continuity. Phrase "whenever the Laplace of $f$ exists" does not include the hypothesis "$f$ continuous." --Gbgustafson (talk) 15:29, 28 February 2022 (UTC)


 * So how would you fix the proof so that it does not assume continuity?
 * The exposition and proof are as given in Spiegel, but he is notorious for being lax, and there are often mistakes in his publications. --prime mover (talk) 20:46, 28 February 2022 (UTC)

March 1, 2022 reply to prime mover.

This result should be available:

Lemma. Let $f$ be piecewise continuous with $|f(t)| < Me^{\alpha t}$ for $t\ge 0$ and some fixed constants $\alpha$. $M$. Then there exists a sequence $\{f_n\}_{n=1}\infty$ of functions continuous on $t\ge 0$ with $|f_n(t)| < Me^{\alpha t}$ such that $\lim_{t\to\infty} f_n(t)=f(t)$ except at discontinuities of $f$.

The proof of the Lemma uses a construction at each discontinuity $t^*$ of $f$ to define $f_n(t)$ near $t=t^*$. The construction idea is born from the example $$g_\epsilon (t)=\left\{ \begin{array}{ll} 1 & 0 \le t \le 1\\ 1-(t-1)/\epsilon & 1 \le t \le 1+\epsilon,\\ 0 & \mbox{elsewhere} \end{array}\right.$$ Continuous function $g_\epsilon$ converges as $\epsilon \to 0$ a.e. to a pulse of one on $0\le t \le 1$. Similar constructions are used in Littlewood's Three Principles in measure theory.

The Spiegel proof on proofwiki is correct with the added assumption of continuity and that $f(t)$ is of exponential order. I don't know if exponential order can be removed from the hypothesis on $f$. Spiegel's statement does not mention it, but instead requires $f$ to have a Laplace transform, which could implicitly mean the same thing.

To extend the proofwiki proof to the expected hypothesis $f$ is piecewise continuous of exponential order would use the Lemma and then apply the Dominated Convergence Theorem. --Gbgustafson (talk) 13:09, 1 March 2022 (UTC)


 * Feel free to post it up. --prime mover (talk) 17:13, 1 March 2022 (UTC)