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)