Laplace Transform of Derivatives of Distributions

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We talk about Distributional Derivatives and impact on the Laplace transform. Interestingly, this is a bit different than what happens for Laplace transforms of functions.

//Additional Reading
Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations

//Exercises
- Show that differentiation is a continuous operation on L.
- Show that 1/n phi(t/n) e^{-st} goes to zero in L (i.e. with respect to each seminorm) as n goes to infinity for every s in C.

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0:00 Introduction
0:48 About Laurent Schwartz
1:26 Derivatives of Regular Distributions
2:56 Autobiography of Laurent Schwartz
3:45 Derivatives of Distributions
5:11 Laplace Transforms of Derivatives
6:58 Too Simple yet Too Complicated
7:30 The Transfer Function

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interesting text on the autobiography of L. Schwartz

edal

huge fan! great video on the topic on the rigor side of Laplace transform, and of course, on the bio of Schwartz.
It seems that most control textbooks are choosing the 0- initial condition, and all math textbooks are (implicitly) choosing the 0+ convention (say, book by Beal as mentioned by you, or books on Fourier analysis by Folland), and of course, this 0- or 0+ is meaningless for the definition of an integral. Is there some secret agreement in the control society? Many thanks!

chaoshengzhe

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