Uploaded January 2022 | Updated September 2026, 1 day ago
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
(JSTOR) jstor.org/stable/2027808?seq=1#metadata_info_tab_contents
(SIAM) epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread
//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.
//Watch Next
The Schwartz Space (Full Lecture) youtu.be/StzEO8B1pFY
Undergraduate Intro to the Laplace Transform youtu.be/mJSqt7nchRQ
Introduction to Control Theory youtu.be/0v4WFmOm764
Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE youtu.be/_I790SqNDjs
//Music Provided by Epidemic Sound
Vegas - Onda Norte
Late Nights - Daxtan
Twist - Cushy
We Come Alive - Waycap
Opposite - Jones Meadows
Video Call from Las Angeles - Trevor Kowalski
From the Cliffside - Tape Machines
Can't Wait for You - Deanz
Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:
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//Books
Advanced Mathematical Analysis - Beals amzn.to/3fn2jdP
From Vector Spaces to Function Spaces - Yutaka Yamamoto amzn.to/3sU2CVj
Feedback Control Theory - Doyle, Francis, Tannenbaum amzn.to/3HFvuov
Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy amzn.to/3EOJU3K
Nonlinear Systems - Khalil amzn.to/32Zyk8E
Modern Control Systems - Dorf and Bishop amzn.to/331JQAm
Fundamentals of Differential Equations - Nagle, Saff, and Snider amzn.to/3qeGD9C
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: amzn.to/3D3upUu
DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
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
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
(JSTOR) jstor.org/stable/2027808?seq=1#metadata_info_tab_contents
(SIAM) epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread
//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.
//Watch Next
The Schwartz Space (Full Lecture) youtu.be/StzEO8B1pFY
Undergraduate Intro to the Laplace Transform youtu.be/mJSqt7nchRQ
Introduction to Control Theory youtu.be/0v4WFmOm764
Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE youtu.be/_I790SqNDjs
//Music Provided by Epidemic Sound
Vegas - Onda Norte
Late Nights - Daxtan
Twist - Cushy
We Come Alive - Waycap
Opposite - Jones Meadows
Video Call from Las Angeles - Trevor Kowalski
From the Cliffside - Tape Machines
Can't Wait for You - Deanz
Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:
epidemicsound.com/referral/644nao
//Books
Advanced Mathematical Analysis - Beals amzn.to/3fn2jdP
From Vector Spaces to Function Spaces - Yutaka Yamamoto amzn.to/3sU2CVj
Feedback Control Theory - Doyle, Francis, Tannenbaum amzn.to/3HFvuov
Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy amzn.to/3EOJU3K
Nonlinear Systems - Khalil amzn.to/32Zyk8E
Modern Control Systems - Dorf and Bishop amzn.to/331JQAm
Fundamentals of Differential Equations - Nagle, Saff, and Snider amzn.to/3qeGD9C
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: amzn.to/3D3upUu
DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
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
