Uploaded January 2022 | Updated September 2026, 22 hours ago
The definition of the convolution of two distributions arises from that of regular distributions and in turn, the convolution of functions. We leverage this definition to give a definition of an antiderivative of a distribution, and we determine the related Laplace transform property of both of these operations.
//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
Late Nights - Daxten
bam bam - Yomoti
Tiny Things - Giant's Nest
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
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Blue Yeti Microphone: amzn.to/3I1y88N
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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!
The definition of the convolution of two distributions arises from that of regular distributions and in turn, the convolution of functions. We leverage this definition to give a definition of an antiderivative of a distribution, and we determine the related Laplace transform property of both of these operations.
//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
Late Nights - Daxten
bam bam - Yomoti
Tiny Things - Giant's Nest
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!
