Uploaded November 2021 | Updated September 2026, 14 hours ago
Here we break down what exactly a delta function is, and how it can sometimes be represented by an actual function. We build up from notions in differential equations up through functional analysis and measure theory to ultimately culminate with a discussion of reproducing kernels and the Shannon Nyquist sampling theorem. These ideas are core to sampling theory.
The delta function is also known as the Dirac delta function, and has appeared in Differential Equations, Partial Differential Equations, Quantum Mechanics, Signal Processing, and Machine Learning. Come find out exactly what it is!
//Watch Next
Tempered Distributions (Full Lecture) youtu.be/bsbTF4VC6qk
The Paley Wiener Space youtu.be/jcY5Tq1JsVc
Kernel Based Dynamic Mode Decompositions youtu.be/BoIQvGoCAmA
Continuous time Dynamic Mode Decompositions youtu.be/xfZG0mhKd0s
Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE youtu.be/_I790SqNDjs
//Music Provided by Epidemic Sound
Dancing Ted - Moorland Songs
Fortune of Hearts - Hallman
Something Different - CLNGR
Opposite - Jones Meadow
Summer Night - Wai
//Books
Sampling Theory in Fourier and Signal Analysis: Volume 1: Foundations: amzn.to/3IcO1tt
Analysis on Fock Spaces: amzn.to/3D14v3X
Fundamentals of Differential Equations and Boundary Value Problems amzn.to/32wnA1e
//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
//MATLAB Code via BitBucket
bitbucket.org/joelrosenfeld/delta-function-video-code/src/master
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 Start
0:14 Impulsive Discussions
1:33 What the heck is a distribution?
2:05 Introducing Function Spaces
3:00 Tempered Distributions and Delta Sequences
4:00 Riesz and Measures
5:04 The Delta function *is* a function!
6:09 Reproducing Kernel Hilbert Spaces
6:37 Shannon Nyquist Theorem and G. H. Hardy
7:28 Machine Learning and Kernels
Here we break down what exactly a delta function is, and how it can sometimes be represented by an actual function. We build up from notions in differential equations up through functional analysis and measure theory to ultimately culminate with a discussion of reproducing kernels and the Shannon Nyquist sampling theorem. These ideas are core to sampling theory.
The delta function is also known as the Dirac delta function, and has appeared in Differential Equations, Partial Differential Equations, Quantum Mechanics, Signal Processing, and Machine Learning. Come find out exactly what it is!
//Watch Next
Tempered Distributions (Full Lecture) youtu.be/bsbTF4VC6qk
The Paley Wiener Space youtu.be/jcY5Tq1JsVc
Kernel Based Dynamic Mode Decompositions youtu.be/BoIQvGoCAmA
Continuous time Dynamic Mode Decompositions youtu.be/xfZG0mhKd0s
Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE youtu.be/_I790SqNDjs
//Music Provided by Epidemic Sound
Dancing Ted - Moorland Songs
Fortune of Hearts - Hallman
Something Different - CLNGR
Opposite - Jones Meadow
Summer Night - Wai
//Books
Sampling Theory in Fourier and Signal Analysis: Volume 1: Foundations: amzn.to/3IcO1tt
Analysis on Fock Spaces: amzn.to/3D14v3X
Fundamentals of Differential Equations and Boundary Value Problems amzn.to/32wnA1e
//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
//MATLAB Code via BitBucket
bitbucket.org/joelrosenfeld/delta-function-video-code/src/master
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 Start
0:14 Impulsive Discussions
1:33 What the heck is a distribution?
2:05 Introducing Function Spaces
3:00 Tempered Distributions and Delta Sequences
4:00 Riesz and Measures
5:04 The Delta function *is* a function!
6:09 Reproducing Kernel Hilbert Spaces
6:37 Shannon Nyquist Theorem and G. H. Hardy
7:28 Machine Learning and Kernels
