Dirac Delta Construction

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Dirac Delta Construction

We construct the Dirac Delta functional, by starting with a super smooth function of compact support and then stretching and compressing it, and finally taking the limit. Once you watch this video, you'll see that it's as easy as baking bread! It's a must-see for any differential equations and analysis students, enjoy!

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Distributions, like Dirac δ, are defined as linear functionals over bump functions.
Often the action of a distribution u on a bump function φ is denoted <u, φ>.
A locally integrable function f induces a distribution by <f, φ> = ∫ f(x) φ(x) dx, where the integral is over all of space.
Dirac δ is defined by <δ, φ> = φ(0) for every bump function φ.
The limit of a sequence of distributions is simply defined by <lim uₖ, φ> = lim <uₖ, φ> for every bump function φ.
What Dr Peyam has constructed is a sequence of bump functions converging to δ as a sequence of distributions.

mdperpe

Another intersection between math and gastronomy is the oily maccaroni constant 😶

alipourzand

Shout out my boy Dirac “The Rock” Johnson!

reportedstolen

I've always found this to be a pointed subject.

douglasstrother

Interesting that we don't have to explicitly define the bump function for it to converge in the limit process to the Dirac delta. 🤔 Nice video!

ZilchZilch

The last point you said the integral is 1 since all the integrals were 1 and this is the limit of that. But in general the integral of a limit is not equal to the limit of the integrals.

Happy_Abe

Great! Dr.!
Question: how do you mathematically capture different dynamics in time series? 😀

marcelob.

In what sence do we take that limit? And why the integral is not just 0, since the delta function is 0 a.s.?

ДенисЛогвинов-зе

Dr. Peyam, I am a little bit disappointed: As you know, the Dirac Delta distribution is NOT a function. It is also NOT the pointwise limit of a sequence of functions. I hoped this video would define what Delta really is… But anyway, keep up the good work! I love your videos (except this one)

MaximilianWorner

Gastromath. It's what's for dinner.

mikecaetano

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