Distributions 6 | The Delta Distribution Is Not Regular

[@juliogodel]

at last! :) also waiting for the stochastic processes playlist. thank you!!!

[@TheArtemis3rd]

This is a very nice series. Please keep it going. Very grateful for this.

[@shlimsady]

Thank you, I’m already excited for part 7!

[@derickd6150]

Same! When will it come!?

[@antoniodeoliveiranginamaub2845]

I would like to learn more about singular distributions, pseudo function, cauchy principal value and Hadamard finite part integrals. Thanks

[@antoniodeoliveiranginamaub2845]

waiting for part seven, is very important, thanks

[@sarojsi890]

I am fully impressed by your unit ball explanation . Thanks a lot upload more videos on distribution theory and sobolev space.

[@brightsideofmaths]

Thank you, I will

[@mateen8793]

Eagerly waiting for your next videos of the series

[@TheAlbertMiralles]

Super nice!! Love all the serires so far :))))

[@japedr]

If I remember correctly, this is the reason why the regular distributions are not "complete". The idea is that one can build a sequence of functions that do not actually converge (at least, not to a proper function), but the associated sequence of regular distributions converges to the delta distribution. So this is a Banach sequence that converges to something more general, so the set of regular distributions is not complete (same idea why a sequence of rationals converging to pi proves that rationals are not complete). In fact, IIRC, the set of all distributions is complete in this sense. Is all of this correct?

[@dietmarkammel3731]

@The Bright Side of Mathematics Awesome explanations! Thx a lot. When will you continue this series? Can't wait for the next one... :)

[@brightsideofmaths]

Thank you very much. Working on it! :)

[@dietmarkammel3731]

@@brightsideofmaths Klingt gut. ;) Dank Dir.

[@amiltonmoreira2341]

@The Bright Side of Mathematics thank you for the lecturer . Could you introduce the concept of wavefront set in this series ? thank you

[@douglasstrother6584]

Now I am inspired to go out and play some ε-ball!

[@brightsideofmaths]

Take some friends and play this game :)

[@juandavidrodriguezcastillo9190]

diracs delta dont asing to a function its value at 0, when one composed the delta with another function delta(g(x))=sum {delta(x-xi)/g'(x-xi)} with xi the roots of g

[@brightsideofmaths]

Yes, but with the composition you are talking about a different distribution :)

[@nafrost2787]

I think there is a simpler proof why the delta distribution is not a regular distribution. Can't we just take any sequence of test functions such that the f_n(0) = 1 -> δ(f_n) = 1, the sequence of the diameters of the supports approaches 0, and that there exists r > 0 such that f_n(x)

[@brightsideofmaths]

I really like the idea! However, as you already noted, for the Lebesgue integral, also unbounded functions can be integrable. Still there are a lot of proofs you can find for showing that delta is not regular. I did it in this way because it fits in my lecture :)

[@rit1237]

Haha I don't get anything

[@oleksiishekhovtsov1564]

same

[@Vietcongster]

I recommend watching the whole distribution series before this one

[@carl3260]

Intuitive summary: - For any integrable function f, the integral ("I_p") of f(x)p(x) over a compact space (i.e. connected region) is the max value of p(x) over the space ("p_max") times a constant ("k"), ie I_p = k p_max. [p(x) is any test function]. - Reducing the space, k gets smaller and eventually k I_p < p_max. That is general, but choosing any p(x) s.t. p_max = p(0) => I_p < p(0). - However, for the delta map, I_p = p(0) [by def] => Contradiction