What is convolution?

What is convolution?

Here is the concept of convolution, which is the correct way of multiplying two functions. Not in an algebra way but in an analysis way. We also state the convolution theorem, which says that the laplace transform of f * g is the product of laplace transforms. This is useful to find inverse Laplace transforms as well. I also show a cool demo illustrating why there is a shift and an integral in the definition.

[iabervon]

One day in my ~applied odes class, the TA made the off-hand comment that convolution is like a dot product in an uncountably infinite number of dimensions. Thinking about how that's true and how it's different was really useful to me for developing an intuition for convolutions. (I was also amazed by how much it sounds like technobabble but actually helps to remember the formula.)

[elyepes]

Many, many deconvoluted thanks for your valuable videos Dr Peyam

[nestorv]

Convolution is an operation that determines how one function influences another function on a point-by-point basis, especially useful when a function is not discrete (i.e. continuous). Like someone else mentioned, it's like an infinite-dimensional dot product.
Examples include: impulse response functions in electrical engineering signals.
the distribution of the addition of two random variables in probability

[hehehepaitachato]

Thanks for the class, Dr Peyam! I've already seen this before on ODE classes, but never understood so profoundly

[theproofessayist]

Thank you for the demo. You are going above and beyond. Also love how the point of this video is hmmm we have a theorem or statement that does not work (or is nice) - let's tweak our definition of multiplication so that it is true and indeed convolution is our gal we need! :)

[YouTube_username_not_found]

What's more interesting is the fact that multiplication of time-functions gets transformed into convolution of frequency-functions.

Edit:
I omitted a factor of 1/(2i×pi) and other details, they could be found in the Laplace transform article on Wikipedia, in the section "properties and theorems".

Also, doesn't the star symbol represent cross-correlation? Isn't convolution denoted by an asterisk? At least that's what I noticed in other sources.

Last thing, it would be a good exercise to find all pairs of functions that make the LT of multiplication equal the multiplication of LTs.

[shieldmytears]

Yayyy you actually did it !! Thank you Dr. Peyam💕

[Giantcrabz]

oof that thumbnail had me do a double take 😅😅😅