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0 pointsVLM13y ago0 comments

"That is, suppose I describe some (infinite) set of a functions.... Is there an intuitive way to picture why the set of sin and cos functions forms a suitable basis?"

There are infinite expansions of trig functions to solve the finite vs infinite apparent mismatch, so don't worry about that.

An intuitive way to look at it, is on a 2-d graph there's no spot on the graph that can't be hit by a point of a triangle aka trig function collection, so it doesn't particularly matter how you pick any one spot, a triangle can always hit that one spot because it can hit all spots.

Comp sci analogy would be something like x=x+1 starting at x=0 will hit all the positive integers eventually...

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ColinWright13y ago

That doesn't really work, because you have to be able to hit them all simultaneously, and in fact, you can't. The theorem says that for any given epsilon you can choose enough to ensure that you are within epsilon everywhere except in a very small section of the real line, and you can make the parts where it's not within epsilon very small.

The problem with your comment is that yes, for any given point you can get as close as you like, but the hard part is getting close nearly everywhere all at once. More, you then have to show that there's some sort of convergence, and that where you are close now is a superset of where you are close when you demand to be closer.

It's not straight forward.

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