Lol you might wanna check out your post again... Whatever you copied of physics forum has been saved as an image so won't show up I guess.
Let be distinct points in the interval [a,b] and .
We show that for any given there exists a polynomial p such that
and for all
I know and I wonder if the polynomial they are asking for is the Lagrangian polynomial interpolating f at the nodes . If yes, I am not sure how to prove the problem.
I'll give it a shot I guess...
Assume WLOG that both a and b are in the , and replace f with f-g, where g is the polynomial which interpolates the nodes . Clearly if f-g can be approximated by a polynomial, f can be approximated by that polynomial plus g. So we have to find a polynomial which is within of f and vanishes at .
Chuse as g a polynomial which is everywhere within of f by the Stone–Weierstrass theorem. Let which is no more than in magnitude. We will chuse polynomials to add to g to correct these errors so that the final approximating polynomial is . Then if all the .
Therefore we have only to prove the following claim: for every and ( ), there is a polynomial h which is everywhere less than in magnitude on [a,b] and h(x)=y.
To do this, chuse a cosine wave centered at x and scale it down to amplitude y. We can take its Taylor polynomial to enough terms that inside [a,b] it is closer to zero at every point than the cosine wave [perhaps someone should check this, as I only have a rough sketch], so that this Taylor polynomial is y at x and it is on [a,b], as desired.
...god, what a mess. Its probably wrong too. Be warned!