Thread: Polynomial Approximations

The following is a worked example:




In the following, for example



What I don't understand is that why they got .

Since

And

But what do we need to substitute for ? Because this function is piece-wise, how do I know which one of the two functions I have to put into the formula?

Well, what are the definitions of , , etc.?

Originally Posted by HallsofIvy

Well, what are the definitions of , , etc.?

Those are inner products. Since and are continious functions on the given interval then their inner product is defined by



I belive that's the definition. The resulting polynomial we get in this problem is called "Legendre's polynomial", but unfortunently there is not much about it in my textbook.

For this:



What should I substitute for "g(x)"? The g(x) is defined as a piece wise function. Do I need to substitute "-1" or "2x-1"? (apparently neither of them give the correct answer)