Unknown

7.2. Approximation on an Interval 213

(k=O,1,2,... , n). Verify the table

1 1

2 t2

3 (3t2 + 1)/4

4 (-4t4 + 7P)/3

5 (-125t4 + 290t2 + 27)/192

6 (81t6 - 135t4 + 74t2)/20

Argue that, in general, fn(t) is an even function and that f;(O) = 0.

Investigate various strategies for conveniently obtaining the polynomial

f,,(t). For example, one might first determine the polynomial gn(t) of degree

not exceeding n for which g,,(t) = max(t,O) at the specified points and

note that fn(t) = gn(t) + gn(-t) = 2g,(t) - t. Alternatively, if n = 2m

and h,(t) = ItI for t = -m, -m + 1,... , -l,O, 1,... ,m, then h,+l(t) =

h,(t) + c,t2(t2 - l)(t2 - 4). e. (t2 - m2) for some constant c,,, and fn(t) =

kh,(mt).

If t is a point in the closed interval [-1, 11, is it necessarily the case that

linh,, fn(t) = f (V

7.2 Approximation on an Interval

Suppose that f(t) is a real-valued function defined for at least some values
oft with a 5 t 5 b. For example f(t) could be a table of values used by
an insurance company to determine premiums or could be a nonpolyno-
mial function given by a formula. How can we accurately compute f(t) for
certain values of t between a and b? As we have seen in Chapter 2, the
values of polynomials are straightforward to compute, so it is worth trying
to find a polynomial which closely approximates the function on the given
interval.
It is natural to take a polynomial which agrees with f where the value of
f can be explicitly determined. But this is fraught with danger. If f(t) =
sin(2rnt) for 0 5 t 5 1, the polynomial of least degree which coincides with
f when t = 0, l/n, 2/n,... , 1 is 0, but this hardly reflects the behaviour of
f. Furthermore, a slight change in the value of f at one of the evaluation
points may dramatically alter the polynomial which interpolates it. We
have no guarantee that making a polynomial close in value to f at one
place will ensure that it is close overall.
This difficulty can be circumvented by giving up the requirement of exact
agreement at some points in favour of gaining some flexibility for making
the approximation close everywhere.
What does it mean for functions to be close on a whole interval? There is
not one right answer which applies in all contexts. The “distance” between