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7.2. Approximation on an Interval^215

measure the degree of closeness between f(t) and p(t). This measures

how far apart the values of f(t) and p(t) can get over the whole

interval. The approximation problem is to choose p from a set of

desired approximants to make this maximum as small as possible. (It

is re-emphasized that this is but one of many possible measures of

closeness we could have chosen.)

(a) Suppose we ask for our approximant p(t) to be a constant poly-

nomial c. Then, the question is: what value of c will make

max{lf(t) - cl : a < t 5 b}

as small as possible?

Consider the following graphical representation. Argue that, for

the optimum value of c, the function f(t) - c should have a

maximum of the same absolute value but opposite sign as its

minimum.

I I

I (a, 0)

w

I(W t

(b) Find the best constant polynomial approximation on the closed

interval [0, l] to each of the following functions:

(i) sin(d)

(ii) t” (k a positive integer)

(iii) l/(1 + t).

(c) Consider polynomial approximations of degree not exceeding 1.

Then we wish to choose a and b in such a way as to minimize

max{lf(t) -(at + b)l : 0 5 t 5 1).

By considering the following sketch, argue that for the optimum

choice of a and 6, f(t) - (at + b) assumes an extreme value at

least three times, that all of these extreme values have the same

absolute value and that there are at least two changes in sign as

we move from one extreme value to the next.