By Richard A. Silverman

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**Example text**

Find a similar generating function for Un(x). 10 Find the polynomial of degree n and leading coefficient unity which vanishes at x = 0, x = 1 and which deviates least from zero in the interval 0 ::;; x ::;; 1. 11 Show that for n = 1, 2, 3, ... we have ITn(x) I ::;; 1 , I Un(x) I : ; ; n + 1 , - 1 ::;; X ::;; 1. Discuss the equality cases. 12 Evaluate ]/2 { ! ,(± 1). 14 Find a polynomial a x 1 + b x in [ - 1, 1], a, b being arbitrary. 14 for a x 2 + x + b. 16 Establish the minimum deviation of T 2 (x), using elementary arguments.

For each p(x) there is a greatest T = -r(p) > 0 such that - T ::;; x ::;; 0 implies I p(x) I ::;; 1. Find the polynomial Ek(x) in Pk for which T = Tk is maximum. What is the value of Tk? Write down Ek(x) fork= 2, 3, 4. 3) (I) show that if the are the zeros of T n+i(x), then Xt (i = 0, I, ... 22 Obtain the linear polynomial of best approximation to (I + x)- 1 in [O, 1]. , v2 [! - (a - 2 V2) (2 x - I)]= ! 4853 x . 23 Show that the polynomial of degree n of best approximation to (I + x)- 1 in [O, 1] is, if c = 3 - 2 ]f2 , 3in(x) = V2 {[ !

G. 0001 in p in NATIONAL BUREAU OF STANDARDS (1948). It can be shown that such tables are unnecessary and that n + 1-point interpolation can be arranged as a succession of linear interpolations. This scheme is due to A. C. 8. g. NAUTICAL ALMANAC OFFICE (1956), NATIONAL PHYSICAL LABORATORY (1961), Fox (1956), and we shall confine our attention mainly to some rather theoretical questions. 2) assuming the existence of /M(x) in an interval including the distinct nodes x1 , x 8 , • • • , Xn, and the current point x; Cis a point in this interval, whose position depends on x.