Suppose that it is desired to construct a set of polynomials f0( x), f1( x), f2( x), .

Chapter 11, Problem 7

(choose chapter or problem)

Suppose that it is desired to construct a set of polynomials f0( x), f1( x), f2( x), . . . , fk ( x), . . . , where fk ( x) is of degree k, that are orthonormal on the interval 0 x 1. That is, the set of polynomials must satisfy ( f j , fk ) = _ 1 0 f j ( x) fk ( x)dx = jk . a. Find f0( x) by choosing the polynomial of degree zero such that ( f0, f0) = 1. b. Find f1( x) by determining the polynomial of degree one such that ( f0, f1) = 0 and ( f1, f1) = 1. c. Find f2( x). d. The normalization condition ( fk , fk ) = 1 is somewhat awkward to apply. Let g0( x), g1( x), . . . , gk ( x), . . . be the sequence of polynomials that are orthogonal on 0 x 1 and that are normalized by the condition gk(1) = 1. Find g0( x), g1( x), and g2( x), and compare them with f0( x), f1( x), and f2( x).