2011年丘成桐大学生数学竞赛——计算与概率卷.pdfVIP

S.-T. Yau College Student Mathematics Contests 2011 Applied Math., Computational Math., Probability and Statistics Individual 6:30 –9:00pm, July 9, 2011 (Please select 5 problems to solve) 1. Given a weight function ρ(x ) 0, let the inner-product correspond- ing to ρ(x ) be de?ned as follows: b (f, g) := ρ(x )f (x ) g(x )d x, a and let f := (f, f ) . (1) De?ne a sequence of polynomials as follows: p (x ) = 1, p (x ) = x - a , 0 1 1 p (x ) = (x - a ) p (x ) - b p (x ) , n = 2, 3, ··· n n n - 1 n n - 2 where (xp n- 1 , pn - 1 ) an = , n = 1, 2, ··· ( pn- 1 , pn - 1) (xp n- 1 , pn - 2 ) b = , n = 2, 3, ···. n ( pn- 2 , pn - 2 ) Show that { pn (x )} is an orthogonal sequence of monic polyno- mials. (2) Let { q (x ) } be an orthogonal sequence of monic polynomials n corresponding to the ρinner product. (A polynomial is called monic if its leading coe?cient is 1.) Show that { q (x ) } is unique