An Interpolation Process on the Roots of Hermite Polynomials on Infinite Interval

摘要

For given arbitrary numbers . . . . * , 1 1 1, , 1 1 k k d k n d k l n . . . . and . . ** , 1 1 , k d k n . we seek to determine explicity polynomials . . n Rx of degree atmost 3n-3 (n even), given by: (1) . . . . . . . . 1 1 * * * 1 1 1 , n n n n k k k k k k k k k R x d U x d V x d W x . . . . . . . . . . . Such that . . . . . . . . . . . . ' * ** , 1 1 1, , 1 1 1 a n d , 1 1 , n k k n k k n k k R y d k n R y d k n R x d k n . . . . . . . . where . . 1 n k k x . are the zeros of th n Hermite polynomial . . n Hx and . . 1 1 n k k y . . are the zeros of . . ' . n Hx Let the interpolated function f be continuously differentiable satisfying the conditions: . . . . 2 0 , 0 ,1, 2 . , ..... Lim v x x f x x . . . . . . . and . . . . . . 2 2 / 2 ' 0 , w h e r e , L im v x x x x f x x e . . . . . . . . further in (1) . . . . , 1 1, x k d f y k n . . . . . . . 2 * 1 ' , ( ', , 1 1 1, 1 yk k k d f y e f k n n . . . . . . . . . . . . . . . . . . . . . . . ** ' , 1 1 , k k d f x k n . . then for the sequence of inter polynomials . . 2 , 4 , ... , n Rn . we have the estimate . . . . 2 1 3 , 0 ( ( '; )l o g n ), 2 vx n e f x R f x f v n . . . . . Which holds the whole real line, O does not depend on a n d a n d nx . is the modulus of continuity of ' f introduce by G. Freud.