Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

Abstract

We investigate limiting behavior as gamma tends to infinity of the best polynomial approximations in the Sobolev-Laguerre space W-N,W-2([0, infinity); e(-x)) and the Sobolev-Legendre space W-N,W-2([-1, 1]) with respect to the Sobolev-Laguerre inner-product phi(f,g): = Sigma(k=0)(N-1)a(k) integral(0)(infinity) f((k))(x)g((k))(x)e(-x) dx + gamma integral(0)(infinity) f((N))(x)g((N))(x)e(-x) dx and with respect to the Sobolev-Legendre inner product phi(1)(f,g): = Sigma(k=0)(N-1)a(k) integral(-1)(1) f((k))(x)g((k))(x) dx + gamma integral(-1)(1) f((N))(x)g((N))(x)dx, respectively, where a(0) = 1, a(k) greater than or equal to 0, 1 less than or equal to k less than or equal to N - 1, gamma > 0, and N greater than or equal to 1 is an integer.