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

Abstract

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

$$\varphi (f,g): = \sum\limits_{k = 0}^{N - 1} {a_k } \int_0^\infty {f^{(k)} (x)g^{(k)} (x)e^{ - x} dx + \gamma } \int_0^\infty {f^{(N)} (x)g^{(N)} (x)e^{ - x} dx} $$

and with respect to the Sobolev-Legendre inner product

$$\varphi _1 (f,g): = \sum\limits_{k = 0}^{N - 1} {a_k } \int_{ - 1}^1 {f^{(k)} (x)g^{(k)} (x)dx + \gamma } \int_{ - 1}^1 {f^{(N)} (x)g^{(N)} (x)dx,} $$

respectively, where a₀ = 1, a_k ≥0, 1 ≤k ≤N −1, γ > 0, and N ≥1 is an integer.