Sobolev orthogonality and best polynomial approximations

Abstract

First, we introduce the concept and general theory for Sobolev orthogonal polynomials with respect to ◁수식 삽입▷(원문을 참조하세요) where $λ_0=1$, $λ_i≥0$ for 1≤i≤N, and $(μ_i)^N_{i=0}$ are positive finite Borel measures. When σ is a quasi-definite moment functional on $\Bbb{P}$, the space of polynomials in one variable with the monic orthogonal polynomial system ${P_n(x)}^∞_{n=0}$ we consider a symmetric bilinear form φ(ㆍ,ㆍ) on $\Bbb{P}×\Bbb{P}$ defined by ◁수식 삽입▷(원문을 참조하세요) where λ, μ, a, b are complex numbers and r,s are non-negative integers. We find a necessary and sufficient condition under which there is an orthogonal polynomial system ${P_n(x)}^∞_{n=0}$ relative to φ(ㆍ,ㆍ) and discuss algebraic properties of ${P_n(x)}^∞_{n=0}$. When σ semi-classical, we show that ${P_n(x)}^∞_{n=0}$ must satisfy a second order differential equation with polynomial coefficients. When σ is positive-definite and λ, μ, a, b are real, we investigate the relations between zeros of ${P_n(x)}^∞_{n=0}$ and ${P_n(x)}^∞_{n=0}$. We consider a point masses perturbation τ of σ given by ◁수식 삽입▷(원문을 참조하세요) where λ, Ulk, and $c_l$ are constants with $c_i ≠ c_j$ for i ≠ j. That is, τ is a generalized Uvarov transform of σ satisfying A(x)τ = A(x)σ where $A(x)=\prod\limits_{l=1}^{m}(x-c_{l})^{m_{l}+1}.$ We find necessary and sufficient conditions for τ to be quasi-definite. We also discuss various properties of monic orthogonal polynomial system ${P_n(x)}^∞_{n=0}$ relative to τ including two examples. We investigate the limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space $W^{N,2}([0,∞);e^{-x})$ and Sobolev-Legendre space $W^{N,2}([-1,1])$ with respect to the Sobolev-Laguerre inner product ◁수식 삽입▷(원문을 참조하세요) and Sobolev-Legendre inner product ◁수식 삽입▷(원문을 참조하세요) respectively, where $a_0=1$ , $a_k ≥ 0$, 1 ≤ k ≤ N - 1, γ ＞ 0 and N ≥ 1 is an integer. Finally, we consider a Sobolev-Jacobi inner product ◁수식 삽입▷(원문을 참조하세요) We show that a Sobolev orthogonal po...