Rational Approximation and Sobolev-type Orthogonality

2019/07/29 by Abel Díaz-González, Díaz-González, Abel, Héctor Pijeira-Cabrera +3
Mathematics · #30E10 #33C47 #41A20 #42C05 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:30E10 #msc:33C47 #msc:41A20 #msc:42C05

paper · pdf · doi:10.48550/arxiv.1907.12243

arxiv created 2019/07/29 · arxiv updated 2019/07/30

Abstract

In this paper, we study the sequence of orthogonal polynomials \S_n\_n=0^∞ with respect to the Sobolev-type inner product ⟨ f,g ⟩= ∫_-1¹ f(x) g(x) dμ(x) +∑_j=1^N η_j f^(d_j)(c_j) g^(d_j)(c_j), where μ is in the Nevai class M(0,1), η_j >0, N,d_j ∈ ℤ₊ and \c₁,…,c_N\⊂ ℝ ∖ [-1,1]. Under some restriction of order in the discrete part of ⟨⋅,⋅ ⟩, we prove that for sufficiently large n the zeros of S_n are real, simple, n-N of them lie on (-1,1) and each of the mass points c_j ``attracts'' one of the remaining N zeros. The sequences of associated polynomials \S<sub>n</sub><sup>[k]</sup>\<sub>n=0</sub><sup>∞</sup> are defined for each k∈ ℤ<sub>+</sub>. We prove an analogous of Markov's Theorem on rational approximation to a function of certain class of holomorphic functions and we give an estimate of the ``speed'' of convergence.

Related

• Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product
• A new approach to the asymptotics for Sobolev orthogonal polynomials
• On a Theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics
• Sobolev orthogonal polynomials: Connection formulae