A note on Taylor expansion of real function

2020/04/01 by Shun Tang, Tang, Shun
Mathematics · #41A10 #41A52 #41A58 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Iterative Methods for Nonlinear Equations #Mathematical and Theoretical Analysis #math.CA #msc:41A10 #msc:41A52 #msc:41A58

paper · pdf · doi:10.48550/arxiv.2004.00474

8 pages, title was changed to a more specific one

openalex publication_date 2020/04/01 · arxiv created 2020/05/09 · arxiv updated 2020/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f(x) be a real function which has (n+1)-th derivative on an interval [a, b]. For any point x₀∈ (a, b) and any integer 0≤ k≤ n, denote by S_k,x0(x) the k-th truncation of the Taylor expansion of f(x) at x₀, i.e. S_k,x0(x)=∑_i=0^k\fracf⁽ⁱ⁾(x₀)i!(x-x₀)ⁱ. In this note, we consider the L₂-approximation of f(x) by polynomials of degree ≤ k, we show that S_k,x0(x) is the limit of the best approximations of f(x) on [x₀-ε, x₀+ε] as ε→ 0.

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