Best approximation by polynomials on the conic domains

2025/06/28 by Yan Ge, Yuan Xu, Ge, Yan +1 · 1 citation
Mathematics · #41A10 #41A63 #42C10 #42C40 #Analytic and geometric function theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2506.22916

openalex publication_date 2025/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A new modulus of smoothness and its equivalent K-function are defined on the conic domains in ℝ^d, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the characterization are established via the modulus of smoothness. For the conic surface \mathbbV₀^d+1 = \(x,t): ‖x‖ = t≤ 1\, the natural weight function is t^-1(1-t)^γ, which has a singularity at the apex, the rotational part of the modulus of smoothness is defined in terms of the difference operator in Euler angles with an increment h/√(t), akin to the Ditzian-Totik modulus on the interval but with √(t) in the denominator, which captures the singularity at the apex.

Citations

• Approximation and localized polynomial frame on conic domains
• Orthogonal polynomials and Fourier orthogonal series on a cone
• Localized polynomial frames on the interval with Jacobi weights
• Weighted Approximation of functions on the unit sphere

Cited by

• Approximation and orthogonality on fully symmetric domains

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