Approximation and orthogonality on fully symmetric domains

2025/09/12 by Yuan Xu, Xu, Yuan
Mathematics · #33C45 #42C05 #42C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2509.10342

openalex publication_date 2025/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study orthogonal polynomials on a fully symmetric planar domain Ω that is generated by a certain triangle in the first quadrant. For a family of weight functions on Ω, we show that orthogonal polynomials that are even in the second variable on Ω can be identified with orthogonal polynomials on the unit disk composed with a quadratic map, and the same phenomenon can be extended to the domain generated by the rotation of Ω in higher dimensions. The connection allows an immediate deduction of results for approximation and Fourier orthogonal expansions on these fully symmetric domains. It applies, for example, to analysis on a double cone or a double hyperboloid.

Citations

• Best approximation by polynomials on the conic domains
• Orthogonal polynomials on domains of revolution
• Maximal functions and multiplier theorem for Fourier orthogonal series
• Approximation and localized polynomial frame on double hyperbolic and conic domains
• Approximation and localized polynomial frame on conic domains
• Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid
• Orthogonal polynomials in and on a quadratic surface of revolution
• Orthogonal polynomials and Fourier orthogonal series on a cone
• Recurrence relations for orthogonal polynomials on a triangle
• Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure
• Localized polynomial frames on the ball
• Weighted Approximation of functions on the unit sphere

Related

• Orthogonal polynomials on domains of revolution
• Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid
• Approximation and orthogonality in Sobolev spaces on a triangle
• Orthogonal polynomials in and on a quadratic surface of revolution
• Orthogonal structure on a wedge and on the boundary of a square