Polynomial approximation of functions in Sobolev spaces

1980/01/01 by Todd Dupont, Ridgway Scott · 14 citations
Mathematics · Engineering · #Mathematical Approximation and Integration #Advanced Numerical Methods in Computational Mathematics #Mathematical functions and polynomials

paper · doi:10.1090/s0025-5718-1980-0559195-7

Abstract

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

Cited by

• Numerical approximation of the integral fractional Laplacian
• Finite Element Methods for Interface Problems: Robust Residual-Based A Posteriori Error Estimates
• Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev\n spaces and applications
• Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method
• Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods
• The virtual element method for resistive magnetohydrodynamics
• Finite element quasi-interpolation and best approximation
• Adaptive Least-Squares Finite Element Methods for Linear Transport Equations Based on an H(div) Flux Reformulation
• Inverse estimates for elliptic boundary integral operators and their\n application to the adaptive coupling of FEM and BEM
• Universal Scalable Robust Solvers from Computational Information Games\n and fast eigenspace adapted Multiresolution Analysis
• The Bramble-Hilbert Lemma
• Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D
• A Survey of Indicators for Mesh Quality Assessment
• A note on error analysis for a nonconforming discretisation of the tri-Helmholtz equation with singular data

Related

• Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball
• Approximation by polynomials in Sobolev spaces with Jacobi weight
• Constructive description of Hardy-Sobolev spaces in ℂⁿ
• On approximation of functions by polynomials and by entire functions of exponential type
• How to recognize polynomials in higher order Sobolev spaces