# A fully discretised filtered polynomial approximation on spherical shells

## TL;DR

This paper introduces a fully discretised filtered polynomial approximation method for spherical shells, demonstrating algebraic decay of approximation error and supported by numerical experiments.

## Contribution

It presents a new quadrature-based filtered polynomial approximation method specifically designed for spherical shells, with separate treatment of radial and angular directions.

## Key findings

- Error decays algebraically for smooth functions
- Numerical experiments confirm theoretical results
- Method is fully implementable

## Abstract

A fully implementable filtered polynomial approximation on spherical shells is considered. The method proposed is a quadrature-based version of a filtered polynomial approximation. The radial direction and the angular direction of the shells are treated separately with constructive filtered polynomial approximation. The approximation error with respect to the supremum norm is shown to decay algebraically for functions in suitable differentiability classes. Numerical experiments support the results.