# Box-splines orthogonal projections

## TL;DR

This paper generalizes a known result about orthogonal projections of polynomial functions to the broader context of box-splines, enabling new ways to define Sobolev space seminorms via projection error asymptotics.

## Contribution

It extends Sweldens and Piessens's result from polynomial splines to box-splines, linking projection errors to Sobolev space seminorms.

## Key findings

- Generalization of Bernoulli polynomial relation to box-splines
- New characterization of Sobolev seminorms via projection error asymptotics
- Potential applications in approximation theory and numerical analysis

## Abstract

Let $P$ be orthogonal projection on B-splines of degree $r-1$ with equally spaced knots. Sweldens and Piessens proved that $P(x^r)-x^r$ is Bernoulli polynomial. We generalize Sweldens ans Piessens's result for box-splines. It gives the opportunity to define the seminorm of Sobolev space in terms of the asymptotic formula for the error in orthogonal projection.