Box-splines orthogonal projections

TL;DR

This paper generalizes the classical result that the projection error $P(x^r) - x^r$ is a Bernoulli polynomial to the setting of box splines, linking the asymptotic projection error to Sobolev seminorms. It shows that $L_\beta(x) = P([]^\beta)(x) - x^\beta$ is a linear combination of Bernoulli splines and provides an explicit asymptotic error formula for $f$ in $W_p^{\varrho_V+1}$, with a precise $L^2$-theory recovery. Extending the analysis to BV functions, the work introduces Marcinkiewicz averaging of projections and derives a boundary-integral asymptotic formula $\int_{\partial^* E} F(\nu(x)) \, dH^{d-1}(x)$ that characterizes the BV semi-norm in terms of box-spline projections. The results offer a mechanism to define Sobolev seminorms via projection-errors and illuminate the BV behavior under dyadic averaging, with implications for harmonic analysis on box splines and approximation theory.

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.

Figures (2)

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Theorems & Definitions (20)

• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6
• Theorem 2.7