Local average sampling and reconstruction with fundamental splines of fractional order

Abstract

We analyse sampling and average sampling techniques for fractional spline subspaces of $L^2(\mathbb{R}).$ Fractional B-splines ${\beta }_{\sigma }$ are extensions of Schoenberg's polynomial splines of integral order to real order $\sigma >-1$. We present the interpolation with fundamental splines of fractional order for $\sigma \ge 1$ and the average sampling with fundamental splines of fractional order for $\sigma \ge \frac{3}{2}.$ Further, we generalise Kramer's lemma in the context of local average sampling.