# Local average sampling and reconstruction with fundamental splines of fractional order

## TL;DR

This paper investigates sampling and reconstruction methods using fractional spline functions of real order, extending classical spline theory to fractional orders and generalizing Kramer's lemma for local average sampling.

## Contribution

It introduces interpolation and average sampling techniques with fractional B-splines of real order and extends Kramer's lemma within this fractional spline framework.

## Key findings

- Established interpolation with fractional order splines for σ ≥ 1
- Developed average sampling methods for σ ≥ 1.5
- Generalized Kramer's lemma for local average sampling in fractional spline spaces

## 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 $\sigma \geq 1$ and the average sampling with fundamental splines of fractional order for $\sigma \geq \frac{3}{2}.$ Further, we generalise Kramer's lemma in the context of local average sampling.