Local Average Sampling and Reconstruction with Fundamental Splines of Fractional Order

Abstract

We analyze 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 generalize Kramer’s lemma in the context of local average sampling.

Keywords:

• Average sampling
• fractional spline
• fractional spline interpolation
• fundamental cardinal spline

Acknowledgment

The third author thanks the management of Sri Sivasubramaniya Nadar College of Engineering.

Additional information

Funding

The second author was partially supported by DFG grant MA 5801/2-1.