On a new multivariate sampling paradigm and a polyspline Shannon function
This work extends Shannon sampling theory to multivariate settings using polysplines, offering a novel theoretical framework for signal processing and approximation in higher dimensions.
The paper introduces a new multivariate sampling paradigm based on polysplines, deriving a Shannon-type formula for functions on annuli. It provides a Shannon-type function S(x) and a formula that reconstructs a function from its values on concentric spheres.
In the monograph Kounchev, O. I., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, San Diego-London, 2001, and in the paper Kounchev O., Render, H., Cardinal interpolation with polysplines on annuli, Journal of Approximation Theory 137 (2005) 89--107, we have introduced and studied a new paradigm for cardinal interpolation which is related to the theory of multivariate polysplines. In the present paper we show that this is related to a new sampling paradigm in the multivariate case, whereas we obtain a Shannon type function $S(x) $ and the following Shannon type formula: $f(rθ) =\sum_{j=-\infty}^{\infty}\int_{\QTR{Bbb}{S}^{n-1}}S(e^{-j}rθ) f(e^{j}θ) dθ.$ This formula relies upon infinitely many Shannon type formulas for the exponential splines arising from the radial part of the polyharmonic operator $Δ^{p}$ for fixed $p\geq 1$. Acknowledgement. The first and the second author have been partially supported by the Institutes partnership project with the Alexander von Humboldt Foundation. The first has been partially sponsored by the Greek-Bulgarian bilateral project BGr-17, and the second author by Grant MTM2006-13000-C03-03 of the D.G.I. of Spain.