2008/09/30 by Ognyan Kounchev, Kounchev, Ognyan, Hermann Render +1
Computer Science · Engineering · Mathematics · #32A50 #60G35 #68U10 #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Information Theory (cs.IT) #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.0809.5153
openalex publication_date 2008/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) =∑j=-∞∞∫_\QTRBbbSn-1S(e-jrθ) f(ejθ) dθ. This formula relies upon infinitely many Shannon type formulas for the exponential splines arising from the radial part of the polyharmonic operator Δp for fixed p≥ 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.