Mathematics – Numerical Analysis
Scientific paper
2008-09-30
Mathematics
Numerical Analysis
Submitted to the conference proceedings of SAMPTA07 held in Thessaloniki, Greece, 2007
Scientific paper
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: $f(r\theta) =\sum_{j=-\infty}^{\infty}\int_{\QTR{Bbb}{S}^{n-1}}S(e^{-j}r\theta ) f(e^{j}\theta) d\theta .$ This formula relies upon infinitely many Shannon type formulas for the exponential splines arising from the radial part of the polyharmonic operator $\Delta ^{p}$ 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.
Kounchev Ognyan
Render Hermann
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