关于一种新的多元采样范式与多项式样条香农函数
On a new multivariate sampling paradigm and a polyspline Shannon function
AI总结：
本文提出一种基于多元多项式样条理论的新的多元采样范式，并推导出香农型公式。
摘要：
在专著Kounchev, O. I., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, San Diego-London, 2001以及论文Kounchev O., Render, H., Cardinal interpolation with polysplines on annuli, Journal of Approximation Theory 137 (2005) 89--107中，我们引入并研究了一种与多元多项式样条理论相关的新基数插值范式。在本文中，我们展示了这与多元情况下的新采样范式相关，同时我们得到了一个香农型函数$S(x)$以及以下香农型公式：$f(rθ) =\sum_{j=-\infty}^{\infty}\int_{\QTR{Bbb}{S}^{n-1}}S(e^{-j}rθ) f(e^{j}θ) dθ.$ 该公式依赖于由固定$p\geq 1$的多调和算子$Δ^{p}$的径向部分产生的指数样条的无穷多个香农型公式。致谢。第一作者和第二作者得到了亚历山大·冯·洪堡基金会研究所合作项目的部分支持。第一作者得到了希腊-保加利亚双边项目BGr-17的部分资助，第二作者得到了西班牙D.G.I.的MTM2006-13000-C03-03项目的部分资助。
英文摘要：
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.
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