Complex B-splines
Complex B-splines
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DOI：
10.1016/j.acha.2005.07.003
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发表时间：
2006-03-01
影响因子：
2.5
通讯作者：
Unser, M
中科院分区：
文献类型：
--
作者：
Forster, B;Blu, T;Unser, M
我们提出勋伯格基数样条的复杂概括。为此，我们回到 B 样条的傅立叶域定义，并将其扩展到复数值度。我们证明所得到的复杂 B 样条是分段调制多项式，并且它们保留了经典多项式的大部分重要属性：平滑度。递归和二尺度关系、Riesz 基生成器、导数的显式公式（包括分数阶）等。我们还表明它们可以生成 L-2 (R) 的多分辨率分析，并且可以生成小波基。当次数不是整数时，我们描述这些函数的衰减，这些函数不再紧支持。最后，我们证明了复 B 样条随着次数的增加而收敛于调制高斯，并且根据海森堡不确定性原理，它们在时频平面上渐近最优局部化。 (c) 2005 Elsevier Inc. 保留所有权利。
We propose a complex generalization of Schoenberg's cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show that the resulting complex B-splines are piecewise modulated polynomials, and that they retain most of the important properties of the classical ones: smoothness. recurrence, and two-scale relations, Riesz basis generator, explicit formulae for derivatives, including fractional orders, etc. We also show that they generate multiresolution analyses of L-2 (R) and that they can yield wavelet bases. We characterize the decay of these functions which are no-longer compactly supported when the degree is not an integer. Finally, we prove that the complex B-splines converge to modulated Gaussians as their degree increases, and that they are asymptotically optimally localized in the time-frequency plane in the sense of Heisenberg's uncertainty principle. (c) 2005 Elsevier Inc. All rights reserved.