Both the Hilbert transform and the B-spline wavelets are important tools in signal processing, which makes a study of relation between these two subjects be of practical significance. In particular, because the B-spline wavelets have good properties including vanishing moments, symmetry, compact support and so on, we focus on the Hilbert transform of B-spline wavelets in this letter. For this purpose, the B-spline wavelets of order $m$ is described in a piecewise polynomial form firstly. An important property of the Pascal triangle transform is then explored. Based on these results, an explicit form of the Hilbert transform of B-spline wavelet of order $m$ is established. Furthermore, the vanishing moments, symmetry and asymptote behavior of the Hilbert transform of B-spline wavelets are also discussed. To demonstrate the effectiveness of these results, two examples in the case of $m=3$ and $m=4$ are given and the graphs of these two B-spline wavelets as well as their Hilbert transforms are presented. These two cases of $m=3$ and $m=4$ provide two Hilbert transform pairs of wavelets, which can be used in digital image processing.

中文翻译：

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B样条小波的希尔伯特变换

希尔伯特变换和B样条小波都是信号处理中的重要工具，这使这两个主题之间的关系研究具有实际意义。特别地，由于B样条小波具有良好的特性，包括消失矩，对称性，紧致支撑等，因此在本文中我们关注B样条小波的希尔伯特变换。为此，有序的B样条小波$ m $首先以分段多项式形式进行描述。然后探索Pascal三角变换的重要属性。基于这些结果，阶B样条小波的希尔伯特变换的显式形式$ m $成立。此外，还讨论了B样条小波的希尔伯特变换的消失矩，对称性和渐近性。为了证明这些结果的有效性，以下是两个例子$ m = 3 $ 和 $ m = 4 $给出了这两个B样条小波的图以及它们的希尔伯特变换。这两种情况$ m = 3 $ 和 $ m = 4 $ 提供两个小波的希尔伯特变换对，可用于数字图像处理。