Splines of complex order, fractional operators and applications to signal and image processing

勋伯格的样条已经进入了大量的应用数学领域，例如，信号处理、计算机图形学、计算机辅助几何设计、偏微分方程式的数值等等。它们具有灵活的参数，可以适应手头的特定任务，并且因为它们的表示简单，所以很容易实现。这些属性对于成功的应用程序是必不可少的。此外，由于样条线是分段多项式，它们便于对数值结果的解释。近年来，分数阶样条和复数阶样条被定义，由于附加参数的存在，使得它具有更强的适应性。连续序参数允许对问题的规律性和相位和幅度的提取的复杂程度进行精确调整。从分数阶微分算子、Dirichlet平均、Riesz变换和高维相位，到用于图像分析的曲率检测算子，这些样条线产生了新的数学互连。对于具体应用，关于复数阶样条所生成空间的逼近理论性质的完整的理论和数值分析仍然缺乏。我们希望通过提供复杂阶样条曲线的全面分析来弥合这一差距。为此，我们将着重于理论结果和数值分析，以及算法的实现。

Schoenberg's splines have found their way into a large number of applied mathematical areas, for example, signal processing, computer graphics, computer assisted geometric design, the numerics of partial differential equations and many more. With their flexible parameters they are adaptable to the particular task at hand and because of their simple representation they are easily implementable. These are properties that are imperative for a successful application. In addition, as splines are piecewise polynomials, they facilitate the interpretation of numerical results. In recent years, splines of fractional and complex order were defined, which, due to the existence of additional parameters, are even more adaptable. The continuous order parameter allows a precise adjustment to the regularity of the problem and the complex degree the extraction of phase and amplitude. These splines generate new mathematical interconnections ranging from fractional differential operators, Dirichlet averages, Riesz transformations and phase in higher dimensions, to curvature detection operators for image analysis.For concrete applications, the complete theoretical and numerical analyses of the approximation-theoretic properties of the spaces generated by splines of complex order are still missing. We want to close this gap by providing a comprehensive analysis of splines of complex order. For this purpose, we will emphasize the theoretical results and numerical analysis, as well as the algorithmic implementation.