这篇论文揭示了广义样条与高斯过程在无限维逆问题中的等价关系，能连接多种已知数学方法。
该论文展示了有限维线性逆问题中的最小均方误差估计器如何扩展到无限维设置。广义样条作为线性回归器，核空间S上的广义高斯过程对应高斯随机向量。研究引入了白化/正则化算子L，其连续扩展诱导出希尔伯特空间H。该方法能恢复所有已知等价实例，包括Kailath及其学生开发的创新方法和核希尔伯特空间方法，以及分数样条与Mandelbrot分数布朗运动之间的数学对应关系。
Generalized Splines and Gaussian Processes
For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.