Approximating a function with a finite series, e.g., involving polynomials or trigonometric functions, is a critical tool in computing and data analysis. The construction of such approximations via now-standard approaches like least squares or compressive sampling does not ensure that the approximation adheres to certain convex linear structural constraints, such as positivity or monotonicity. Existing approaches that ensure such structure are norm-dissipative and this can have a deleterious impact when applying these approaches, e.g., when numerical solving partial differential equations. We present a new framework that enforces via optimization such structure on approximations and is simultaneously norm-preserving. This results in a conceptually simple convex optimization problem on the sphere, but the feasible set for such problems can be very complex. We establish well-posedness of the optimization problem through results on spherical convexity and design several spherical-projection-based algorithms to numerically compute the solution. Finally, we demonstrate the effectiveness of this approach through several numerical examples.

翻译：以有限序列(例如,涉及多数值或三角函数)对等函数进行匹配,是计算和数据分析的一个关键工具。通过现在的标准方法(如最小正方或压缩抽样)构建这种近似,并不能确保近近似符合某些二次曲线线性结构限制,例如主动性或单一性。现有的方法确保这种结构具有标准差异性,在应用这些方法时可能会产生有害影响,例如,在数字解决部分差异方程式时。我们提出了一个新框架,通过优化近似结构来实施,同时保留规范。这在概念上造成一个简单矩形优化问题,但针对此类问题的可行办法可能非常复杂。我们通过球形凝结结果来确定优化问题的准确性,并设计若干基于球形预测的算法,以便用数字来计算解决方案。最后,我们通过几个数字示例来展示这一方法的有效性。