Let $X_{\rho }$ be a modulated convergence space, that is, a modular space equipped with a sequential convergence structure. Given an element $x$ of $X_{\rho }$, we consider the minimisation problem of finding $x_0\in C$ such that $\rho (x-x_o)=inf\{\rho (x-y):y\in C\}$, where $\rho$ is a convex modular and $C$ is a closed convex subset of $X_{\rho }$. Such an element $x_0$ is called a best approximant. We prove existence and uniqueness of such a best approximant in a large classes of modulated convergence spaces, provided $\rho$ is uniformly convex. We also touch upon an interesting subject of semicontinuity of the related modular projection. Problems of finding best approximants are important in approximation theory and probability theory. In particular, we show how our results can be applied to the nonlinear prediction theory.

让 $X_{\rho }$是一个调制的会聚空间，也就是配备有顺序会聚结构的模块化空间。给定一个元素$X$ 的 $X_{\rho }$，我们考虑了发现的最小化问题 $X_0\in C$ 这样 $\rho （X-X_{Ø}）=信息\{\rho （X-ÿ）：ÿ\in C\}$，在哪里 $\rho$ 是凸模块 $C$ 是...的闭合凸子集 $X_{\rho }$。这样的元素$X_0$被称为最佳近似值。我们证明了在一类最佳的调制收敛空间中，这种最佳近似的存在性和唯一性$\rho$均匀地凸。我们还谈到了相关模块投影的半连续性的有趣主题。寻找最佳近似值的问题在近似理论和概率论中很重要。特别是，我们展示了如何将我们的结果应用于非线性预测理论。