This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in $L^{2}$ as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the $H^{1}$-norm for a sequence of $L^{2}$-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

翻译：本文研究某类凸化方法中一个级数的收敛性问题。该类方法由第一作者所在研究团队近期为求解系数反问题而提出。凸化方法的核心在于构造一个包含卡尔曼权函数的全局凸Tikhonov型泛函。在以往工作中，这种严格凸加权Tikhonov型泛函的构造需假设总波场生成函数对应截断傅里叶级数（即有限项级数而非无穷级数）。本文证明了该截断傅里叶级数近似的收敛性质。具体而言，我们证明了当截断傅里叶级数的截断指标趋于无穷时，由截断傅里叶级数得到的偏微分方程近似解残差在$L^{2}$范数下趋于零。该证明依赖于一个特殊傅里叶基元素张成的有限维子空间上$L^{2}$正交投影序列在$H^{1}$范数下的收敛性结论。然而，由于系数反问题的病态本质，我们无法证明通过极小化该Tikhonov型泛函获得的近似偏微分方程解会收敛到真解。