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：本文关注与某种版本的混凝法相关的一系列序列的趋同性。 该版本是第一位作者的研究组最近为解决反系数问题而开发的。 混凝土法旨在与其中的Carleman Weight 函数构建一个类似于 Tikhonov 的全球性 convex Tikhonov- 类似功能。 在先前的工程中, 完全civec加权的 Tikhoonov 类似功能的构建假设是一个短短的 Fourier 序列( 即一个有限序列, 而不是一个无限的系列), 用于由整个波字段生成的函数。 在本文中, 我们证明, 这个扭曲的 Fourier 序列的趋同性近似性。 更确切地说, 我们显示, 使用调的 Fourier 序列的近似 PDE 函数函数的剩余值为零 $L<unk> 2} 。 在短调的 Fourervier 序列中, 最小的调指数指数值指数值指数的构建为无限。 证据取决于 $H_1} NOthalnalnalnalalalalal resal rotispal resmission sal resmissue resmissueal romaismismismismismismismismismismismism 。 y prom</s>