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.

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与系数反问题的凸化方法相关的级数的收敛

本文关注与特定版本的凸化方法相关的级数的收敛。该版本最近由第一作者的研究小组开发，用于解决系数逆问题。凸化方法旨在构建一个全局凸的类 Tikhonov 函数，其中包含 Carleman 权重函数。在之前的工作中，严格凸加权的类吉洪诺夫泛函的构造假设了一个由总波场产生的函数的截断傅立叶级数（即有限级数而不是无限级数）。在本文中，我们证明了这种截断傅立叶级数近似的收敛性。更确切地说，我们表明使用截断傅立叶级数获得的近似 PDE 的残差在 $L^{2}$ 中趋于零，因为截断傅立叶级数中的截断指数趋于无穷大。该证明依赖于 $H^{1}$-norm 中的收敛结果，用于在由特殊傅立叶基的元素跨越的有限维子空间上的 $L^{2}$-正交投影序列。然而，由于系数逆问题的不适定性质，我们无法证明由类 Tikhonov 函数的最小化产生的近似 PDE 的解收敛到正确解。该证明依赖于 $H^{1}$-norm 中的收敛结果，用于在由特殊傅立叶基的元素跨越的有限维子空间上的 $L^{2}$-正交投影序列。然而，由于系数逆问题的不适定性质，我们无法证明由类 Tikhonov 函数的最小化产生的近似 PDE 的解收敛到正确解。该证明依赖于 $H^{1}$-norm 中的收敛结果，用于在由特殊傅立叶基的元素跨越的有限维子空间上的 $L^{2}$-正交投影序列。然而，由于系数逆问题的不适定性质，我们无法证明由类 Tikhonov 函数的最小化产生的近似 PDE 的解收敛到正确解。