  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.
