Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $% \frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let $S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim_{n\rightarrow \infty }a_{n}=0$, then% \begin{equation*} \lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0 \text{ if and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0. \end{equation*}% We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi }$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the MVBV condition.

In this paper we consider solutions to the free Schr\" odinger equation in $n+1$ dimensions. When we restrict the last variable to be a smooth function of the first $n$ variables we find that the solution, so restricted, is locally in $L^2$, when the initial data is in an appropriate Sobolev space.