Let any function of the form $x \mapsto 1-\frac{2x}{L}$ for $x \in (kL, (k+1)L)$ for all integers $k$ and some period $L > 0$ be referred to as periodic sawtooth.
Let $\mathcal{S}$ be the space of sawtooth functions of arbitrary periods closed under finite linear combinations.
I'm working on a problem in which I could greatly benefit from a result similar to the following:
Let $f(x)$ be a piecewise continuous periodic function with period $L$. Then there exists a sequence of sawtooths $(s_n)_{n \geq 1}$ drawn from $\mathcal{S}$ such that $$ f(x) = \lim_{n \to \infty}s_n(x) $$ for all $x \in \mathbb{R}$, i.e. the convergence is pointwise.
This question seemed to have already been asked, but on a closer inspection it seems that the author of that post didn't have the additional requirement of the sawtooth being periodic, and thus had the freedom of building piece-wise linear approximations by scaling, reverting and translating individual sawtooths.
My attempt: Periodicity suggests that Fourier Convergence Theorem could be useful. Indeed, if I could approximate just $\cos(x)$, I think I could make the rest of it work. I tried building a "periodic step function" (a step function that repeats itself periodically) out of sawtooths, but either I'm missing something, or it's really hard.
Questions:
- Would anyone happen to have any insight on the feasibility of the above blockquoted problem?
- Does anyone have any idea whether periodic sawtooth could be used to approximate at least $\cos(x)$? Uniform convergence is not necessary, in fact I would readily settle for pointwise convergence.
- Does anybody know whether it is possible to use periodic sawtooth to build a periodic step function? From that, the rest would follow -- or so it seems to me.