Another relation between the cosine and Dirac's delta function

Question

I am looking for a smooth function $f_\sigma(x)$ with parameter $\sigma \in [0,1]$, such that

$$\lim_{\sigma \rightarrow 1}\ f_\sigma(x) = \cos(\pi x)$$

and

$$\lim_{\sigma \rightarrow 0}\ f_\sigma(x) = \sum_{i = -\infty}^{\infty}(-1)^i\ \delta(x-i) =: D(x)$$

i.e. a series of alternating delta peaks, superelevating the peaks of the cosine.

The relation between cosine and Dirac's delta function $\delta$ was given by Fourier:

$$\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ dp\ \cos(px)$$

You can also consider $\delta$ as the limit

$$\delta(x) = \lim_{\sigma \rightarrow 0} \frac{1}{\sqrt{2\pi}\sigma} e^{-x^2/2\sigma^2}$$

Given this definition of $\delta$ there is a straight-forward and somehow "trivial" solution:

$$f_\sigma(x) = \sigma\cos(\pi x) + (1-\sigma)\sum_{i = -\infty}^{\infty}(-1)^i\ \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-i)^2/2\sigma^2}$$

Question 1: Is there a more "simple", "natural" family of functions $f_\sigma(x)$, esp. without infinite sums?

Question 2: What is the Fourier transform of $D(x)$?

Answer 1

Take $\varphi(x) = e^{-\pi x^2}$ or any function you like

Let $$\psi_\alpha(x) = \frac{1}{\alpha }\sum_n (-1)^n \varphi( (x- n)/\alpha)$$ $$F_\alpha(x) = (1-\alpha)\psi_\alpha(x) +\alpha \cos(\pi x), \qquad\qquad f_\sigma = F_{\displaystyle e^{-\sigma}}$$

Answer 2

Thanks to @reuns' request (what I mean with "no infinite sum") I could clarify what I mean, and this is what I came up with: The infinite sum is only necessary to make the function periodical. But the function is a periodical repetition of one half-period $f_\sigma(x)$, $x \in [0,1]$ which is "simple" and "natural":

$$f_\sigma(x) = \sigma\cos(\pi x) + \frac{1-\sigma}{\sqrt{2\pi}\sigma} (e^{-x^2/2\sigma^2} - e^{-(x-1)^2/2\sigma^2})$$

with $f_\sigma(0.5) \equiv 0$. (Obviously, considering only the quarter-period $[0,0.5]$ would suffice.)

The quest for the Fourier transform of $D(x)$ (see original question) remains open, i.e. for the most "simple" presentation – the presentations of the Fourier transforms of the cosine and Dirac's delta function being examples of being "simple" (resp. "trivial").