Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.

**1)** Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times \mathbb{R}\to\mathbb{R},\ \varphi\in C^{1}$ such that the sequence of curves:

$$\gamma_n:\mathbb{R}\to\mathbb{C},\ \gamma_{n}(t)=\sum_{k=0}^n r_k e^{i\varphi(k,t)}$$

converges uniformly to $\gamma$ on $\mathbb{R}$? Moreover can we choose $(r_n)_{n\in\mathbb{N}}$ to be decreasing or strictly decreasing? Can we choose $\varphi (x,t)=2x\pi t+f(x),\forall (x,t)\in\mathbb{R}\times \mathbb{R}$, where $f$ is a $C^1$ function?

**2)** Is it true that there is a function $r:\mathbb{R}\to \mathbb{C},\ r\in L^1 (\mathbb{R})$ such that $r(k)=r_k e^{if(k)},\ \forall k\in\mathbb{N}$ and

$$\gamma(t)=\int_{-\infty}^{\infty}r(x)e^{i2\pi x t} dx\ ?$$

I'm trying to prove this theorem: http://www.u.arizona.edu/~aversa/scholastic/Mathematical%20Power%20of%20Epicyclical%20Astronomy%20%28Hanson%29.pdf

See the begining of the last page. You can take a look also, at:https://www.youtube.com/watch?v=QVuU2YCwHjw

thatcontext, or at least talking in a way that allows reference to it, would help people answer usefully... $\endgroup$