Is it possible to have a sine series with $b_1 \sin(x/2) + b_2 \sin(3x/2) + \cdots$ instead of $b_1 \sin(x) + b_2 \sin(2x) + \cdots$?

Question

If we have an odd function $f(x)$ defined on $(-\pi, \pi)$, we have a sine series for it $$f(x) \sim \sum_{k=1}^\infty b_k \sin(kx),$$ with $$b_k = \frac{2}{\pi}\int_0^\pi f(x) \sin(kx)\,dx.$$ Is it possible to have a sine series for the same function $f(x)$ of the form $$f(x) \sim \sum_{k=1}^\infty d_k \sin\left(\frac{2k+1}{2}x\right)?$$

Answer 1

Let $g : \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ be defined as $g(x) = f(2x)$. No expand it to $(-\pi, \pi)$ such that $g(x) = g(\pi-x)$ for $x > 0$ and $g(x) = g(-\pi-x)$ for $x < 0$. These conditions guarantee that in the expansion

$$g(x) \sim \sum_{k=1}^{\infty} b_k \sin(kx)$$

the even terms are zero, i.e.

$$f(2x) \sim \sum_{k=0}^{\infty} b_{2k+1} \sin( (2k+1) x )$$

so

$$f(x) \sim \sum_{k=0}^{\infty} b_{2k+1} \sin \left( \frac{2k+1}{2} \cdot x \right).$$