If the fourier sine transform of $f(x)$ is
$$F_s(f(x))=\int_0^\infty f(x) \sin(\omega x) \, dx$$
and the fourier sine inverse of a function is $$f(x)=\frac 2 \pi \int_0^\infty F(\omega) \sin(\omega x) \, d\omega$$
I want to know whether the convolution and multiplication rules of Fourier Transform can be applied using Fourier sine and cosine also .. I mean if I want the inverse Fourier sine of $$F(\omega)G(\omega)$$ then will it be $$f(t)*g(t)=\int_{-\infty}^\infty f(\tau)g(x-\tau) \, d\tau$$ ??