Given that $f,g$ are trigonometric polynomial of this form $a_0 + \sum \limits_{n=1}^N a_n \sin( 2 \pi n x)+ b_n \cos(2 \pi n x)$,
Prove that $f*g (x) = \int \limits_{0}^{1} f(x-t) g(t)\,dt$ is also trigonometric polynomial
My proof : from the linearity of the integral and the trigonometric polynomials, its suffice to prove that $ a*b , a*\sin(2 \pi n x) , a*\cos(2 \pi n x) , \sin(2 \pi k x)* \sin (2 \pi n x) , \cos(2 \pi k x) * \cos(2 \pi n x) , \sin(2 \pi k x) * \cos(2 \pi n x)$ are trigonometric polynomials .
I have 2 questions , is this proof correct ?
and also what is $\sin(2 \pi k x) * \cos(2 \pi n x)$ ?
Note : $a,b$ are real numbers and $k,n$ are positive integers.