Consider the following functions: for $x \in [-1/2,1/2] \subset \mathbb{R}$, $f_{0}(x) = 1$, $f_{2k-1}(x) = \sqrt{2}\sin((2k-1)\pi x)$ and $f_{2k}(x) = \sqrt{2}\cos(2k\pi x)$, for $k \in \mathbb{N}^{+}$. This family of function is an orthonormal in $L^{2}([-1/2,1/2])$.
Using these functions, I obtained the following rule:$$\int dx f_{k}(x) f_{t}(x+y) = \delta_{k,t}f_{t}(y),$$where $\delta_{k,t}$ is the Kronecker delta. To prove this, I simply used the orthonormality of the functions. For instance:$$\int dx \cos(k \pi x)\sin(t\pi (x+y)) = \int dx \cos(k \pi x)(\sin(t\pi x)\cos(t\pi y)+\sin(t\pi y)\cos(t\pi x)) = \sin(t \pi y)\int dx \cos(k\pi x)\cos(t\pi x) = \delta_{k,t}\sin(t \pi y).$$
Is my reasoning correct? It is my first time proving results like this.