By restricting the generalized Fourier transform and convolution on time scales \(h\mathbb{Z}\), \(h>0\) to even and odd functions \(f: h\mathbb{N}^0\to\mathbb{C}\), \(\mathbb{N}^0=\mathbb{N}\cup\{0\}\), we can define the \(h\)-Fourier sine and cosine transforms \(F_s\{f\}\) and \(F_c\{f\}\) of \(f\) as follows. \(F_s\{f\}(\omega)=2h\sum_{m=1}^\infty f(nh)\sin(\omega nh)\) for \(\omega\in[0,\frac{\pi}{h}]\), and \(F_c\{f\}\) is given by replacing \(\sin(\omega nh)\) by \(\cos(\omega nh)\) and adding \(hf(0)\). Moreover, if we define the \(h\)-Fourier sine generalized convolution of \(f*g\) on \(h\mathbb{N}^0\) by \[(f*g)(t)=h\sum_{m=1}^\infty f(mh)\big(g(\vert t-mh\vert)-g(t+mh)\big),\] then \(F_s\{f*g\}(\omega)=F_s\{f\}(\omega)F_c\{g\}(\omega)\) provided that \(f, g\in\ell_1(h\mathbb{N}^0)\) and \(f(0)=0\). As an application, a unique solution of a discrete version of the Toepliz-Hankel type integral equation \(x*k=z\) on \(h\mathbb{N}^0\) is given by \(z*u\), where \(u\) is the inverse \(u\) of \(1/F_c\{k\}\), provided that \(k, z\in\ell_1(h\mathbb{N}^0)\), \(F_c\{k\}(\omega)\neq0\) for all \(\omega\in[0,\frac{\pi}{h}]\), and \(u\in\ell_1(h\mathbb{N}^0)\). Similarly, a unique solution of the equation with \(x(nh)\) added to the left hand side is given.