Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$
Regard the unitary transformation: $$U:\mathcal{L}^2(-\pi,\pi)\to\ell^2(\mathbb{Z}):(Uu)_n:=\int_{-\pi}^\pi u(y)\frac{1}{\sqrt{2\pi}}e^{iny}\mathrm{d}y$$ One obtains the representation as: $$(U^*\Delta Uu)(x)=\cos(x)u(x)$$
Now, consider the discrete Laplace with boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ How to find a representation as multiplication operator here?