Extended Sobolev scale on \(\mathbb {Z}^n\)

In analogy with the definition of “extended Sobolev scale" on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ by Mikhailets and Murach, working in the setting of the lattice \(\mathbb {Z}^n\) ${\mathbb{Z}}^n$ , we define the “extended Sobolev scale" \(H^{\varphi }(\mathbb {Z}^n)\) $H^{\phi }({\mathbb{Z}}^n)$ , where \(\varphi \) $\phi$ is a function which is RO-varying at infinity. Using the scale \(H^{\varphi }(\mathbb {Z}^n)\) $H^{\phi }({\mathbb{Z}}^n)$ , we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces \([H^{(s_0)}(\mathbb {Z}^n), H^{(s_1)}(\mathbb {Z}^n)]\) $[H^{(s_0)}({\mathbb{Z}}^n),H^{(s_1)}({\mathbb{Z}}^n)]$ , with \(s_0<s_1\) $s_0<s_1$ . We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale \(H^{\varphi }(\mathbb {Z}^n)\) $H^{\phi }({\mathbb{Z}}^n)$ . Furthermore, starting from a first-order positive-definite (discrete) PDO A of elliptic type, we define the “extended discrete A-scale" \(H^{\varphi }_{A}(\mathbb {Z}^n)\) $H_A^{\phi }({\mathbb{Z}}^n)$ and show that it coincides, up to norm equivalence, with the scale \(H^{\varphi }(\mathbb {Z}^n)\) $H^{\phi }({\mathbb{Z}}^n)$ . Additionally, we establish the \(\mathbb {Z}^n\) ${\mathbb{Z}}^n$ -analogues of several other properties of the scale \(H^{\varphi }(\mathbb {R}^n)\) $H^{\phi }({\mathbb{R}}^n)$ .