Definition. A circle mesh with normalized pairing [fdip-004H]

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Supply the unit circle \(\mathbb R/\mathbb Z\), its coordinate \(x\) modulo one and length measure \(dx\). For an integer \(n\geq 3\), the circle mesh consists of the points \(j/n\), with \(j\in \mathbb Z/n\mathbb Z\), and spacing \(h_n=1/n\). Its real field space and reference pairing are \[E_n=\mathbb R^{\mathbb Z/n\mathbb Z},\qquad g_{0,n}(f,g)=\frac 1n\sum _{j=0}^{n-1}f_jg_j.\] On the complexification use \(\langle f,g\rangle _n=n^{-1}\sum _j\overline {f_j}g_j\); its restriction to real fields is \(g_{0,n}\).

Fix a mass squared \(\mu >0\), independent of \(n\). Define the cyclic shift, scaled nearest-neighbor Laplacian and precision by \[\begin {aligned} (T_nf)_j&=f_{j+1},\\ L_n&=n^2(2I-T_n-T_n^{-1}),\\ Q_n&=\mu I+L_n,\qquad q_n(f)=g_{0,n}(f,Q_nf). \end {aligned}\] Indices are taken modulo \(n\). These specialize the cyclic shift and difference energy, with the supplied scale \(n^2\). The scalar \(L_n\) is the nearest-neighbor operator compared with, and generally different from, the centered Dirac square in Proposition [fdip-000T].

Periodic reindexing gives \(T_n^*=T_n^{-1}\) in the normalized pairing. Consequently \[\langle f,L_nf\rangle _n =n\sum _{j=0}^{n-1}|f_{j+1}-f_j|^2\geq 0.\] Thus \(Q_n\geq \mu I\) and \(Q_n\) is invertible, including on constant fields. This is the positive precision convention with the declared \(g_{0,n}\), rather than a change of the action form.

The step-function interpolation \(I_n\) assigns to \(f\) the function with value \(f_j\) whenever \(j/n\leq x<(j+1)/n\), using the representative \(0\leq x<1\). It maps real or complex fields into \(L^2(\mathbb R/\mathbb Z,dx)\), the square-integrable functions modulo almost-everywhere equality. Each interval has length \(1/n\), so \[\|I_nf\|_{L^2}^2=\frac 1n\sum _j|f_j|^2=\|f\|_n^2.\] Thus the comparison preserves the specified norm.