Example. Fourier modes and their Gaussian moments on a circle mesh [fdip-004I]

✍️sourceAGENTDRAFTED

For the circle mesh, let \(k\in \mathbb Z\) and set \[e_k(x)=e^{2\pi ikx},\qquad e_{n,k}(j)=e^{2\pi ikj/n}.\] Every entry has modulus one, hence \(\|e_{n,k}\|_n=1\). Applying the shift and its inverse directly gives \[\begin {aligned} T_ne_{n,k}&=e^{2\pi ik/n}e_{n,k},\\ L_ne_{n,k}&=\lambda _{n,k}e_{n,k},\\ \lambda _{n,k} &=n^2(2-e^{2\pi ik/n}-e^{-2\pi ik/n})\\ &=4n^2\sin ^2(\pi k/n). \end {aligned}\] This derives the scaled version of the nearest-neighbor symbol. Frequencies differing by \(n\) give the same samples. For fixed \(k\ne 0\), the condition \(n>2|k|\) separates \(k\) from \(-k\) modulo \(n\).

For \(k=0\), \(e_{n,0}=\mathbf 1\) and \(\lambda _{n,0}=0\). For \(k=1\), a four-point mesh has \(\lambda _{4,1}=32\); an eight-point mesh has \(\lambda _{8,1}=64(2-\sqrt 2)\). The picture shows the supplied unit circle sampled at 4 equally spaced points. Its arrows indicate increasing \(j\); they do not specify a dynamics.

Now use real fields and the Gaussian law \[d\nu _n(f)=Z_n^{-1}e^{-q_n(f)/2}\,d\operatorname {vol}_{g_{0,n}}(f).\] In raw coordinates the volume and normalization are \[\begin {aligned} d\operatorname {vol}_{g_{0,n}}&=n^{-n/2}\,df_0\cdots df_{n-1},\\ Z_n&=(2\pi )^{n/2}(\det Q_n)^{-1/2}. \end {aligned}\] The volume factor follows because \(f\mapsto f/\sqrt n\) is an orthonormal chart; the partition function is Theorem [fdip-002T]. The covariance operator relative to \(g_{0,n}\) is \(Q_n^{-1}\) by Theorem [fdip-000B]. Since the exponent is \(f^TQ_nf/(2n)\), the raw coordinate covariance matrix is \(nQ_n^{-1}\).

Define the complex Fourier observable \(a_{n,k}=n^{-1}\sum _j e^{-2\pi ikj/n}f_j\). It has zero mean. Writing \(v=e_{n,k}\) and \(v^*\) for its conjugate transpose, the coordinate covariance gives \[\begin {aligned} \mathbb E_{\nu _n}|a_{n,k}|^2 &=n^{-2}v^*(nQ_n^{-1})v\\ &=\frac 1{\mu +\lambda _{n,k}}, \end {aligned}\] because \(v^*v=n\). All real and imaginary parts and their products are integrable polynomials by Proposition [fdip-002W]. In particular the real mean \(a_{n,0}\) has variance \(1/\mu \). At \(k=1\), the four-point and eight-point second moments are respectively \((\mu +32)^{-1}\) and \((\mu +64(2-\sqrt 2))^{-1}\).

For \(k\ne 0\) and \(n>2|k|\), put \[\begin {aligned} c_{n,k}&=\operatorname {Re}a_{n,k} =\frac 1n\sum _j\cos (2\pi kj/n)f_j,\\ s_{n,k}&=-\operatorname {Im}a_{n,k} =\frac 1n\sum _j\sin (2\pi kj/n)f_j. \end {aligned}\] The geometric sum \(\sum _j e^{4\pi ikj/n}=0\): its ratio is not one, its \(n\)-th power is one, and multiplying the sum by one minus the ratio gives zero. Taking real and imaginary parts, and using the double-angle identities, gives normalized squared norms \(1/2\) for both the cosine and sine vectors and their orthogonality. Both vectors have precision eigenvalue \(\mu +\lambda _{n,k}\), so Theorem [fdip-000B] yields \[\operatorname {Var}(c_{n,k})=\operatorname {Var}(s_{n,k}) =\frac 1{2(\mu +\lambda _{n,k})}.\] The coefficients of the orthogonal projection \(A\cos (2\pi kj/n)+B\sin (2\pi kj/n)\) are \(A=2c_{n,k}\), \(B=2s_{n,k}\). Their variances are 4 times the displayed projection variances.

The character calculation and normalization above are direct deductions from the cited finite operators and Gaussian results. Mathlib's ZMod.dft uses the counting-measure forward sum; \(a_{n,k}\) has the additional factor \(1/n\). This formal Fourier transform is a supporting construction, not a formalization of the mesh limit.