Regular-grid spatial periodogram and smoothed estimator

Let $Y=(Y_j{)}_{j\in {\mathbb{Z}}^d}$ satisfy the constant-mean, lag-dependent-covariance conditions from Lagged lattice covariance and variogram estimators, with known mean $\mu$ (set $\mu =0$ for a centered field), and use the frequency torus ${\mathbb{T}}^d$ identified with $\widehat{{\mathbb{Z}}^d}$ by The Pontryagin dual of the integer lattice is the torus. Represent frequency coordinates $\lambda \in [-\pi ,\pi {)}^d$ by the circle tuple $(e^{i{\lambda }_1},\dots ,e^{i{\lambda }_d})$; coordinates and integration are periodic modulo $2\pi$ in each coordinate. For $n\ge 1$, set ${\mathrm{\Lambda }}_n:=\{0,1,\dots ,n-1{\}}^d,N_n:=n^d,$ and let the $n^d$ Fourier-grid frequencies be ${\lambda }_{n,q}:=2\pi q\mathrm{/}n(\mathrm{m}\mathrm{o}\mathrm{d}2\pi ),q\in {\mathrm{\Lambda }}_n\mathrm{.}$ The known-mean regular-grid periodogram is $I_n(\lambda ):=\frac{1}{(2\pi {)}^d{N}_n}{\mid {\sum }_{j\in {\mathrm{\Lambda }}_n}(Y_j-\mu )e^{-ij\cdot \lambda }\mid }^2,\lambda \in {\mathbb{T}}^d\mathrm{.}$ Integration over ${\mathbb{T}}^d$ uses product angular Lebesgue measure on the fundamental cube $[-\pi ,\pi {)}^d$ (The product measure of two sigma-finite measure spaces).

For smoothing, let $K:{\mathbb{R}}^d\to [0,\mathrm{\infty })$ be continuous, even, compactly supported, and satisfy $K(0)>0$. For $b>0$, periodize the rescaled kernel by $K_b^{\mathrm{p}\mathrm{e}\mathrm{r}}(x):={\sum }_{m\in {\mathbb{Z}}^d}{b}^{-d}K((x+2\pi m)\mathrm{/}b)\mathrm{.}$ Because $K$ has compact support, only finitely many summands are nonzero for each fixed $x$, so this periodization is well-defined. For $\lambda \in {\mathbb{T}}^d$, when the denominator is positive, set $w_{n,q,b}(\lambda ):=\frac{K_b^{\mathrm{p}\mathrm{e}\mathrm{r}}(\lambda -{\lambda }_{n,q})}{{\sum }_{r\in {\mathrm{\Lambda }}_n}{K}_b^{\mathrm{p}\mathrm{e}\mathrm{r}}(\lambda -{\lambda }_{n,r})},{\tilde{f}}_{n,b}(\lambda ):={\sum }_{q\in {\mathrm{\Lambda }}_n}{w}_{n,q,b}(\lambda )I_n({\lambda }_{n,q})\mathrm{.}$ The smoother is undefined when that denominator vanishes. Its target, when a spectral density exists, is the density of the lattice-sampled field on the dual torus; no unaliased continuous-space spectral-density target is part of this definition.