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The Fourier Ratio on Resistance Spaces and Applications to Electrical Networks
Christopher Housholder, Alex Iosevich, Steven J. Miller, Eyvindur Pálsson
informal web version / working manuscript
Abstract
We study the Fourier Ratio for spectral expansions associated with nonnegative self-adjoint operators and relate it to eigenvalue growth, sampling, and resistance geometry. A weighted Cauchy–Schwarz argument shows that Weyl-type counting estimates control the Fourier Ratio of spectrally regular functions. Combined with standard bounded-orthonormal-system recovery results, this yields stable reconstruction from incomplete random point samples. On compact measured resistance spaces, the spectral term admits a sharper geometric interpretation: the full reciprocal eigenvalue sum is exactly one half of the average resistance distance. This identity gives Fourier Ratio bounds independent of the spectral cutoff, implies sublinear eigenvalue counting, and forces spectral dimension strictly below two whenever a modified Weyl law is available. We apply the resulting estimates to the Sierpinski gasket and related self-similar fractals, and then to finite resistive electrical networks, where energy becomes dissipated power and resistance distance becomes effective resistance. We also record an independent equidistribution principle for Fourier Ratios of finite families.
Contents
This page follows the manuscript closely. It is written as mathematical exposition rather than as a summary page.
1 Introduction
The Fourier Ratio has recently been used as a quantitative measure of spectral complexity in problems involving recovery, approximation, uncertainty, localization, learning, and continuous Fourier analysis [1, 2, 3, 4, 5]. In the discrete setting, if \(\hat f\) denotes the coefficient vector of a signal in an orthonormal Fourier expansion, one sets \[\operatorname{FR}(f) \,=\, \frac{\norm{\hat f}_1}{\norm{\hat f}_2}.\] The ratio behaves like the square root of an effective support size. In particular, it can remain small even when the coefficient vector is not literally sparse, and it interfaces naturally with the standard \(\ell^1\) recovery theory of compressed sensing [6, 7, 8, 10, 11, 12].
The question considered here is how classical spectral and geometric information can produce Fourier Ratio bounds without imposing sparsity as an independent hypothesis. The basic mechanism is simple. For a nonnegative self-adjoint operator \(A\) with eigenvalues \(\lambda_j\), weighted Cauchy–Schwarz separates the nonconstant coefficient sum into a regularity term involving \(A^{s/2}f\) and a reciprocal spectral sum \[\sum_{0<\lambda_j\leq L}\lambda_j^{-s}.\] Eigenvalue counting then controls the second factor. Thus Weyl-type information, usually used to describe the geometry or dimension of the underlying space, also gives a quantitative bound on spectral compressibility.
Writing \[N_A(\Lambda) \,=\, \#\{j:\lambda_j\leq\Lambda\},\] we show that an estimate \(N_A(\Lambda)\lesssim \Lambda^\alpha\) leads to three regimes according as \(s>\alpha\), \(s=\alpha\), or \(s<\alpha\). The borderline case produces logarithmic growth, while the supercritical regularity range gives a Fourier Ratio bound independent of the cutoff. This is the spectral analogue of the principle that regularity enforces compressibility in the discretization results of [4].
The resistance-space setting gives a stronger conclusion. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space and let \(A=-\Delta_\mu\) be the associated resistance Laplacian. The form identity identifies \(\norm{A^{1/2}f}_2^2\) with the energy \(\mathcal E(f,f)\). More importantly, we prove \[\sum_{j=1}^{\infty}\frac{1}{\lambda_j} \,=\, \frac{1}{2}\int_X\int_X R(x,y)\dd\mu(x)\dd\mu(y).\] Thus the reciprocal spectral term is exactly one half of the average resistance distance. The resulting Fourier Ratio estimate is uniform in the spectral cutoff and depends only on the mean of \(f\), its \(L^2\) norm, its energy, and the resistance geometry of the space. The same resistance metric controls eigenfunction size and perturbations of sampling locations, so the geometric input reappears on the measurement side of the recovery problem.
There is also a spectral consequence. Since the reciprocal eigenvalue sum is finite on every compact measured resistance space, one has \[N_A(\Lambda)=o(\Lambda).\] Consequently, if a self-similar fractal admits a modified Weyl law \[N_A(\Lambda)=\bigl(G(\log\Lambda)+o(1)\bigr)\Lambda^{d_s/2},\] with \(G\) bounded above and bounded away from zero, then necessarily \(d_s<2\). For the standard Sierpinski gasket this recovers \[N_A(\Lambda)\asymp \Lambda^{\log 3/\log 5}, \qquad d_s=\frac{2\log 3}{\log 5}<2,\] up to the usual bounded log-periodic oscillation [22, 21, 23]. The resistance identity nevertheless gives the stronger conclusion needed here: the Fourier Ratio bound itself is already independent of the cutoff. We record analogous consequences for the Sierpinski tetrahedron and the Minkowski curve.
Once a Fourier Ratio bound is available, standard bounded-orthonormal-system estimates convert it into a sampling statement. We use this only as a recovery mechanism, rather than as a new reconstruction algorithm: the contribution is to supply explicit complexity and coherence bounds from spectral growth and resistance geometry. In particular, random point samples recover spectrally truncated functions by \(\ell^1\) minimization, and perturbing a sampling location by resistance distance at most \(\delta\) changes a finite-energy signal by at most \(O(\delta^{1/2}\mathcal E(f,f)^{1/2})\).
Finite resistive electrical networks provide a direct specialization. With the uniform measure on the vertices, the resistance-space operator is a normalized weighted graph Laplacian, the resistance metric is effective resistance, and the energy is the dissipated power of a voltage profile [31, 32, 33]. The resistance constant is expressed through the Kirchhoff index, while the network equation \(L_Gv=i\) gives an additional inverse-Laplacian filtering effect for current-driven voltages. This yields Fourier Ratio, reconstruction, approximation, and sensor-stability estimates in electrical variables.
Finally, we record an independent equidistribution result for finite coefficient families. If a fixed continuous function is evaluated on point sets whose empirical measures converge weakly, then the normalized Fourier Ratio converges to an explicit ratio of \(L^1\) and \(L^2\) integrals against the limiting measure. This part is separate from the spectral theory and will be used as a basic comparison principle.
The paper is organized as follows. Section 2 collects the equidistribution and resistance-space preliminaries. Section 3 develops the spectral Fourier Ratio estimates and the reciprocal-spectrum identity. Section 4 combines these bounds with random sampling and \(\ell^1\) recovery. Section 5 treats spectral dimension and self-similar fractals. Section 6 gives the electrical-network interpretation.
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2 Preliminaries
We begin with two preliminary observations. The first concerns finite coefficient families obtained by evaluating a fixed function along equidistributed point sets. The second records the compactness properties of resistance spaces needed to obtain a discrete spectral decomposition for the associated Laplacian.
2.1 Equidistribution and Fourier Ratio
We first consider coefficient vectors of the form \(\{h(\beta_{D,j})\}_{j=1}^D\). The elementary finite-dimensional bounds identify \(\sqrt D\) as the maximal scale of the Fourier Ratio, while weak convergence of the empirical measures determines the asymptotic constant.
Definition 2.1. For a nonzero vector \(a\in\mathbb C^D\), define \[\operatorname{FR}(a) \ := \ \frac{\norm{a}_1}{\norm{a}_2}.\]
Proposition 2.2. For every nonzero \(a\in\mathbb C^D\), \[1 \ \leq \ \operatorname{FR}(a) \ \leq \ \sqrt D,\] and \[\operatorname{FR}(a) \ \geq \ \frac{\norm{a}_2}{\norm{a}_\infty}.\]
Proof. The first lower bound follows from \(\norm{a}_2\leq\norm{a}_1\), while Cauchy-Schwarz gives \(\norm{a}_1\leq\sqrt D\norm{a}_2\). Also, \[\norm{a}_2^2 \ \leq \ \norm{a}_\infty\norm{a}_1,\] which gives the final bound after dividing by \(\norm{a}_\infty\norm{a}_2\). ◻
The upper bound is attained at the diffuse scale. For structured families arising from a limiting distribution, the leading \(\sqrt D\) coefficient can be computed explicitly.
Theorem 2.3. For every \(D\), let \(\beta_{D,1},\ldots,\beta_{D,D}\) lie in some fixed compact set \(K\subset\mathbb C\), and suppose that the empirical measures \[\nu_D \ = \ \frac{1}{D}\sum_{j=1}^D\delta_{\beta_{D,j}}\] converge weakly to a probability measure \(\nu\) on \(K\). Let \(h:K\rightarrow\mathbb C\) be continuous, and suppose that \[\int_K\abs{h(z)}^2d\nu(z) \ > \ 0.\] Then \[\frac{\sum_{j=1}^D\abs{h(\beta_{D,j})}}{\left(\sum_{j=1}^D\abs{h(\beta_{D,j})}^2\right)^{1/2}} \ = \ \sqrt{D}\frac{\int_K\abs{h(z)}d\nu(z)}{\left(\int_K\abs{h(z)}^2d\nu(z)\right)^{1/2}}+o(\sqrt{D}).\]
Proof. Weak convergence of \(\nu_D\) applies to the bounded continuous functions \(|h|\) and \(|h|^2\), hence \[\frac1D\sum_{j=1}^D|h(\beta_{D,j})|\longrightarrow \int_K|h|\,d\nu, \qquad \frac1D\sum_{j=1}^D|h(\beta_{D,j})|^2\longrightarrow \int_K|h|^2\,d\nu.\] The second limit is positive by hypothesis. Dividing the first convergence by the square root of the second and restoring the powers of \(D\) gives the stated asymptotic. ◻
Thus the limiting measure determines the asymptotic Fourier Ratio completely, apart from the unavoidable factor \(\sqrt D\).
2.2 Resistance-Space Tools
For the spectral applications below, the relevant Fourier system will be supplied by the eigenfunctions of a resistance Laplacian. We therefore recall the resistance inequality and the compactness statement needed for the spectral theorem.
Definition 2.4. Let \(X\) be a set. A pair \((\mathcal E,\mathcal F)\) is a resistance form on \(X\) if the following properties hold.
\(\mathcal F\) is a linear space of real-valued functions on \(X\) containing the constants, \(\mathcal E\) is a nonnegative symmetric bilinear form on \(\mathcal F\), and \(\mathcal E(u,u) \ = \ 0\) if and only if \(u\) is constant.
After functions differing by a constant are identified, \((\mathcal F/\!\sim,\mathcal E)\) is a Hilbert space.
For every finite set \(V\subset X\) and every function \(v:V\rightarrow\mathbb R\), there is some \(u\in\mathcal F\) such that \(u|_V \ = \ v\).
For every \(x,y\in X\), \[R(x,y) \ := \ \sup_{\substack{u\in\mathcal F\\ \mathcal E(u,u)>0}} \frac{\abs{u(x)-u(y)}^2}{\mathcal E(u,u)} \ < \ \infty.\]
If \(u\in\mathcal F\) and \[\overline u \ = \ \max\{0,\min\{1,u\}\},\] then \(\overline u\in\mathcal F\) and \[\mathcal E(\overline u,\overline u) \ \leq \ \mathcal E(u,u).\]
The metric \(R\) is called the resistance metric [16].
By definition of the resistance metric, \[\abs{u(x)-u(y)}^2 \ \leq \ R(x,y)\mathcal E(u,u)\] for every \(u\in\mathcal F\) and \(x,y\in X\). This is the basic pointwise estimate used throughout.
Definition 2.5. A compact measured resistance space is a tuple \[(X,R,\mu,\mathcal E,\mathcal F),\] where \((\mathcal E,\mathcal F)\) is a resistance form on \(X\), \(R\) is its associated resistance metric, \((X,R)\) is compact, and \(\mu\) is a Borel probability measure on \(X\) with full support. We write \[D_R \ = \ \operatorname{diam}_R(X)\] for the resistance diameter of \(X\).
We use the complexification of \(\mathcal F\) and \(\mathcal E\) without changing notation. Compactness of \((X,R)\) and the resistance inequality give the following uniform estimate.
Lemma 2.6. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\), and let \(D_R \ = \ \operatorname{diam}_R(X)\). For \(u\in\mathcal F\), let \(u_X \ = \ \int_Xu\dd\mu\). Then \[\norm{u-u_X}_{L^\infty(X)} \ \leq \ D_R^{1/2}\mathcal E(u,u)^{1/2},\] and consequently \[\norm{u}_{L^\infty(X)} \ \leq \ \norm{u}_{L^2(X,\mu)}+D_R^{1/2}\mathcal E(u,u)^{1/2}.\]
Proof. For fixed \(x\in X\), \[u(x)-u_X=\int_X\bigl(u(x)-u(y)\bigr)\,d\mu(y).\] The resistance inequality and \(R(x,y)\leq D_R\) imply \[|u(x)-u_X|\leq D_R^{1/2}\mathcal E(u,u)^{1/2}.\] Taking the supremum gives the first estimate. The second follows from \(\norm{u}_\infty\leq \norm{u-u_X}_\infty+|u_X|\) and \(|u_X|\leq\norm{u}_2\). ◻
Lemma 6, together with the resistance inequality, gives uniform boundedness and equicontinuity for bounded subsets of \(\mathcal F\). The compact embedding is therefore an immediate Arzelà–Ascoli consequence.
Theorem 2.7. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\), and equip \(\mathcal F\) with the norm \[\norm{u}_{\mathcal F}^2 \ = \ \mathcal E(u,u)+\norm{u}_{L^2(X,\mu)}^2.\] Then every bounded sequence in \(\mathcal F\) has a uniformly convergent subsequence. In particular, \[\mathcal F \ \hookrightarrow \ L^2(X,\mu)\] is compact.
Proof. Let \(\{u_n\}\) be bounded in \(\mathcal F\). Lemma 6 gives a uniform \(L^\infty\) bound, while \[|u_n(x)-u_n(y)|\leq R(x,y)^{1/2}\mathcal E(u_n,u_n)^{1/2}\] shows equicontinuity in the resistance metric. Since \((X,R)\) is compact, Arzelà–Ascoli gives a uniformly convergent subsequence. Uniform convergence implies \(L^2(X,\mu)\) convergence because \(\mu(X)=1\). ◻
The compact embedding will be used below to show that the resistance Laplacian has compact resolvent. This supplies the discrete eigenbasis required for the spectral Fourier Ratio estimates.
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3 Spectral and Resistance Spaces
We now turn to the spectral estimates. The main observation is that a weighted Cauchy–Schwarz inequality separates the Fourier Ratio into a reciprocal spectral sum and a regularity norm of the function. Eigenvalue counting controls the first term, while on resistance spaces the second becomes the energy.
3.1 Spectral Fourier Ratio Bounds
We first state the estimate with arbitrary positive weights. The choice \(w_j=\lambda_j^{s/2}\) then converts the weighted coefficient norm into \(\norm{A^{s/2}f}_{L^2}\).
Theorem 3.1. Let \((X,\mu)\) be a probability space and let \(A\) be a densely defined nonnegative self-adjoint operator on \(L^2(X,\mu)\) with compact resolvent and \(\ker A \ = \ \operatorname{span}\{\mathbf 1\}\). Fix an orthonormal eigenbasis \(\{e_j\}_{j=0}^{\infty}\) such that \[Ae_j \ = \ \lambda_j e_j, \qquad 0 \ = \ \lambda_0<\lambda_1\leq\lambda_2\leq\cdots,\] with \(e_0 \ = \ \mathbf 1\). For \(L\geq\lambda_1\), let \(V_L \ = \ \operatorname{span}\{e_j:\lambda_j\leq L\}\), and for nonzero \(f\in V_L\) let \(\hat f(j) \ = \ \langle f,e_j\rangle\) and \[\operatorname{FR}_L(f) \ = \ \frac{\sum_{\lambda_j\leq L}\abs{\hat f(j)}}{\norm{f}_{L^2(X,\mu)}}.\] Then for every choice of positive weights \(w_j\) on the nonzero eigenvalues below \(L\), \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+\left(\sum_{0<\lambda_j\leq L}w_j^{-2}\right)^{1/2}\frac{\left(\sum_{0<\lambda_j\leq L}w_j^2\abs{\hat f(j)}^2\right)^{1/2}}{\norm{f}_{L^2}}.\] In particular, for every \(s>0\), \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+\left(\sum_{0<\lambda_j\leq L}\lambda_j^{-s}\right)^{1/2}\frac{\norm{A^{s/2}f}_{L^2}}{\norm{f}_{L^2}}.\]
Proof. Since \(e_0=\mathbf 1\) and \(\mu(X)=1\), the zero mode is \[\widehat f(0)=\int_X f\,d\mu.\] For the positive modes, Cauchy–Schwarz gives \[\sum_{0<\lambda_j\leq L}|\widehat f(j)| \leq \left(\sum_{0<\lambda_j\leq L}w_j^{-2}\right)^{1/2} \left(\sum_{0<\lambda_j\leq L}w_j^2|\widehat f(j)|^2\right)^{1/2}.\] Adding the zero mode and dividing by \(\norm{f}_2\) proves the first estimate. Taking \(w_j=\lambda_j^{s/2}\) and using the spectral calculus identity \[\sum_{0<\lambda_j\leq L}\lambda_j^s|\widehat f(j)|^2=\norm{A^{s/2}f}_2^2\] proves the second. ◻
The possible cutoff growth is therefore contained entirely in the reciprocal spectral sum. A dyadic decomposition converts a counting estimate into the three familiar regimes below.
Theorem 3.2. Under the hypotheses and notation of Theorem 8, define \(N_A(\Lambda) \ = \ \#\{j:\lambda_j\leq\Lambda\}\). Suppose that for some \(\alpha>0\) and \(C_N>0\) we have \[N_A(\Lambda) \ \leq \ C_N\Lambda^\alpha\] for every \(\Lambda\geq\lambda_1\). Then for every \(s>0\) and \(L\geq\lambda_1\), \[\sum_{0<\lambda_j\leq L}\lambda_j^{-s} \ \leq \ C_{\alpha,s,A}\begin{cases}1,&s>\alpha,\\1+\log(1+L/\lambda_1),&s=\alpha,\\L^{\alpha-s},&0<s<\alpha.\end{cases}\] Consequently every nonzero \(f\in V_L\) satisfies \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+C_{\alpha,s,A}\frac{\norm{A^{s/2}f}_{L^2}}{\norm{f}_{L^2}}\begin{cases}1,&s>\alpha,\\\left(1+\log(1+L/\lambda_1)\right)^{1/2},&s=\alpha,\\L^{(\alpha-s)/2},&0<s<\alpha.\end{cases}\]
Proof. The proof is a standard dyadic summation. Let \(M\) be the largest integer with \(2^M\lambda_1\leq L\) and decompose the positive spectrum into shells \(I_n=[2^n\lambda_1,2^{n+1}\lambda_1)\). On \(I_n\), \[\sum_{\lambda_j\in I_n}\lambda_j^{-s} \leq (2^n\lambda_1)^{-s}N_A(2^{n+1}\lambda_1) \leq C_N2^\alpha\lambda_1^{\alpha-s}2^{n(\alpha-s)}.\] Summing this geometric sequence gives a uniform bound for \(s>\alpha\), logarithmic growth for \(s=\alpha\), and \(O(L^{\alpha-s})\) for \(s<\alpha\). The Fourier Ratio estimate follows by substituting this bound into Theorem 8. ◻
For resistance forms the natural choice is \(s=1\), since the \(A^{1/2}\) norm is exactly the square root of the energy. We next verify the operator hypotheses needed to make this specialization.
3.2 Resistance Laplacians and Energy
The compact embedding from Theorem 7 gives compact resolvent for the resistance Laplacian. The representation theorem for closed forms then identifies its square-root domain with the energy space.
Theorem 3.3. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\). Then the nonnegative self-adjoint operator \(A \ = \ -\Delta_\mu\) associated with the closed form \((\mathcal E,\mathcal F)\) has compact resolvent and an orthonormal eigenbasis \(\{e_j\}_{j=0}^{\infty}\) satisfying \[Ae_j \ = \ \lambda_j e_j, \qquad 0 \ = \ \lambda_0<\lambda_1\leq\lambda_2\leq\cdots, \qquad \lambda_j\rightarrow\infty.\] The zero eigenspace consists exactly of the constants, every eigenfunction lies in \(\mathcal F\subset C(X)\), and for every \(f\in\mathcal F\), \[\mathcal E(f,f) \ = \ \norm{A^{1/2}f}_{L^2(X,\mu)}^2.\]
Proof. The representation theorem for closed nonnegative forms gives a unique nonnegative self-adjoint operator \(A\) associated with \((\mathcal E,\mathcal F)\). If \(u=(A+I)^{-1}g\), then \[\norm{u}_{\mathcal F}^2 =\mathcal E(u,u)+\norm{u}_2^2 =\langle g,u\rangle \leq \norm{g}_2\norm{u}_{\mathcal F},\] so the resolvent maps bounded subsets of \(L^2\) into bounded subsets of \(\mathcal F\). Theorem 7 therefore makes \((A+I)^{-1}\) compact. The spectral theorem gives the discrete eigenbasis and \(\lambda_j\to\infty\). The kernel consists of the constants because \(\mathcal E(u,u)=0\) if and only if \(u\) is constant. Finally, the standard square-root identity for the operator associated with the form gives \[\mathcal E(f,f)=\norm{A^{1/2}f}_2^2.\] The resistance inequality implies continuity of finite-energy functions. ◻
Taking \(s=1\) in Theorem 8 now converts regularity directly into energy. The only remaining spectral quantity is the partial reciprocal sum.
Theorem 3.4. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\), let \(A \ = \ -\Delta_\mu\), and use the eigenbasis of Theorem 10. For \(L\geq\lambda_1\), let \(V_L \ = \ \operatorname{span}\{e_j:\lambda_j\leq L\}\). Then every nonzero \(f\in V_L\) satisfies \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+\left(\sum_{0<\lambda_j\leq L}\frac{1}{\lambda_j}\right)^{1/2}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}}.\] In particular, \[\operatorname{FR}_L(f) \ \leq \ 1+\left(\sum_{0<\lambda_j\leq L}\frac{1}{\lambda_j}\right)^{1/2}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}}.\]
Proof. Apply Theorem 8 with \(s=1\) and use \(\norm{A^{1/2}f}_2^2=\mathcal E(f,f)\) from Theorem 10. The second estimate follows from \(|\int_Xf\,d\mu|\leq\norm{f}_2\). ◻
In a general spectral problem the reciprocal sum would be estimated from eigenvalue counting. On a resistance space it has an exact geometric interpretation, obtained by expanding the difference of point evaluations in the energy basis.
3.3 The Reciprocal Spectrum and Resistance
Theorem 3.5. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\), let \(A \ = \ -\Delta_\mu\), and let \(\{e_j\}_{j=0}^{\infty}\) and \(\{\lambda_j\}_{j=0}^{\infty}\) be as in Theorem 10. Then for every \(x,y\in X\), \[R(x,y) \ = \ \sum_{j=1}^{\infty}\frac{\abs{e_j(x)-e_j(y)}^2}{\lambda_j},\] and consequently \[\sum_{j=1}^{\infty}\frac{1}{\lambda_j} \ = \ \frac{1}{2}\int_X\int_XR(x,y)\dd\mu(x)\dd\mu(y) \ \leq \ \frac{D_R}{2}.\]
Proof. Work in the mean-zero energy space, where \(\mathcal E\) is an inner product. For \(j\geq1\) set \[\phi_j=\lambda_j^{-1/2}e_j.\] By Theorem 10, \(\{\phi_j\}_{j\geq1}\) is an orthonormal basis of this space. For fixed \(x,y\in X\), the functional \(L_{x,y}(u)=u(x)-u(y)\) has squared norm \(R(x,y)\) by definition of the resistance metric. Parseval therefore gives \[R(x,y)=\sum_{j=1}^{\infty}|L_{x,y}(\phi_j)|^2 =\sum_{j=1}^{\infty}\frac{|e_j(x)-e_j(y)|^2}{\lambda_j}.\] The summands are nonnegative, so monotone convergence permits integration over \(X\times X\). Since \(e_j\perp\mathbf 1\) and \(\norm{e_j}_2=1\) for \(j\geq1\), \[\int_X\int_X|e_j(x)-e_j(y)|^2\,d\mu(x)d\mu(y)=2.\] Hence \[\int_X\int_XR(x,y)\,d\mu(x)d\mu(y) =2\sum_{j=1}^{\infty}\frac1{\lambda_j}.\] The diameter bound follows from \(R(x,y)\leq D_R\) and \(\mu(X)=1\). ◻
Theorem 12 removes the cutoff dependence in Theorem 11 without any separate Weyl estimate.
Corollary 3.6. Under the hypotheses of Theorem 12, every nonzero \(f\in V_L\) satisfies \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+\left(\frac{1}{2}\int_X\int_XR(x,y)\dd\mu(x)\dd\mu(y)\right)^{1/2}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}},\] and consequently \[\operatorname{FR}_L(f) \ \leq \ 1+\left(\frac{D_R}{2}\right)^{1/2}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}}.\] In particular, both bounds are independent of the spectral cutoff \(L\).
Proof. Insert Theorem 12 into Theorem 11. The diameter-only estimate follows from \(R(x,y)\leq D_R\) and \(|\int_Xf\,d\mu|\leq\norm{f}_2\). ◻
The trace identity also forces the eigenvalue counting function to be sublinear.
Corollary 3.7. Under the hypotheses of Theorem 12, \[N_A(\Lambda) \ = \ o(\Lambda)\] as \(\Lambda\rightarrow\infty\).
Proof. Fix \(\varepsilon>0\) and choose \(M\) so that \(\sum_{j>M}\lambda_j^{-1}<\varepsilon\). For \(\Lambda>\lambda_M\), \[\frac{N_A(\Lambda)-1-M}{\Lambda} \leq \sum_{\substack{j>M\\ \lambda_j\leq\Lambda}}\frac1{\lambda_j} <\varepsilon.\] Letting \(\Lambda\to\infty\) and then \(\varepsilon\to0\) proves the claim. ◻
Thus cutoff-independent Fourier Ratio control is intrinsic to compact measured resistance spaces; no fractal counting law is needed for this conclusion.
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4 Sampling and Reconstruction
We now convert the preceding Fourier Ratio estimates into sampling statements. The recovery input is standard: a Fourier Ratio bound gives effective sparsity of the coefficient vector, while a uniform \(L^\infty\) bound on the spectral basis controls the coherence of point sampling [2, 9, 10]. The abstract theorem keeps these two parameters separate; on resistance spaces both are controlled geometrically.
4.1 Spectral Recovery
Let \(V_L\) be the spectral subspace from Theorem 8. Point evaluations produce a random sampling matrix with columns given by the eigenfunctions below \(L\). The quantity \(K_L\) below is the corresponding bounded-orthonormal-system constant.
Theorem 4.1. Assume the hypotheses of Theorem 8. For \(L>0\), define \[D_L \ = \ \#\{j:\lambda_j\leq L\}, \qquad K_L \ = \ \sup_{\lambda_j\leq L}\norm{e_j}_{L^\infty(X)},\] and assume \(K_L<\infty\). Let \(0<\varepsilon<1/2\), let \(f\in V_L\) be nonzero, and suppose that \[\operatorname{FR}_L(f) \ \leq \ r.\] Choose \(x_1,\ldots,x_q\) independently according to \(\mu\), write \(\mathcal X=(x_1,\ldots,x_q)\), and define \[\norm{h}_{L^2(\mathcal X)} \ = \ \left(\frac{1}{q}\sum_{i=1}^q\abs{h(x_i)}^2\right)^{1/2}.\] Let \(f^*\in V_L\) solve \[\min_{h\in V_L}\norm{\hat h}_1\] subject to \[\norm{f-h}_{L^2(\mathcal X)} \ \leq \ \varepsilon\norm{f}_{L^2(X,\mu)}.\] There are absolute constants \(C,c,C_{\mathrm{rec}}>0\) such that if \[\label{eq:abstractsamples} q \ \geq \ C K_L^2\frac{r^2}{\varepsilon^2}\log^2\left(2+\frac{r}{\varepsilon}\right)\log(2D_L),\] then with probability at least \(1-D_L^{-c}\), \[\norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(X,\mu)}.\]
Proof. This is the standard bounded-orthonormal-system recovery argument; see [2, 9, 10]. We indicate the reduction. Write \[f=\sum_{\lambda_j\leq L}a_je_j.\] Then \(\norm{a}_2=\norm{f}_2\) and \(\norm{a}_1\leq r\norm{a}_2\). Taking \(S\asymp r^2\varepsilon^{-2}\) gives \(\sigma_S(a)_1/\sqrt S\lesssim \varepsilon\norm{a}_2\). The sampling matrix \[E_{ij}=q^{-1/2}e_j(x_i)\] has bounded-orthonormal-system constant \(K_L\). Standard restricted-isometry estimates apply once \[q\gtrsim K_L^2S\log^2(2S)\log(2D_L),\] and stable basis pursuit then yields \[\norm{a^*-a}_2\lesssim \frac{\sigma_S(a)_1}{\sqrt S}+\varepsilon\norm{a}_2 \lesssim \varepsilon\norm{a}_2.\] Parseval gives the stated function-space estimate. The displayed sample bound is obtained by substituting \(S\asymp r^2\varepsilon^{-2}\) and adjusting the absolute constants. ◻
Theorem 15 separates sampling coherence from coefficient complexity. The spectral estimate from Theorem 8 may therefore be inserted directly.
Corollary 4.2. Under the hypotheses of Theorem 8, fix \(s>0\) and define \[r_{L,s}(f) \ = \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+\left(\sum_{0<\lambda_j\leq L}\lambda_j^{-s}\right)^{1/2}\frac{\norm{A^{s/2}f}_{L^2}}{\norm{f}_{L^2}}.\] Then the reconstruction in Theorem 15 satisfies \[\norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(X,\mu)}\] with probability at least \(1-D_L^{-c}\) whenever \[q \ \geq \ C K_L^2\frac{r_{L,s}(f)^2}{\varepsilon^2}\log^2\left(2+\frac{r_{L,s}(f)}{\varepsilon}\right)\log(2D_L).\]
4.2 Recovery on Compact Resistance Spaces
On a compact measured resistance space, the same geometry controls both parameters in the recovery theorem. The Fourier Ratio is bounded by Corollary 13, while the resistance inequality controls the pointwise size of the eigenfunctions.
Proposition 4.3. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with resistance diameter \(D_R\), and let \(A=-\Delta_\mu\) have the eigenbasis of Theorem 10. Then for every \(j\geq1\), \[\norm{e_j}_{L^\infty(X)} \ \leq \ \sqrt{D_R\lambda_j}.\] Consequently, \[K_L^2 \ \leq \ \max\{1,D_RL\}.\]
Proof. For \(j\geq1\), orthogonality to the constant eigenfunction says \(\int_Xe_j\dd\mu=0\). Hence for every \(x\in X\), \[e_j(x) \ = \ \int_X\bigl(e_j(x)-e_j(y)\bigr)\dd\mu(y).\] The resistance inequality and Theorem 10 give \[\abs{e_j(x)-e_j(y)} \ \leq \ R(x,y)^{1/2}\mathcal E(e_j,e_j)^{1/2} \ = \ R(x,y)^{1/2}\lambda_j^{1/2}.\] Since \(R(x,y)\leq D_R\), integration in \(y\) yields \[\abs{e_j(x)} \ \leq \ \sqrt{D_R\lambda_j}.\] Taking the supremum over \(x\) proves the first claim. Since \(e_0=\mathbf 1\) has \(L^\infty\) norm \(1\), the cutoff coherence satisfies the second bound. ◻
Combining Proposition 17 with Corollary 13 eliminates the abstract parameters in Theorem 15.
Theorem 4.4. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X)=1\), resistance diameter \(D_R\), and resistance Laplacian \(A=-\Delta_\mu\). Define \[C_R \ = \ \left(\frac12\int_X\int_XR(x,y)\dd\mu(x)\dd\mu(y)\right)^{1/2}\] and for nonzero \(f\in V_L\) define \[r_R(f) \ = \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2}}+C_R\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}}.\] Choose \(x_1,\ldots,x_q\) independently according to \(\mu\) and let \(f^*\) be the reconstruction from Theorem 15. If \[\label{eq:resistancesamples} q \ \geq \ C\max\{1,D_RL\}\frac{r_R(f)^2}{\varepsilon^2}\log^2\left(2+\frac{r_R(f)}{\varepsilon}\right)\log(2D_L),\] then with probability at least \(1-D_L^{-c}\), \[\norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(X,\mu)}.\] Moreover, \[C_R \ \leq \ \left(\frac{D_R}{2}\right)^{1/2},\] so the same conclusion follows from the diameter-only complexity bound \[r_R(f) \ \leq \ 1+\left(\frac{D_R}{2}\right)^{1/2}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}}.\]
Proof. Corollary 13 gives \(\operatorname{FR}_L(f)\leq r_R(f)\), while Proposition 17 gives \(K_L^2\leq\max\{1,D_RL\}\). Substitution into Theorem 15 proves the recovery statement. The diameter-only bound follows from \(C_R\leq(D_R/2)^{1/2}\) and \(|\int_Xf\,d\mu|\leq\norm{f}_2\). ◻
4.3 Sensor Stability in the Resistance Metric
The resistance metric also gives a natural perturbation scale for sampling locations. The defining resistance inequality converts geometric displacement directly into measurement error.
Theorem 4.5. Let \(f\in\mathcal F\), and let intended sensor locations \(x_1,\ldots,x_q\) and actual sensor locations \(y_1,\ldots,y_q\) satisfy \[R(x_i,y_i) \ \leq \ \delta\] for every \(i\). Then \[\label{eq:sensorerror} \left(\frac1q\sum_{i=1}^q\abs{f(y_i)-f(x_i)}^2\right)^{1/2} \ \leq \ \delta^{1/2}\mathcal E(f,f)^{1/2}.\] Thus the placement error is at most \(\varepsilon\norm{f}_{L^2(X,\mu)}\) whenever \[\delta \ \leq \ \varepsilon^2\frac{\norm{f}_{L^2(X,\mu)}^2}{\mathcal E(f,f)}.\] If in addition \(f\in V_L\), then the simpler sufficient condition \[\delta \ \leq \ \frac{\varepsilon^2}{L}\] guarantees the same relative error bound.
Proof. For every \(i\), the resistance inequality gives \[\abs{f(y_i)-f(x_i)}^2 \ \leq \ R(x_i,y_i)\mathcal E(f,f) \ \leq \ \delta\mathcal E(f,f).\] Averaging over the \(q\) measurements and taking square roots gives [eq:sensorerror]. For \(f\in V_L\), the spectral expansion and Theorem 10 give \[\mathcal E(f,f) \ = \ \sum_{\lambda_j\leq L}\lambda_j\abs{\hat f(j)}^2 \ \leq \ L\sum_{\lambda_j\leq L}\abs{\hat f(j)}^2 \ = \ L\norm{f}_{L^2}^2.\] Substituting this into [eq:sensorerror] gives the final condition. ◻
Since basis pursuit is stable under measurement noise, the same perturbation estimate propagates to the reconstruction error.
Theorem 4.6. Let \(f\in V_L\) be nonzero and suppose \(\operatorname{FR}_L(f)\leq r\). Choose intended locations \(x_1,\ldots,x_q\) independently according to \(\mu\). Suppose the actual locations \(y_1,\ldots,y_q\) satisfy \(R(x_i,y_i)\leq\delta\). Define sampled data on the intended locations by \[g_t(x_i) \ = \ f(y_i),\] and let \(f^*\in V_L\) solve \[\min_{h\in V_L}\norm{\hat h}_1\] subject to \[\left(\frac1q\sum_{i=1}^q\abs{h(x_i)-g_t(x_i)}^2\right)^{1/2} \ \leq \ \delta^{1/2}\mathcal E(f,f)^{1/2}.\] If \[q \ \geq \ C K_L^2\frac{r^2}{\varepsilon^2}\log^2\left(2+\frac{r}{\varepsilon}\right)\log(2D_L),\] then with probability at least \(1-D_L^{-c}\), \[\label{eq:perturbedrecoveryerror} \norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C_{\mathrm{rec}}\left(\varepsilon\norm{f}_{L^2(X,\mu)}+\delta^{1/2}\mathcal E(f,f)^{1/2}\right).\] In particular, if \(\delta\leq\varepsilon^2/L\), then \[\norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C'_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(X,\mu)}.\]
Proof. The proof is the same bounded-orthonormal-system argument as Theorem 15, with the sensor perturbation treated as additive noise. If \[\xi_i=q^{-1/2}\bigl(f(y_i)-f(x_i)\bigr),\] then Theorem 19 gives \(\norm{\xi}_2\leq\delta^{1/2}\mathcal E(f,f)^{1/2}\). Stable basis pursuit therefore yields \[\norm{f^*-f}_2 \leq C_{\mathrm{rec}}\left(\varepsilon\norm{f}_2+\delta^{1/2}\mathcal E(f,f)^{1/2}\right).\] If \(\delta\leq\varepsilon^2/L\), then \(\mathcal E(f,f)\leq L\norm{f}_2^2\), giving the final estimate. ◻
Equivalently, sensors may move inside cells of small resistance diameter without changing the form of the stability estimate.
Corollary 4.7. Let \(Q_1,\ldots,Q_K\subset X\) satisfy \[\operatorname{diam}_R(Q_k) \ \leq \ h\] for every \(k\). If an intended sensor \(x_i\) and its actual location \(y_i\) lie in the same cell, then \[\abs{f(y_i)-f(x_i)} \ \leq \ h^{1/2}\mathcal E(f,f)^{1/2}.\] Consequently the conclusions of Theorems 19 and 20 hold with \(\delta=h\).
Proof. If \(x_i,y_i\in Q_k\), then \(R(x_i,y_i)\leq\operatorname{diam}_R(Q_k)\leq h\). Apply Theorems 19 and 20 with \(\delta=h\). ◻
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5 Fractals
We next specialize the general results to self-similar fractals. The resistance identity already gives cutoff-independent Fourier Ratio bounds; the spectral dimension provides a complementary explanation through eigenvalue counting.
5.1 Spectral Dimension
For many self-similar fractals the counting function is a power \(\Lambda^{d_s/2}\) multiplied by a bounded log-periodic factor [19, 21, 24]. Corollary 14 then forces the exponent to be strictly smaller than one.
Theorem 5.1. Let \((X,R,\mu,\mathcal E,\mathcal F)\) be a compact measured resistance space with \(\mu(X) \ = \ 1\) and resistance Laplacian \(A \ = \ -\Delta_\mu\). Suppose that for some \(d_s>0\) the eigenvalue counting function satisfies a modified Weyl law \[N_A(\Lambda) \ = \ \left(G(\log\Lambda)+o(1)\right)\Lambda^{d_s/2},\] where \(G\) is bounded above and bounded away from zero. Then \[d_s \ < \ 2.\] Consequently the counting estimate from Theorem 9 with \(s \ = \ 1\) is in its cutoff-independent case, and the reciprocal eigenvalue sum converges.
Proof. By Corollary 14, \(N_A(\Lambda)/\Lambda\to0\). If \(d_s>2\), the modified Weyl law makes this quotient grow like a positive bounded factor times \(\Lambda^{d_s/2-1}\); if \(d_s=2\), it is \(G(\log\Lambda)+o(1)\) and is bounded away from zero. Both cases are impossible. Hence \(d_s<2\). The final statement follows from Theorem 9 with \(s=1\). ◻
The point is that \(d_s<2\) is not needed as an additional hypothesis for the uniform Fourier Ratio bound: the resistance identity gives that bound first, and the modified Weyl law is then forced into the same range.
5.2 The Sierpinski Gasket
The standard Sierpinski gasket provides the basic example. Its spectral decimation law makes the counting exponent explicit.
Proposition 5.2. For the standard Laplacian on the Sierpinski gasket, \[N_A(\Lambda) \ \asymp \ \Lambda^{\log 3/\log 5}\] up to bounded log-periodic oscillation [22, 23]. Hence its spectral dimension satisfies \[\frac{d_s}{2} \ = \ \frac{\log 3}{\log 5}<1, \qquad d_s \ = \ \frac{2\log 3}{\log 5}<2.\]
Proof. This is the standard spectral-decimation asymptotic for the Sierpinski gasket; see [22, 21, 23]. At generation \(m\), eigenvalues occur on the scale \(5^m\) while the counting function grows on the scale \(3^m\), yielding \[N_A(\Lambda)\asymp \Lambda^{\log 3/\log 5}\] up to bounded log-periodic oscillation. Comparing with \(N_A(\Lambda)\asymp\Lambda^{d_s/2}\) gives \(d_s=2\log 3/\log 5<2\). ◻
Proposition 23 makes the spectral-dimension restriction from Theorem 22 explicit for the gasket, while Corollary 13 gives the Fourier Ratio estimate directly from its resistance geometry.
Corollary 5.3. Let \(SG\) be the standard Sierpinski gasket with its standard resistance form and self-similar probability measure. Then for every spectral cutoff \(L\) and every nonzero \(f\in V_L\), \[\operatorname{FR}_L(f) \ \leq \ \frac{\abs{\int_{SG}f\dd\mu}}{\norm{f}_{L^2(SG,\mu)}}+C_{SG}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2(SG,\mu)}} \ \leq \ 1+C_{SG}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2(SG,\mu)}},\] where \[C_{SG} \ = \ \left(\frac{1}{2}\int_{SG}\int_{SG}R(x,y)\dd\mu(x)\dd\mu(y)\right)^{1/2} \ \leq \ \left(\frac{D_R}{2}\right)^{1/2},\] and in particular \(C_{SG}\) does not depend on \(L\).
Proof. Apply Corollary 13 to the standard measured resistance structure on \(SG\). Proposition 23 gives the complementary spectral-dimension interpretation. ◻
Thus on the gasket the resistance geometry yields a Fourier Ratio bound independent of \(L\), with function dependence only through the mean, \(L^2\) norm, and energy.
5.3 Further Fractal Examples
The same discussion applies to other self-similar resistance spaces. We record the Sierpinski tetrahedron and Minkowski curve as two standard examples [19, 28, 29].
Proposition 5.4. For the standard Laplacians on the Sierpinski tetrahedron \(ST\) and the Minkowski curve \(MC\), the eigenvalue counting functions satisfy, up to bounded log-periodic factors, \[N_{ST}(\Lambda) \ \asymp \ \Lambda^{\log 4/\log 6}, \qquad N_{MC}(\Lambda) \ \asymp \ \Lambda^{1/2},\] and hence \[d_s(ST) \ = \ \frac{2\log 4}{\log 6}<2, \qquad d_s(MC) \ = \ 1<2.\]
Proof. The quoted counting exponents are the standard modified Weyl exponents for these two examples; see [28, 29]. Comparing with \(N_A(\Lambda)\asymp\Lambda^{d_s/2}\) gives the stated spectral dimensions. ◻
Corollary 5.5. Both the Sierpinski tetrahedron and the Minkowski curve satisfy Fourier Ratio bounds of the form \[\operatorname{FR}_L(f) \ \leq \ 1+C_X\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2}},\] with a constant \(C_X\) independent of \(L\) for the corresponding compact measured resistance space.
Proof. Apply Corollary 13. Proposition 25 gives the corresponding spectral-dimension interpretation. ◻
These examples exhibit the two complementary viewpoints of the paper. Resistance geometry gives the cutoff-independent estimate directly, while the modified Weyl law explains the same phenomenon through spectral growth.
5.4 Reconstruction on Fractals
The cutoff-independent Fourier Ratio bounds combine with Proposition 17 to give reconstruction statements on these fractals. We begin with the Sierpinski gasket.
Corollary 5.6. Let \(SG\) be the standard Sierpinski gasket with its standard resistance form and self-similar probability measure, and let \(C_{SG}\) be the constant from Corollary 24. For nonzero \(f\in V_L\), define \[r_{SG}(f) \ = \ \frac{\abs{\int_{SG}f\dd\mu}}{\norm{f}_{L^2(SG,\mu)}}+C_{SG}\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2(SG,\mu)}}.\] If \(x_1,\ldots,x_q\) are chosen independently according to \(\mu\) and \[q \ \geq \ C\max\{1,D_RL\}\frac{r_{SG}(f)^2}{\varepsilon^2}\log^2\left(2+\frac{r_{SG}(f)}{\varepsilon}\right)\log(2D_L),\] then with probability at least \(1-D_L^{-c}\) the \(\ell^1\) reconstruction \(f^*\) from Theorem 15 satisfies \[\norm{f^*-f}_{L^2(SG,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(SG,\mu)}.\]
Proof. Corollary 24 gives \(\operatorname{FR}_L(f)\leq r_{SG}(f)\), and Proposition 17 gives \(K_L^2\leq\max\{1,D_RL\}\). Apply Theorem 15. ◻
Nothing in the recovery argument is specific to the gasket. The same specialization applies to the Sierpinski tetrahedron and Minkowski curve.
Corollary 5.7. Let \(X\) be either the Sierpinski tetrahedron or the Minkowski curve with the resistance form and probability measure used above, and set \[C_X \ = \ \left(\frac12\int_X\int_XR(x,y)\dd\mu(x)\dd\mu(y)\right)^{1/2}.\] For nonzero \(f\in V_L\), define \[r_X(f) \ = \ \frac{\abs{\int_Xf\dd\mu}}{\norm{f}_{L^2(X,\mu)}}+C_X\frac{\mathcal E(f,f)^{1/2}}{\norm{f}_{L^2(X,\mu)}}.\] Then stable random reconstruction holds with probability at least \(1-D_L^{-c}\) whenever \[q \ \geq \ C\max\{1,D_RL\}\frac{r_X(f)^2}{\varepsilon^2}\log^2\left(2+\frac{r_X(f)}{\varepsilon}\right)\log(2D_L),\] and the reconstruction satisfies \[\norm{f^*-f}_{L^2(X,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{f}_{L^2(X,\mu)}.\]
Proof. Corollary 13 controls the Fourier Ratio and Proposition 17 controls the sampling coherence. Apply Theorem 15. Proposition 25 supplies the spectral-dimension interpretation. ◻
Thus the same resistance metric controls coefficient complexity, sampling coherence, and sensor perturbations on the fractal examples considered above.
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6 Electrical Networks
Finite resistive networks are finite measured resistance spaces: the resistance metric is effective resistance, the energy is Joule dissipation, and the associated operator is a weighted graph Laplacian [31, 32, 33]. The first results are therefore direct specializations of the general theory. Current-driven voltages have additional structure, since \(v=L_G^+i\) on the mean-zero subspace; this inverse-Laplacian relation gives the filtering and approximation estimates used later.
6.1 Resistive Networks and Effective Resistance
Let \(G=(V,E,c)\) be a finite connected conductance network with \(n=\abs{V}\) and symmetric conductances \(c_{xy}=c_{yx}>0\) on the edges. The physical weighted graph Laplacian is \[(L_Gv)(x) \ = \ \sum_{y\sim x}c_{xy}\bigl(v(x)-v(y)\bigr),\] and its Dirichlet energy is \[\mathcal E_G(v,v) \ = \ \sum_{\{x,y\}\in E}c_{xy}\abs{v(x)-v(y)}^2 \ = \ v^{*}L_Gv.\] For a voltage potential \(v\), this is exactly the dissipated power, which we write as \(P_{\mathrm{diss}}(v)\). We place the uniform probability measure \(\mu(x)=1/n\) on \(V\). With this normalization the self-adjoint operator associated with \(\mathcal E_G\) in \(L^2(V,\mu)\) is not \(L_G\) itself but \[A \ = \ nL_G,\] since \(\langle Au,v\rangle_{L^2(V,\mu)}=u^{*}L_Gv=\mathcal E_G(u,v)\). The corresponding resistance metric is the usual effective resistance \[R_{\mathrm{eff}}(x,y) \ = \ (\mathbf e_x-\mathbf e_y)^{*}L_G^{+}(\mathbf e_x-\mathbf e_y),\] where \(L_G^{+}\) denotes the Moore-Penrose pseudoinverse [31, 32, 33].
The only normalization point to record is that the resistance-space operator is \(A=nL_G\). With this convention, the geometric constant in the Fourier Ratio bound is expressed by the classical Kirchhoff index.
Proposition 6.1. Let \[0 \ = \ \nu_0<\nu_1\leq\cdots\leq\nu_{n-1}\] be the eigenvalues of \(L_G\), and let \(\lambda_j\) be the eigenvalues of the resistance-space operator \(A=nL_G\). Then \(\lambda_j=n\nu_j\) and \[\sum_{j=1}^{n-1}\frac{1}{\lambda_j} \ = \ \frac{1}{n}\sum_{j=1}^{n-1}\frac{1}{\nu_j} \ = \ \frac{Kf(G)}{n^2} \ = \ \frac{1}{2n^2}\sum_{x,y\in V}R_{\mathrm{eff}}(x,y),\] where \[Kf(G) \ = \ \sum_{x<y}R_{\mathrm{eff}}(x,y)\] is the Kirchhoff index of the network. Consequently the resistance constant from Theorem 18 is \[C_R \ = \ \frac{\sqrt{Kf(G)}}{n}.\]
Proof. Since \(A=nL_G\), one has \(\lambda_j=n\nu_j\). Theorem 12, with the uniform measure on \(V\), gives \[\sum_{j=1}^{n-1}\frac1{\lambda_j} =\frac1{2n^2}\sum_{x,y\in V}R_{\mathrm{eff}}(x,y) =\frac{Kf(G)}{n^2}.\] Substituting \(\lambda_j=n\nu_j\) recovers \(Kf(G)=n\sum_{j=1}^{n-1}\nu_j^{-1}\), and taking square roots gives \(C_R=\sqrt{Kf(G)}/n\). ◻
6.2 Voltage Potentials and Current Injection
We now rewrite the resistance estimates in electrical variables. Let \(\phi_0,\ldots,\phi_{n-1}\) be an orthonormal eigenbasis of \(L_G\) in the Euclidean inner product, with \(\phi_0=n^{-1/2}\mathbf 1\), and write \[\widetilde v(j) \ = \ \langle v,\phi_j\rangle_{\ell^2(V)}, \qquad \operatorname{FR}_G(v) \ = \ \frac{\sum_{j=0}^{n-1}\abs{\widetilde v(j)}}{\left(\sum_{j=0}^{n-1}\abs{\widetilde v(j)}^2\right)^{1/2}}.\] The \(L^2(V,\mu)\) eigenfunctions are \(e_j=\sqrt n\,\phi_j\), so the common factor relating the two coefficient sequences cancels in the ratio and \(\operatorname{FR}_G\) is exactly the full-cutoff Fourier Ratio used above. Define \[V_{\mathrm{mean}} \ = \ \frac1n\sum_{x\in V}v(x), \qquad V_{\mathrm{RMS}} \ = \ \left(\frac1n\sum_{x\in V}\abs{v(x)}^2\right)^{1/2}.\]
Corollary 6.2. Every nonzero voltage profile \(v\) satisfies \[\operatorname{FR}_G(v) \ \leq \ \frac{\abs{V_{\mathrm{mean}}}}{V_{\mathrm{RMS}}}+\frac{\sqrt{Kf(G)}}{n}\frac{P_{\mathrm{diss}}(v)^{1/2}}{V_{\mathrm{RMS}}}.\] In the mean-zero voltage gauge, this simplifies to \[\operatorname{FR}_G(v) \ \leq \ \frac{\sqrt{Kf(G)P_{\mathrm{diss}}(v)}}{nV_{\mathrm{RMS}}}.\]
Proof. Apply Corollary 13 with the identifications \[\norm{v}_{L^2(V,\mu)}=V_{\mathrm{RMS}},\qquad \int v\,d\mu=V_{\mathrm{mean}},\] \[\mathcal E_G(v,v)=P_{\mathrm{diss}}(v),\qquad C_R=\frac{\sqrt{Kf(G)}}{n},\] where the last identity is Proposition 29. ◻
A current-driven voltage has additional structure through Kirchhoff’s law. Let \(i:V\to\mathbb R\) have total current zero, \[\sum_{x\in V}i(x) \ = \ 0,\] and choose the mean-zero voltage gauge. The network equation is \[L_Gv \ = \ i.\] Since \(L_G\) is invertible on the mean-zero subspace, this determines \[v \ = \ L_G^{+}i.\]
Proposition 6.3. Let \(\widetilde i(j)=\langle i,\phi_j\rangle_{\ell^2(V)}\). For the mean-zero voltage potential generated by \(i\), \[v \ = \ \sum_{j=1}^{n-1}\frac{\widetilde i(j)}{\nu_j}\phi_j,\] and \[P_{\mathrm{diss}}(v) \ = \ i^{*}L_G^{+}i \ = \ \sum_{j=1}^{n-1}\frac{\abs{\widetilde i(j)}^2}{\nu_j}.\] Moreover, \[V_{\mathrm{RMS}}^2 \ = \ \frac1n\sum_{j=1}^{n-1}\frac{\abs{\widetilde i(j)}^2}{\nu_j^2}\] and the Fourier Ratio of the voltage has the exact current-source representation \[\operatorname{FR}_G(v) \ = \ \frac{\sum_{j=1}^{n-1}\abs{\widetilde i(j)}/\nu_j}{\left(\sum_{j=1}^{n-1}\abs{\widetilde i(j)}^2/\nu_j^2\right)^{1/2}}.\]
Proof. Expanding \(i\) in the eigenbasis of \(L_G\) and using \(\widetilde i(0)=0\) gives \[i \ = \ \sum_{j=1}^{n-1}\widetilde i(j)\phi_j.\] Applying \(L_G^{+}\) divides the \(j\)th nonconstant coefficient by \(\nu_j\), which gives the expansion of \(v\). The power identity follows from \[P_{\mathrm{diss}}(v) \ = \ v^{*}L_Gv \ = \ v^{*}i \ = \ i^{*}L_G^{+}i,\] and Parseval gives both spectral sums. The final formula is then the definition of \(\operatorname{FR}_G(v)\) applied to the coefficients \(\widetilde i(j)/\nu_j\). ◻
Thus the current-to-voltage map acts as an inverse-Laplacian spectral filter. On a fixed spectral band, its distortion of the Fourier Ratio is controlled by the spectral condition number.
Corollary 6.4. Suppose the current injection \(i\) has spectral support only on eigenvalues satisfying \(0<\nu_a\leq\nu_j\leq\nu_b\). Then \[\frac{\nu_a}{\nu_b}\operatorname{FR}_G(i) \ \leq \ \operatorname{FR}_G(v) \ \leq \ \frac{\nu_b}{\nu_a}\operatorname{FR}_G(i).\]
Proof. On the spectral support of \(i\), \(\nu_b^{-1}\leq\nu_j^{-1}\leq\nu_a^{-1}\). Hence \[\nu_b^{-1}\norm{\widetilde i}_p \leq \norm{\widetilde v}_p \leq \nu_a^{-1}\norm{\widetilde i}_p, \qquad p\in\{1,2\}.\] Taking the ratio of the \(p=1\) and \(p=2\) estimates gives the claim. ◻
The same inverse-Laplacian structure yields a deterministic low-frequency approximation for arbitrary current sources.
Theorem 6.5. Let \(i\) have total current zero, let \(v=L_G^{+}i\) be the mean-zero voltage potential, and define \[I_{\mathrm{RMS}} \ = \ \left(\frac1n\sum_{x\in V}\abs{i(x)}^2\right)^{1/2}.\] For \(\Lambda>0\), let \(P_{\Lambda}\) denote the orthogonal projection onto the graph-spectral subspace spanned by the eigenvectors with \(\nu_j\leq\Lambda\), and set \(v_{\Lambda}=P_{\Lambda}v\). Then \[\norm{v-v_{\Lambda}}_{L^2(V,\mu)} \ \leq \ \frac{I_{\mathrm{RMS}}}{\Lambda},\] \[\mathcal E_G(v-v_{\Lambda},v-v_{\Lambda}) \ \leq \ \frac{nI_{\mathrm{RMS}}^2}{\Lambda},\] and, if \(D_R=\max_{x,y}R_{\mathrm{eff}}(x,y)\), \[\norm{v-v_{\Lambda}}_{L^{\infty}(V)} \ \leq \ \left(\frac{nD_R}{\Lambda}\right)^{1/2}I_{\mathrm{RMS}}.\] In particular, if \(i\) is graph-spectrally supported below \(\Lambda\), then \(v=v_{\Lambda}\).
Proof. By Proposition 31, \[v-v_{\Lambda} \ = \ \sum_{\nu_j>\Lambda}\frac{\widetilde i(j)}{\nu_j}\phi_j.\] Therefore Parseval and \(\nu_j>\Lambda\) give \[\norm{v-v_{\Lambda}}_{L^2(V,\mu)}^2 \ = \ \frac1n\sum_{\nu_j>\Lambda}\frac{\abs{\widetilde i(j)}^2}{\nu_j^2} \ \leq \ \frac{1}{n\Lambda^2}\sum_{j=1}^{n-1}\abs{\widetilde i(j)}^2 \ = \ \frac{I_{\mathrm{RMS}}^2}{\Lambda^2}.\] Similarly, \[\mathcal E_G(v-v_{\Lambda},v-v_{\Lambda}) \ = \ \sum_{\nu_j>\Lambda}\frac{\abs{\widetilde i(j)}^2}{\nu_j} \ \leq \ \frac{1}{\Lambda}\sum_{j=1}^{n-1}\abs{\widetilde i(j)}^2 \ = \ \frac{nI_{\mathrm{RMS}}^2}{\Lambda}.\] The tail has mean zero, so for every \(x\in V\) we may average its difference against the uniform measure and use the resistance inequality exactly as in Proposition 17 to obtain \[\abs{(v-v_{\Lambda})(x)} \ \leq \ D_R^{1/2}\mathcal E_G(v-v_{\Lambda},v-v_{\Lambda})^{1/2} \ \leq \ \left(\frac{nD_R}{\Lambda}\right)^{1/2}I_{\mathrm{RMS}}.\] If \(i\) has no spectral coefficients above \(\Lambda\), then the expansion of Proposition 31 shows the same for \(v\) and the tail vanishes. ◻
6.3 Voltage Reconstruction and Sensor Stability
We now restate the recovery theorem in electrical variables. For \(\Lambda>0\), let \[W_{\Lambda} \ = \ \operatorname{span}\{\phi_j:\nu_j\leq\Lambda\}, \qquad D_{\Lambda}^{G} \ = \ \#\{j:\nu_j\leq\Lambda\}.\] Since the corresponding cutoff for the resistance-space operator \(A=nL_G\) is \(n\Lambda\), Proposition 17 gives the coherence factor \(\max\{1,nD_R\Lambda\}\).
Theorem 6.6. Let \(v\in W_{\Lambda}\) be nonzero and define \[r_{\mathrm{el}}(v) \ = \ \frac{\abs{V_{\mathrm{mean}}}}{V_{\mathrm{RMS}}}+\frac{\sqrt{Kf(G)}}{n}\frac{P_{\mathrm{diss}}(v)^{1/2}}{V_{\mathrm{RMS}}}.\] Choose \(x_1,\ldots,x_q\) independently and uniformly from the nodes of \(G\), and let \(v^*\in W_{\Lambda}\) minimize the \(\ell^1\) norm of its graph-spectral coefficients subject to \[\left(\frac1q\sum_{k=1}^q\abs{v^*(x_k)-v(x_k)}^2\right)^{1/2} \ \leq \ \varepsilon V_{\mathrm{RMS}}.\] If \[q \ \geq \ C\max\{1,nD_R\Lambda\}\frac{r_{\mathrm{el}}(v)^2}{\varepsilon^2}\log^2\left(2+\frac{r_{\mathrm{el}}(v)}{\varepsilon}\right)\log(2D_{\Lambda}^{G}),\] then with probability at least \(1-(D_{\Lambda}^{G})^{-c}\), \[\norm{v^*-v}_{L^2(V,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon V_{\mathrm{RMS}}.\] For mean-zero voltage potentials the first term in \(r_{\mathrm{el}}(v)\) vanishes.
Proof. Apply Theorem 18 to the finite measured resistance space associated with \(G\). Proposition 29 gives \(C_R=\sqrt{Kf(G)}/n\), while the resistance-space cutoff is \(L=n\Lambda\) and \(D_L=D_\Lambda^G\). This gives the stated complexity, sample count, and recovery estimate. ◻
For current-driven voltages, Theorem 33 permits the same argument beyond exact bandlimiting: the high-frequency voltage tail is deterministic and may be treated as measurement/modeling error.
Theorem 6.7. Let \(i\) have total current zero, let \(v=L_G^{+}i\), and let \(v_{\Lambda}=P_{\Lambda}v\neq0\). Define \[\eta_{\Lambda} \ = \ \left(\frac{nD_R}{\Lambda}\right)^{1/2}I_{\mathrm{RMS}}, \qquad r_{\mathrm{el},\Lambda} \ = \ \frac{\sqrt{Kf(G)}}{n}\frac{P_{\mathrm{diss}}(v)^{1/2}}{\norm{v_{\Lambda}}_{L^2(V,\mu)}}.\] Choose \(x_1,\ldots,x_q\) independently and uniformly from \(V\), and let \(v^*\in W_{\Lambda}\) minimize the \(\ell^1\) norm of its graph-spectral coefficients subject to \[\left(\frac1q\sum_{k=1}^q\abs{v^*(x_k)-v(x_k)}^2\right)^{1/2} \ \leq \ \eta_{\Lambda}.\] If \[q \ \geq \ C\max\{1,nD_R\Lambda\}\frac{r_{\mathrm{el},\Lambda}^2}{\varepsilon^2}\log^2\left(2+\frac{r_{\mathrm{el},\Lambda}}{\varepsilon}\right)\log(2D_{\Lambda}^{G}),\] then with probability at least \(1-(D_{\Lambda}^{G})^{-c}\), \[\norm{v^*-v}_{L^2(V,\mu)} \ \leq \ C_{\mathrm{rec}}\varepsilon\norm{v_{\Lambda}}_{L^2(V,\mu)}+C_{\mathrm{rec}}\eta_{\Lambda}+\frac{I_{\mathrm{RMS}}}{\Lambda}.\]
Proof. The argument is the same as Theorem 20, with the spectral tail playing the role of additive noise. Since \(v_\Lambda\) is mean zero and \(\mathcal E_G(v_\Lambda,v_\Lambda)\leq P_{\mathrm{diss}}(v)\), Corollary 30 gives \(\operatorname{FR}_G(v_\Lambda)\leq r_{\mathrm{el},\Lambda}\). Theorem 33 gives \[\norm{v-v_\Lambda}_\infty\leq\eta_\Lambda, \qquad \norm{v-v_\Lambda}_{L^2(V,\mu)}\leq I_{\mathrm{RMS}}/\Lambda.\] Stable basis pursuit applied to \(v_\Lambda\), with coherence \(K_{n\Lambda}^2\leq\max\{1,nD_R\Lambda\}\), yields \[\norm{v^*-v_\Lambda}_2 \leq C_{\mathrm{rec}}\left(\varepsilon\norm{v_\Lambda}_2+\eta_\Lambda\right).\] The triangle inequality gives the result. ◻
In the network setting, sensor stability is therefore governed by effective resistance rather than graph or Euclidean distance.
Corollary 6.8. Let \(v\) be a voltage potential and suppose intended sensor nodes \(x_1,\ldots,x_q\) and actual sensor nodes \(y_1,\ldots,y_q\) satisfy \[R_{\mathrm{eff}}(x_k,y_k) \ \leq \ \delta\] for every \(k\). Then \[\left(\frac1q\sum_{k=1}^q\abs{v(y_k)-v(x_k)}^2\right)^{1/2} \ \leq \ \delta^{1/2}P_{\mathrm{diss}}(v)^{1/2}.\] If \(v\in W_{\Lambda}\), then the sufficient condition \[\delta \ \leq \ \frac{\varepsilon^2}{n\Lambda}\] guarantees that this placement error is at most \(\varepsilon V_{\mathrm{RMS}}\). Under the sample-size hypothesis of Theorem 34, the corresponding perturbed reconstruction satisfies \[\norm{v^*-v}_{L^2(V,\mu)} \ \leq \ C_{\mathrm{rec}}\left(\varepsilon V_{\mathrm{RMS}}+\delta^{1/2}P_{\mathrm{diss}}(v)^{1/2}\right).\]
Proof. The first estimate is Theorem 19 with \(R=R_{\mathrm{eff}}\) and \(\mathcal E_G(v,v)=P_{\mathrm{diss}}(v)\). If \(v\in W_\Lambda\), then \[P_{\mathrm{diss}}(v) =\sum_{\nu_j\leq\Lambda}\nu_j|\widetilde v(j)|^2 \leq n\Lambda V_{\mathrm{RMS}}^2,\] which gives the condition on \(\delta\). The reconstruction estimate follows from Theorem 20 under the normalization \(A=nL_G\). ◻
The electrical specialization therefore has two ingredients. The resistance-space theory converts effective resistance and dissipated power into Fourier Ratio, recovery, and sensor-stability bounds, while the current-voltage equation adds inverse-Laplacian smoothing. Consequently the Kirchhoff index, resistance diameter, graph spectrum, and current profile enter explicitly into the complexity and recoverability of voltage measurements.
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