The Fourier Ratio on Non-Resistance Fractals

Active research. This page is intentionally a problem statement rather than a progress report. Current lemmas, conjectures, calculations, experiments, and proof strategies are omitted while the project is active.

Background

The Fourier Ratio \[ \operatorname{FR}(f)=\frac{\|\widehat f\|_1}{\|\widehat f\|_2} \] is a quantitative measure of effective spectral support. In operator settings, regularity and eigenvalue growth can sometimes be converted into Fourier-Ratio bounds, and those bounds can then feed into sampling and reconstruction results.

Resistance spaces are especially pleasant because the energy, spectrum, and resistance metric are tied to the same analytic structure. Many fractal spaces do not come with exactly that package. The natural problem is therefore not simply “do the old formulas still work?”, but which parts of the argument are genuinely spectral and which parts are artifacts of resistance geometry.

Problem

Find useful hypotheses under which spectral-complexity estimates remain meaningful on fractal or self-similar spaces outside the standard resistance-space setting. A good answer should be stable enough to survive reasonable choices of Laplacian, finite approximations, and eigenspace multiplicity.

Tools

Before asking what survives outside resistance spaces, I want the basic spectral tools fixed first. These are background statements rather than claims about the active problem.

Spectral Decomposition

Theorem (spectral theorem, compact-resolvent form). Let \(H\) be a separable Hilbert space and let \(A\) be a nonnegative self-adjoint operator with compact resolvent. Then \(H\) admits an orthonormal eigenbasis \(\{e_j\}\) with \[ Ae_j=\lambda_j e_j,\qquad 0\leq\lambda_0\leq\lambda_1\leq\cdots,\qquad \lambda_j\to\infty, \] and every eigenvalue has finite multiplicity.

Thus a function can be handled through spectral coefficients \(\widehat f(j)=\langle f,e_j\rangle\) even when there is no Euclidean Fourier transform.

Fourier Ratio and Regularity

Definition. For a nonzero spectrally truncated function \(f\), define \[ \operatorname{FR}_L(f)=\frac{\sum_{\lambda_j\leq L}|\widehat f(j)|}{\|f\|_2}. \]

Proposition (weighted spectral estimate). Suppose \(e_0=\mathbf 1\), \(\lambda_0=0\), and \(s>0\). Then \[ \operatorname{FR}_L(f) \leq \frac{|\int f\,d\mu|}{\|f\|_2} + \left(\sum_{0<\lambda_j\leq L}\lambda_j^{-s}\right)^{1/2} \frac{\|A^{s/2}f\|_2}{\|f\|_2}. \]

This is weighted Cauchy-Schwarz: the point is that it separates the regularity of the function from the growth of the spectrum.

Eigenvalue Counting

Proposition. Let \(N_A(\Lambda)=\#\{j:\lambda_j\leq\Lambda\}\), and suppose \(N_A(\Lambda)\leq C\Lambda^\alpha\) for large \(\Lambda\). Then \[ \sum_{0<\lambda_j\leq L}\lambda_j^{-s}\lesssim \begin{cases} 1,&s>\alpha,\\ 1+\log(1+L),&s=\alpha,\\ L^{\alpha-s},&0

Heat Semigroup

Definition. For \(t>0\), \[ e^{-tA}f=\sum_j e^{-t\lambda_j}\widehat f(j)e_j. \]

Proposition. For every \(t>0\), \(\|e^{-tA}f\|_2\leq\|f\|_2\).

The active work begins after these preliminaries; I am deliberately not posting which tools are currently being combined or modified.

Direction

The public direction is deliberately broad: identify an operator-level notion of spectral complexity that remains stable when resistance geometry is unavailable, and determine what kind of approximation or invariance principle is needed before that complexity can be used for recovery.

I am intentionally not posting the candidate estimates, preferred examples, or current route through those questions.

Last updated: September 14, 2026.