Adaptive multipliers for extrapolation in frequency

TL;DR

The paper addresses the ill-posed problem of extrapolating high-frequency Fourier content from limited low-frequency data by introducing adaptive frequency multipliers. It develops a rigorous variational framework that seeks worst-case optimal multipliers $m$ to satisfy $D_{\alpha}v \approx m v$ for functions in a collection, demonstrating existence and a canonical projection-like structure. In the finite-collection setting, it shows that optimal multipliers form a Σ-multiplier family parameterized by a Hermitian matrix, enabling a fixed-point algorithm to compute them and revealing deep connections to multiresolution analysis via refinement equations. Numerical experiments on synthetic signals and the MNIST dataset illustrate the practical impact: adaptive multipliers yield multiresolution representations and improved frequency extrapolation, with interpretable spatial filters and controlled extrapolation artifacts.

Abstract

Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned. In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call $Σ$-multipliers, that can be used to perform extrapolation in frequency. We establish connections between $Σ$-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.

Figures (7)

• Figure 1: The multiresolution induced by the collection $\mathcal{F}$ with the single element $f = \operatorname{sinc}^2$ recovers the multiresolution induced by a $B$-spline of degree 1.
• Figure 2: Multiresolution induced by the collection $\mathcal{U}$.
• Figure 3: (a, b) Examples for the discrete data used for the digits 1 and 8. (c) Annular region $\Omega_0$ and its division into regions.
• Figure 4: Performance metrics for the regularized fixed-point iteration.
• Figure 5: Optimal multipliers computed from 500 samples for the digit 1 (a, b) and 8 (c, d) in the frequency domain (a, c) and in the spatial domain (b, d).

Theorems & Definitions (35)

• Proposition 3.1
• proof : Proof of Proposition \ref{['prop:approximationErrorDifferentiable']}
• Proposition 3.2
• proof : Proof of Proposition \ref{['prop:wcErrorIsProperClosedConvex']}
• Proposition 3.3
• proof : Proof of Proposition \ref{['prop:model:eStarNotCoercive']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:wcErrorCoerciveInL2']}
• Proposition 3.4
• Lemma 3.1