Adaptive multipliers for extrapolation in frequency

TL;DR

This paper introduces a new approach using adaptive Fourier multipliers for frequency extrapolation of compactly supported objects, improving stability and providing a framework linked to multiresolution analysis.

Contribution

It develops a theory for optimal Fourier multipliers for frequency extrapolation, including a fixed-point iteration and the concept of -multipliers, connecting to multiresolution analysis.

Findings

Existence of worst-case optimal multipliers for frequency extrapolation.

Representation of optimal multipliers via Hermitian matrices.

Numerical experiments demonstrating practical effectiveness.

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 $\Sigma$-multipliers, that can be used to perform extrapolation in frequency. We establish connections between $\Sigma$-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.