# Stability of nearly optimal decompositions in Fourier Analysis

## TL;DR

This paper investigates the stability of near-minimizers in Fourier analysis, specifically for wavelet projections in the context of interpolation between L^1 and L^p spaces, demonstrating their existence under weak decay conditions.

## Contribution

It establishes the existence of stable near-minimizers for the (L^1, L^p) couple under wavelet projections with minimal decay assumptions.

## Key findings

- Stable near-minimizers exist for wavelet projections with weak decay conditions.
- The results apply to the (L^1, L^p) interpolation couple.
- Existence is proven for near-minimizers under certain operator actions.

## Abstract

The question of existence is treated for near-minimizers for the distance functional (or $E$-functional in the interpolation terminology) that are stable under the action of certain operators. In particular, stable near-minimizers for the couple $(L^1, L^p)$ are shown to exist when the operator is the projection on wavelets and these wavelets possess only some weak conditions of decay at infinity.