Stability of nearly optimal decompositions in Fourier analysis

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.