Geodesic
Stability of nearly optimal decompositions in Fourier analysis
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 191-206

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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. 
@article{ZNSL_2018_467_a15,
     author = {A. S. Tselishchev},
     title = {Stability of nearly optimal decompositions in {Fourier} analysis},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {191--206},
     publisher = {mathdoc},
     volume = {467},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_467_a15/}
}
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A. S. Tselishchev. Stability of nearly optimal decompositions in Fourier analysis. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 191-206. http://geodesic.mathdoc.fr/item/ZNSL_2018_467_a15/