The oscillation spaces \(\mathcal{O}_p^{s, s'} (\mathbb{R}^d )\) introduced by Jaffard are a variation on the definition of Besov spaces for either s ≥ 0 or s ≤ −d/p. On the contrary, the spaces \(\mathcal{O}_p^{s, s'} (\mathbb{R}^d )\) for −d/p < s < 0 cannot be sharply imbedded between Besov spaces with almost the same exponents, and, thus, they are new spaces of really different nature. Their norms take into account correlations between the positions of large wavelet coefficients through the scales. Several numerical studies uncovered such correlations in several settings including turbulence, image processing, traffic, finance, etc. These spaces allow one to capture oscillatory behaviors that are left undetected by Sobolev or Besov spaces. Unlike Sobolev spaces (respectively, Besov spaces Bps,q (ℝd)), which are expressed by simple conditions on wavelet coefficients as ℓp norms (respectively, mixed ℓp − ℓq norms), oscillation spaces are written as ℓp averages of local Cs′ norms. In this paper, we prove the completeness of oscillation spaces in spite of such a mixture of two norms of different kinds.
