Sharp Jackson and Bernstein inequalities for N-term approximation in sequence spaces with applications

Frame Decomposition of Decomposition Spaces

“…The class of smoothness spaces considered here can be adapted to decompositions of curvelettype. The connection between nonlinear approximation properties and sparse expansions has been studied from an abstract point of view in a number of papers, see for example [33,14,25,28,29].…”

Tensor Products of Approximation Spaces Associated with Regular Elliptic Operators

“…In [3], one can find the application of approximation spaces associated with closed operators in Banach spaces to the indicated class of problems. In connection with the application of various approximating classes of functions, we should especially mention the works [8,9] In the present paper, it is shown that the tensor product of approximation spaces associated with regular elliptic operators is an intermediate interpolation space between the tensor products of the spaces L p (Ω) and entire functions of exponential type whose restriction to a bounded domain Ω belongs to the space L p (Ω) . We establish inequalities estimating the distance from a given element of the tensor product of the spaces L p (Ω) to the subspace determined by the tensor product of the spaces of entire functions of exponential type.…”

“…By using inequalities (9) and (11) and the Paley-Wiener theorem, we establish the equivalence of quasinorms:…”

Orthonormal bases for α-modulation spaces

“…This is indeed the case here where (5.1) shows that W(β) induces an isomorphism between M s,α p,p (R 2 ) and p . We can thus apply Theorem 5.3 and its extension in [15] to W(β), using the estimate (5.2), to obtain the first complete characterization of nonlinear approximation classes associated with a bi-variate α-modulation space.…”