A discrete transform and decompositions of distribution spaces

Nonlinear approximation with general wave packets

“…Using the notion of almost diagonal operators we can give a result like in Theorem 3.1 for a larger class of atoms. We recall the following result given by Frazier and Jawerth in [10]. Again, we only state the result for the TriebelLizorkin spaces, but the result holds true for Besov spaces too, see Remark 3.7.…”

“…Consequently, many of the approximation results hold true for a much larger class of generators than just wavelet and tight wavelet frames. Support for this fact can be found in the work of Frazier-Jawerth [10] and Petrushev-Kyriazis [17,15]. Frazier and Jawerth study expansions of functions with wavelet like systems generated by a smooth function with a prescribed number of vanishing moments, while Petrushev and Kyriazis consider approximation with wavelet systems generated by a smooth function with non-zero integral such as the Gaussian.…”

Weighted norm inequalities for general maximal operators

“…the definition above) . We would like to point out that (8) has played a fundamental role in the analysis made in the recent works [5] and [10] . We shall assume throughout the section that X3 is a Muckenhoupt basis.…”

Bilinear decompositions and commutators of singular integral operators

“…It is easy to establish that the matrix {p δ (I, I ′ )} I,I ′ is almost diagonal (by taking ε = δ/4 in the definition (3.1) of Frazier and Jawerth [16]) and thus is bounded onḟ 0,2 1 the space of all sequences (a I ) I such that…”

“…We then use the wavelet characterization of H 1 (R n ) (see Theorem 2.1) and the fact that (cf. [16])…”