Convergence of wavelet frame operators as the sampling density tends to infinity

Convergence of sequences of Calderón‐Zygmund operators with application to wavelet expansions

“…Proof. The case of β = 0 is already shown in 13, Theorem 1.1]. It suffices to prove the conclusion for |β| = ⌊ν⌋.…”

“…In studying the convergence of a class of operators which are introduced by considering the Riemann sums of the inverse wavelet transform 13, we proved that the convergence of wavelet frame operators on L 2 implies the convergence on L p . In this paper, we show that the conclusion is in fact true for a large class of operators.…”

“…In 13, 14, 16, it was shown that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$S_{a,b;\psi ,\tilde{\psi }}$\end{document} converges to the identity in the operator norm on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^p(\mathbb {R}^n)$\end{document} for 1 < p < ∞. In this paper, we show that the convergence holds also on certain Triebel‐Lizorkin spaces, Hardy spaces and Besov spaces.…”

“…Proof of Theorem 1.1(i) Let { S m : m ≥ 1} be a sequence of Calderón‐Zygmund operators in CZO A , ν such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lim _{m,m^{\prime }\rightarrow \infty }\Vert S_m - S_{m^{\prime }}\Vert _{L^2\rightarrow L^2}=0$\end{document}. Using Marcinkiewicz interpolation theorem 20, similar arguments as that done in 13, Theorem 3.2] we can prove that …”

Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

“…Their work was extended to irregular translation grids (still oversampled) by Li and Sun [25], in the restricted case of L p , 1 < p < ∞. Underlying the approaches of Gilbert et al (and Frazier and Jawerth [15, §4], Bui and Paluszyński [3], and Li and Sun [25]) is the fact that a sufficiently dense oversampling of translations and dilations must yield approximate reconstruction in any reasonable function space, for any reasonable analyzer and synthesizer. The reason is that by the Calderón reproducing formula, perfect reconstruction holds in the limit of infinite oversampling (the case of continuous parameter translations and dilations).…”