EXISTENCE OF UNCONDITIONAL WAVELET PACKET BASES FOR THE SPACES $L^{p}(\mathbb{R})$ AND $\mathcal{H}^{1}(\mathbb{R})$

Abstract

It is proved that the system $\left\{ \omega_{\ell,n,k}: \ell = j-m; n = 2^{m}, 2^{m}+1, \ldots, 2^{m+1}-1; \quad j,k \in \mathbb{Z} \right\}$ of wavelet packets is an unconditional basis for $ L^{p}(\mathbb{R})$, $1 \lt p \lt \infty$ and $\mathcal{H}^{1}(\mathbb{R})$, where $m=0$ if $j \leq 0$ and $m = 0,1,2,\ldots,j$ if $j \gt 0$, provided the orthonormal wavelet packets $\omega_{n}$ and its derivative $ \omega'_{n}$ have a common radial decreasing $L^{1}$-majorant satisfying the condition $\int_{0}^{\infty} sH(s) \, ds \lt \infty$.