Characterization of wave front sets by wavelet transforms

Abstract

We consider a special wavelet transform of Moritoh and give new definitions of wave front sets of tempered distributions via that wavelet transform. The major result is that these wave front sets are equal to the wave front sets in the sense of Hörmander in the cases $n=1, 2, 4, 8$. If $n\in \boldsymbol{N} \setminus \{1, 2, 4, 8\}$, then we combine results for dimensions $n=1, 2, 4, 8$ and characterize wave front sets in $\xi$-directions, where $\xi$ are presented as products of non-zero points of $\boldsymbol{R}^{n_1}, \dotsc, \boldsymbol{R}^{n_s}$, $n_1+ \dotsb +n_s=n, n_i \in \{1, 2, 4, 8\}$, $i=1, \dotsc, s$. In particular, the case $n=3$ is discussed through the fourth-dimensional wavelet transform.