## Tohoku Mathematical Journal 2006 September SECOND SERIES VOL. 58, NO. 3

 Tohoku Math. J. 58 (2006), 369-391

Title CHARACTERIZATION OF WAVE FRONT SETS BY WAVELET TRANSFORMS

Author Stevan Pilipović and Mirjana Vuletić

(Received October 21, 2004, revised November 25, 2005)
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.

2000 Mathematics Subject Classification. Primary 46F12; Secondary 43A32.

Key words and phrases. Wavelet transform, wave front.