## Tohoku Mathematical Journal

### 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.

#### Article information

Source
Tohoku Math. J. (2) Volume 58, Number 3 (2006), 369-391.

Dates
First available in Project Euclid: 17 November 2006

https://projecteuclid.org/euclid.tmj/1163775136

Digital Object Identifier
doi:10.2748/tmj/1163775136

Mathematical Reviews number (MathSciNet)
MR2273276

Zentralblatt MATH identifier
1122.46021

Keywords
Wavelet transform wave front

#### Citation

Pilipović, Stevan; Vuletić, Mirjana. Characterization of wave front sets by wavelet transforms. Tohoku Math. J. (2) 58 (2006), no. 3, 369--391. doi:10.2748/tmj/1163775136. https://projecteuclid.org/euclid.tmj/1163775136

#### References

• R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87--89.
• I. Daubechies, Ten lectures on wavelets CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, Penn., 1992.
• M. Holschneider, Wavelets, An analysis tool, Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1995.
• L. Hörmander, The analysis of linear partial differential operators I--IV, Springer, Berlin, 1983--1985.
• L. Hörmander, Lectures on nonlinear hyperbolic differential equations, Math. Appl. (Berlin) 26, Springer, Berlin, 1997.
• Y. Meyer, Wavelets and operators, Cambridge Stud. Adv. Math. 37, Cambridge Univ. Press, Cambridge, 1992.
• S. Moritoh, Wavelet transforms in Euclidean spaces---their relation with wave front sets and Besov, Triebel-Lizorkin spaces, Tôhoku Math. J. (2) 47 (1995), 555--565.
• R. Murenzi, Wavelet transforms associated to the $n$-dimensional Euclidean group with dilations: signal in more than one dimension, Wavelets (Marseille, 1987), 239--246, Inverse Probl. Theoret. Imaging, Springer, Berlin, 1989.
• R. S. Pathak, The wavelet transform of distributions, Tohoku Math. J. (2) 56 (2004), 411--421.
• P. Wagner. Private communication, 2003.