Tohoku Mathematical Journal

Characterization of wave front sets by wavelet transforms

Stevan Pilipović and Mirjana Vuletić

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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

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

First available in Project Euclid: 17 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46F12: Integral transforms in distribution spaces [See also 42-XX, 44-XX]
Secondary: 43A32: Other transforms and operators of Fourier type

Wavelet transform wave front


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.

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