{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:54Z","timestamp":1740134514515,"version":"3.37.3"},"reference-count":28,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>f<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msup><mml:mrow><mml:mi>\ud835\udcae<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, the continuous wavelet transform of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math>with respect to a conical wavelet is defined in such a way that the directional wavelet transform of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math>yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2012\/758694","type":"journal-article","created":{"date-parts":[[2012,7,24]],"date-time":"2012-07-24T17:03:01Z","timestamp":1343149381000},"page":"1-11","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets"],"prefix":"10.1155","volume":"2012","author":[{"given":"Felipe A.","family":"Apolonio","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de Vi\u00e7osa, Avenida Peter Henry Rolfs s\/n, Campus Universit\u00e1rio Vi\u00e7osa, 36570-000 Vio\u00e7sa, MG, Brazil"}]},{"given":"Daniel H. 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