{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:24:43Z","timestamp":1753881883715,"version":"3.41.2"},"reference-count":15,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2023,1]]},"abstract":"<jats:p> In this paper, a special group is used to introduce the two-direction two-scale similarity transform (TDTST) of the two-direction refinable function, and the existence of the solution of the two-direction refinable equation is discussed by the TDTST transform. The approximation order of the two-direction refinable function is discussed by the TDTST transform. Whether the TDTST transform can maintain the orthogonality of the two-direction refinable function is discussed, and it is pointed out that a special real orthogonal matrix can maintain the orthogonality. It is pointed out that the TDTST transform can maintain the symmetry of the two-direction refinable function. Finally, an example is given. <\/jats:p>","DOI":"10.1142\/s0219691322500394","type":"journal-article","created":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T07:25:31Z","timestamp":1661498731000},"source":"Crossref","is-referenced-by-count":0,"title":["Two-direction two-scale similarity transformation of two-direction refinable function"],"prefix":"10.1142","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7424-0600","authenticated-orcid":false,"given":"Gang","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics Science, Xinjiang Normal University, Urumqi, Xinjiang 830054, China"}]}],"member":"219","published-online":{"date-parts":[[2022,9,30]]},"reference":[{"issue":"1","key":"S0219691322500394BIB001","first-page":"69","volume":"6","author":"Ahmad S. F.","year":"2015","journal-title":"Int. J. Nonlinear Anal. 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