{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:51:11Z","timestamp":1766137871213,"version":"3.41.2"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","funder":[{"DOI":"10.13039\/100010661","name":"Horizon 2020 Framework Programme","doi-asserted-by":"publisher","award":["861584"],"award-info":[{"award-number":["861584"]}],"id":[{"id":"10.13039\/100010661","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Taishan Distinguished Professor Fund"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:p> Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions. <\/jats:p>","DOI":"10.1142\/s0219691321500594","type":"journal-article","created":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T16:31:35Z","timestamp":1639672295000},"source":"Crossref","is-referenced-by-count":2,"title":["Frequency domain of weakly translation invariant frame MRAs"],"prefix":"10.1142","volume":"20","author":[{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-6449-4332","authenticated-orcid":false,"given":"Zhihua","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100 China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,12,15]]},"reference":[{"volume-title":"A Wavelet Tour of Signal Processing","year":"2008","author":"Mallat S.","key":"S0219691321500594BIB001"},{"key":"S0219691321500594BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2021.03.003"},{"key":"S0219691321500594BIB003","first-page":"303","volume-title":"Finite Frames: Theory and Applications","author":"Bajwa W. 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