{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T16:28:09Z","timestamp":1782404889954,"version":"3.54.5"},"reference-count":14,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval\u2019s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.<\/jats:p>","DOI":"10.3390\/e24101340","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T04:07:07Z","timestamp":1663906027000},"page":"1340","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Discrete Quadratic-Phase Fourier Transform: Theory and Convolution Structures"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9826-9475","authenticated-orcid":false,"given":"Waseem Z.","family":"Lone","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8461-869X","authenticated-orcid":false,"given":"Firdous A.","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7878-826X","authenticated-orcid":false,"given":"Ahmed I.","family":"Zayed","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,23]]},"reference":[{"key":"ref_1","first-page":"107","article-title":"Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces","volume":"230","author":"Saitoh","year":"2010","journal-title":"Am. Math. Soc. Trans. Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s00009-017-1063-y","article-title":"New convolutions for quadratic-phase Fourier integral operators and their applications","volume":"15","author":"Castro","year":"2018","journal-title":"Mediterr. J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Debnath, L., and Shah, F.A. (2017). Lecture Notes on Wavelet Transforms, Birkh\u00e4user.","DOI":"10.1007\/978-3-319-59433-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"167689","DOI":"10.1016\/j.ijleo.2021.167689","article-title":"Short-time quadratic-phase Fourier transform","volume":"245","author":"Shah","year":"2021","journal-title":"Optik"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"169021","DOI":"10.1016\/j.ijleo.2022.169021","article-title":"An interplay between quadratic-phase Fourier and Zak transforms","volume":"260","author":"Shah","year":"2022","journal-title":"Optik"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.3934\/math.2022111","article-title":"Analytical solutions to generalized differential equations using quadratic-phase Fourier transform","volume":"7","author":"Shah","year":"2022","journal-title":"AIMS Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lone, W.Z., Shah, F.A., Nisar, K.S., Albalawi, W., Alshahrani, B., and Park, C. (2022). Non-ideal sampling in shift-invariant spaces associated with quadratic-phase Fourier transforms. Alex. Eng. J.","DOI":"10.1016\/j.aej.2022.07.065"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18138","DOI":"10.1364\/OE.21.018138","article-title":"Discrete linear canonical transform computation by adaptive method","volume":"21","author":"Zhang","year":"2013","journal-title":"Opt. Express"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1329","DOI":"10.1109\/78.839980","article-title":"The discrete fractional Fourier transform","volume":"48","author":"Candan","year":"2000","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103361","DOI":"10.1016\/j.dsp.2021.103361","article-title":"Discrete quaternion linear canonical transform","volume":"122","author":"Urynbassarovaa","year":"2022","journal-title":"Digit Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1049\/iet-spr.2015.0028","article-title":"Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications","volume":"10","author":"Feng","year":"2016","journal-title":"IET Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"214018","DOI":"10.1155\/2021\/2140189","article-title":"A convolution-based shearlet transform in free metaplectic domains","volume":"2021","author":"Garg","year":"2021","journal-title":"J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1002\/mma.7842","article-title":"Quadratic-phase wavelet transform with applications to generalized differential equations","volume":"45","author":"Shah","year":"2021","journal-title":"Math. Method Appl. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"169063","DOI":"10.1016\/j.ijleo.2022.169063","article-title":"Shift-invariant spaces and dynamical sampling in quadratic-phase Fourier domains","volume":"260","author":"Lone","year":"2022","journal-title":"Optik"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1340\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:37:54Z","timestamp":1760143074000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1340"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":14,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["e24101340"],"URL":"https:\/\/doi.org\/10.3390\/e24101340","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,23]]}}}