{"title":"正交序列代数","authors":"A. N. Pankratov, R. K. Tetuev","doi":"10.1134/s105466182304034x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>An algebra of orthogonal series has been developed for the operations of multiplication and division, differentiation and integration of signals represented by series of classical orthogonal polynomials and functions. It is shown that the considered linear transformations for classical orthogonal polynomials and functions are subject to recurrence relations of a special form, which makes it possible to perform these transformations over series in the space of expansion coefficients. A theorem on the condition for the existence of a recurrence relation for the inverse transformation is proven. A general scheme of an algorithm for calculating the coefficients of the resulting series from the coefficients of the original series through the coefficients of recurrent relations is proposed. It has been proven that the linear transformation of a series of length <span>\\(N\\)</span> is executed by a linear complexity algorithm <span>\\({{\\Theta }}\\left( N \\right)\\)</span>. Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":"147 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Algebra of Orthogonal Series\",\"authors\":\"A. N. Pankratov, R. K. Tetuev\",\"doi\":\"10.1134/s105466182304034x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>An algebra of orthogonal series has been developed for the operations of multiplication and division, differentiation and integration of signals represented by series of classical orthogonal polynomials and functions. It is shown that the considered linear transformations for classical orthogonal polynomials and functions are subject to recurrence relations of a special form, which makes it possible to perform these transformations over series in the space of expansion coefficients. A theorem on the condition for the existence of a recurrence relation for the inverse transformation is proven. A general scheme of an algorithm for calculating the coefficients of the resulting series from the coefficients of the original series through the coefficients of recurrent relations is proposed. It has been proven that the linear transformation of a series of length <span>\\\\(N\\\\)</span> is executed by a linear complexity algorithm <span>\\\\({{\\\\Theta }}\\\\left( N \\\\right)\\\\)</span>. Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.</p>\",\"PeriodicalId\":35400,\"journal\":{\"name\":\"PATTERN RECOGNITION AND IMAGE ANALYSIS\",\"volume\":\"147 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"PATTERN RECOGNITION AND IMAGE ANALYSIS\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1134/s105466182304034x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s105466182304034x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
摘要
摘要 针对经典正交多项式和函数序列所代表信号的乘除、微分和积分运算，开发了一种正交序列代数。研究表明，所考虑的经典正交多项式和函数的线性变换受制于特殊形式的递推关系，这使得在膨胀系数空间中对序列进行这些变换成为可能。证明了关于逆变换递推关系存在条件的定理。提出了通过递推关系系数从原数列系数计算所得数列系数的算法的一般方案。证明了长度为 \(N\) 的数列的线性变换是通过线性复杂性算法 \({{\Theta }}\left( N \right)\)执行的。给出了切比雪夫、雅可比、拉盖尔和赫米特多项式的公式。
An algebra of orthogonal series has been developed for the operations of multiplication and division, differentiation and integration of signals represented by series of classical orthogonal polynomials and functions. It is shown that the considered linear transformations for classical orthogonal polynomials and functions are subject to recurrence relations of a special form, which makes it possible to perform these transformations over series in the space of expansion coefficients. A theorem on the condition for the existence of a recurrence relation for the inverse transformation is proven. A general scheme of an algorithm for calculating the coefficients of the resulting series from the coefficients of the original series through the coefficients of recurrent relations is proposed. It has been proven that the linear transformation of a series of length \(N\) is executed by a linear complexity algorithm \({{\Theta }}\left( N \right)\). Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.
期刊介绍：
The purpose of the journal is to publish high-quality peer-reviewed scientific and technical materials that present the results of fundamental and applied scientific research in the field of image processing, recognition, analysis and understanding, pattern recognition, artificial intelligence, and related fields of theoretical and applied computer science and applied mathematics. The policy of the journal provides for the rapid publication of original scientific articles, analytical reviews, articles of the world''s leading scientists and specialists on the subject of the journal solicited by the editorial board, special thematic issues, proceedings of the world''s leading scientific conferences and seminars, as well as short reports containing new results of fundamental and applied research in the field of mathematical theory and methodology of image analysis, mathematical theory and methodology of image recognition, and mathematical foundations and methodology of artificial intelligence. The journal also publishes articles on the use of the apparatus and methods of the mathematical theory of image analysis and the mathematical theory of image recognition for the development of new information technologies and their supporting software and algorithmic complexes and systems for solving complex and particularly important applied problems. The main scientific areas are the mathematical theory of image analysis and the mathematical theory of pattern recognition. The journal also embraces the problems of analyzing and evaluating poorly formalized, poorly structured, incomplete, contradictory and noisy information, including artificial intelligence, bioinformatics, medical informatics, data mining, big data analysis, machine vision, data representation and modeling, data and knowledge extraction from images, machine learning, forecasting, machine graphics, databases, knowledge bases, medical and technical diagnostics, neural networks, specialized software, specialized computational architectures for information analysis and evaluation, linguistic, psychological, psychophysical, and physiological aspects of image analysis and pattern recognition, applied problems, and related problems. Articles can be submitted either in English or Russian. The English language is preferable. Pattern Recognition and Image Analysis is a hybrid journal that publishes mostly subscription articles that are free of charge for the authors, but also accepts Open Access articles with article processing charges. The journal is one of the top 10 global periodicals on image analysis and pattern recognition and is the only publication on this topic in the Russian Federation, Central and Eastern Europe.
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