Algebra of Orthogonal Series

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 po...

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Veröffentlicht in:Pattern recognition and image analysis Jg. 33; H. 4; S. 1309 - 1314
Hauptverfasser: Pankratov, A. N., Tetuev, R. K.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Moscow Pleiades Publishing 01.12.2023
Schlagworte:
Computer Science
Image Processing and Computer Vision
Moscow Region
Pattern Recognition
Pushchino
Scientific School of the Institute of Mathematical Problems of Biology of the Russian Academy of Sciences–The Branch of Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
the Russian Federation
Hermite
recurrence relations
differentiation
Jacobi
integration
Laguerre
polynomial algebra
orthogonal polynomials of Chebyshev
ISSN:1054-6618, 1555-6212
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Abstract 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 is executed by a linear complexity algorithm . Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.
AbstractList 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 is executed by a linear complexity algorithm . Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.
Author Pankratov, A. N.
Tetuev, R. K.
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Cites_doi 10.1145/362003.362037
10.1016/0010-4655(83)90015-2
10.1137/1.9781611975949
10.1016/s0096-3003(03)00301-1
10.1145/362452.362478
10.1134/s1054661819040102
10.17537/2010.5.30
10.1090/s0025-5718-1955-0071856-0
10.1016/j.cam.2005.10.013
10.1080/13658816.2016.1188932
10.1016/0010-4655(88)90049-5
10.17537/2013.8.398
10.1007/978-3-642-74748-9
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Copyright Pleiades Publishing, Ltd. 2023. ISSN 1054-6618, Pattern Recognition and Image Analysis, 2023, Vol. 33, No. 4, pp. 1309–1314. © Pleiades Publishing, Ltd., 2023.
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Issue 4
Keywords Hermite
recurrence relations
differentiation
Jacobi
integration
Laguerre
polynomial algebra
orthogonal polynomials of Chebyshev
Language English
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PublicationTitle Pattern recognition and image analysis
PublicationTitleAbbrev Pattern Recognit. Image Anal
PublicationYear 2023
Publisher Pleiades Publishing
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SubjectTerms Computer Science
Image Processing and Computer Vision
Moscow Region
Pattern Recognition
Pushchino
Scientific School of the Institute of Mathematical Problems of Biology of the Russian Academy of Sciences–The Branch of Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
the Russian Federation
Title Algebra of Orthogonal Series
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