The continous Legendre transform, its inverse transform, and applications
This paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial Pk(x) by the function Pλ(x) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legen...
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| Format: | Article |
| Language: | English |
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Wiley
1980-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117128000004X |
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| _version_ | 1832552358038470656 |
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| author | P. L. Butzer R. L. Stens M. Wehrens |
| author_facet | P. L. Butzer R. L. Stens M. Wehrens |
| author_sort | P. L. Butzer |
| collection | DOAJ |
| description | This paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial Pk(x) by the function Pλ(x) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping L1(ℝ+) into L2(−1,1), is defined. Its inversion in turn is naturally achieved by the continuous Legendre transform. One application is devoted to the Shannon sampling theorem in the Legendre frame together with a new type of error estimate. The other deals with a new representation of Legendre functions giving information about their behaviour near the point x=−1. |
| format | Article |
| id | doaj-art-5e499c6dffbf42a5a8bdd4b57a1292e8 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1980-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5e499c6dffbf42a5a8bdd4b57a1292e82025-02-03T05:58:49ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251980-01-0131476710.1155/S016117128000004XThe continous Legendre transform, its inverse transform, and applicationsP. L. Butzer0R. L. Stens1M. Wehrens2Lehrstuhl A für Mathematik, Aachen University of Technology, Aachen 51, GermanyLehrstuhl A für Mathematik, Aachen University of Technology, Aachen 51, GermanyLehrstuhl A für Mathematik, Aachen University of Technology, Aachen 51, GermanyThis paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial Pk(x) by the function Pλ(x) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping L1(ℝ+) into L2(−1,1), is defined. Its inversion in turn is naturally achieved by the continuous Legendre transform. One application is devoted to the Shannon sampling theorem in the Legendre frame together with a new type of error estimate. The other deals with a new representation of Legendre functions giving information about their behaviour near the point x=−1.http://dx.doi.org/10.1155/S016117128000004Xcontinuous Legendre transforminverse Legendre transforminversion formulaShannon sampling theoremtruncation error estimatesbehaviour of Legendre functions near singular points. |
| spellingShingle | P. L. Butzer R. L. Stens M. Wehrens The continous Legendre transform, its inverse transform, and applications International Journal of Mathematics and Mathematical Sciences continuous Legendre transform inverse Legendre transform inversion formula Shannon sampling theorem truncation error estimates behaviour of Legendre functions near singular points. |
| title | The continous Legendre transform, its inverse transform, and applications |
| title_full | The continous Legendre transform, its inverse transform, and applications |
| title_fullStr | The continous Legendre transform, its inverse transform, and applications |
| title_full_unstemmed | The continous Legendre transform, its inverse transform, and applications |
| title_short | The continous Legendre transform, its inverse transform, and applications |
| title_sort | continous legendre transform its inverse transform and applications |
| topic | continuous Legendre transform inverse Legendre transform inversion formula Shannon sampling theorem truncation error estimates behaviour of Legendre functions near singular points. |
| url | http://dx.doi.org/10.1155/S016117128000004X |
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