Stable Calculation of Discrete Hahn Functions
Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/3/437 |
| _version_ | 1797441615014395904 |
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| author | Albertus C. den Brinker |
| author_facet | Albertus C. den Brinker |
| author_sort | Albertus C. den Brinker |
| collection | DOAJ |
| description | Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℓ</mo><mn>2</mn></msub></semantics></math></inline-formula> derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power. |
| first_indexed | 2024-03-09T12:26:41Z |
| format | Article |
| id | doaj.art-5420a62bdf7f4e82bb27a90624e42df3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:26:41Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-5420a62bdf7f4e82bb27a90624e42df32023-11-30T22:34:37ZengMDPI AGSymmetry2073-89942022-02-0114343710.3390/sym14030437Stable Calculation of Discrete Hahn FunctionsAlbertus C. den Brinker0Philips Research, 5656 AE Eindhoven, The NetherlandsGenerating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℓ</mo><mn>2</mn></msub></semantics></math></inline-formula> derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power.https://www.mdpi.com/2073-8994/14/3/437orthogonal polynomialsdiscrete Hahn polynomialsdiscrete Hahn functionserror propagationdifference equationthree-term recurrence relation |
| spellingShingle | Albertus C. den Brinker Stable Calculation of Discrete Hahn Functions Symmetry orthogonal polynomials discrete Hahn polynomials discrete Hahn functions error propagation difference equation three-term recurrence relation |
| title | Stable Calculation of Discrete Hahn Functions |
| title_full | Stable Calculation of Discrete Hahn Functions |
| title_fullStr | Stable Calculation of Discrete Hahn Functions |
| title_full_unstemmed | Stable Calculation of Discrete Hahn Functions |
| title_short | Stable Calculation of Discrete Hahn Functions |
| title_sort | stable calculation of discrete hahn functions |
| topic | orthogonal polynomials discrete Hahn polynomials discrete Hahn functions error propagation difference equation three-term recurrence relation |
| url | https://www.mdpi.com/2073-8994/14/3/437 |
| work_keys_str_mv | AT albertuscdenbrinker stablecalculationofdiscretehahnfunctions |