Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds
For μ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Yn (n∈ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t4n(hμ′Φ)(t)=1/w(t), where hμ′ de...
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/242750 |
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| author | Cristian Arteaga Isabel Marrero |
| author_facet | Cristian Arteaga Isabel Marrero |
| author_sort | Cristian Arteaga |
| collection | DOAJ |
| description | For μ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Yn (n∈ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t4n(hμ′Φ)(t)=1/w(t), where hμ′ denotes the generalized Hankel transform of order μ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space ℋμ in order to derive explicit representations of the derivatives SμmΦ and their Hankel transforms, the former ones being valid when m∈ℤ+ is restricted to a suitable interval for which SμmΦ is continuous. Here, Sμm denotes the mth iterate of the Bessel differential operator Sμ if m∈ℕ, while Sμ0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (hμ′Φ)(t)=1/t4nw(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Yn. |
| format | Article |
| id | doaj-art-78259c7ef3e14570b50dac13d4bfa943 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
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| series | The Scientific World Journal |
| spelling | doaj-art-78259c7ef3e14570b50dac13d4bfa9432025-08-20T03:35:11ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/242750242750Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial BoundsCristian Arteaga0Isabel Marrero1Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), SpainDepartamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), SpainFor μ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Yn (n∈ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t4n(hμ′Φ)(t)=1/w(t), where hμ′ denotes the generalized Hankel transform of order μ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space ℋμ in order to derive explicit representations of the derivatives SμmΦ and their Hankel transforms, the former ones being valid when m∈ℤ+ is restricted to a suitable interval for which SμmΦ is continuous. Here, Sμm denotes the mth iterate of the Bessel differential operator Sμ if m∈ℕ, while Sμ0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (hμ′Φ)(t)=1/t4nw(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Yn.http://dx.doi.org/10.1155/2014/242750 |
| spellingShingle | Cristian Arteaga Isabel Marrero Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds The Scientific World Journal |
| title | Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds |
| title_full | Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds |
| title_fullStr | Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds |
| title_full_unstemmed | Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds |
| title_short | Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds |
| title_sort | interpolation by hankel translates of a basis function inversion formulas and polynomial bounds |
| url | http://dx.doi.org/10.1155/2014/242750 |
| work_keys_str_mv | AT cristianarteaga interpolationbyhankeltranslatesofabasisfunctioninversionformulasandpolynomialbounds AT isabelmarrero interpolationbyhankeltranslatesofabasisfunctioninversionformulasandpolynomialbounds |