Algorithms, Convergence and Rate of Convergence for an Interpolation Model Between Lagrange and Hermite
The aim of this piece of work is to study an interpolation problem on the interval [ - 1 , 1 ] , which can be considered an intermediate case between the interpolation problems of Lagrange and Hermite. The nodal points belong to the Chebyshev–Lobatto nodal system and the novelty is that we use the c...
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| Vydané v: | Resultate der Mathematik Ročník 73; číslo 1 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Springer International Publishing
01.03.2018
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| ISSN: | 1422-6383, 1420-9012 |
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| Abstract | The aim of this piece of work is to study an interpolation problem on the interval
[
-
1
,
1
]
, which can be considered an intermediate case between the interpolation problems of Lagrange and Hermite. The nodal points belong to the Chebyshev–Lobatto nodal system and the novelty is that we use the complete system for the Lagrange data and half of the nodes for the derivative data. When the extremal points are not used for the values of the derivative we have a quasi-interpolation problem. First we give two different types of algorithms for computing the interpolation polynomials. One of the expressions is given in terms of the Chebyshev basis of the first kind and the second one is based on a barycentric formula. The second part of the paper is devoted to obtain some results about the convergence and the rate of convergence of the interpolants when interpolating some type of smooth functions. We also consider the case of merely continuous functions obtaining a result of convergence which is closer to the Lagrange problem. Finally, we analyze the quasi-interpolation case. |
|---|---|
| AbstractList | The aim of this piece of work is to study an interpolation problem on the interval
[
-
1
,
1
]
, which can be considered an intermediate case between the interpolation problems of Lagrange and Hermite. The nodal points belong to the Chebyshev–Lobatto nodal system and the novelty is that we use the complete system for the Lagrange data and half of the nodes for the derivative data. When the extremal points are not used for the values of the derivative we have a quasi-interpolation problem. First we give two different types of algorithms for computing the interpolation polynomials. One of the expressions is given in terms of the Chebyshev basis of the first kind and the second one is based on a barycentric formula. The second part of the paper is devoted to obtain some results about the convergence and the rate of convergence of the interpolants when interpolating some type of smooth functions. We also consider the case of merely continuous functions obtaining a result of convergence which is closer to the Lagrange problem. Finally, we analyze the quasi-interpolation case. |
| ArticleNumber | 40 |
| Author | Cachafeiro, Alicia Berriochoa, Elías García-Amor, José M. |
| Author_xml | – sequence: 1 givenname: Elías orcidid: 0000-0001-6170-6160 surname: Berriochoa fullname: Berriochoa, Elías organization: Universidad de Vigo – sequence: 2 givenname: Alicia orcidid: 0000-0002-9413-7352 surname: Cachafeiro fullname: Cachafeiro, Alicia email: acachafe@uvigo.es organization: Universidad de Vigo – sequence: 3 givenname: José M. surname: García-Amor fullname: García-Amor, José M. organization: Xunta de Galicia |
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| Cites_doi | 10.1016/j.cam.2014.10.001 10.1093/imanum/24.4.547 10.1016/0021-9045(79)90057-1 10.1016/j.jat.2016.12.004 |
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| Keywords | Interpolation convergence Lobatto–Chebyshev nodal systems rate of convergence 65D05 41A05 Chebyshev polynomials 42C05 |
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| References | Higham (CR7) 2004; 24 Mason, Handscomb (CR9) 2003 Szegő (CR13) 1975 Berriochoa, Cachafeiro, Díaz (CR2) 2015; 284 Rivlin (CR11) 1974 CR14 Berriochoa, Cachafeiro, Díaz, Illán (CR4) 2014; 234 CR10 Berriochoa, Cachafeiro, García Amor (CR1) 2017; 215 Davis (CR6) 1975 Bojanic, Prasad, Saxena (CR5) 1979; 26 Berriochoa, Cachafeiro, Díaz (CR3) 2015; 253 Jackson (CR8) 1930 Saxena (CR12) 1974; 9 E Berriochoa (802_CR4) 2014; 234 G Szegő (802_CR13) 1975 T Rivlin (802_CR11) 1974 E Berriochoa (802_CR3) 2015; 253 802_CR10 RB Saxena (802_CR12) 1974; 9 802_CR14 R Bojanic (802_CR5) 1979; 26 D Jackson (802_CR8) 1930 JC Mason (802_CR9) 2003 PJ Davis (802_CR6) 1975 E Berriochoa (802_CR2) 2015; 284 E Berriochoa (802_CR1) 2017; 215 NJ Higham (802_CR7) 2004; 24 |
| References_xml | – ident: CR14 – volume: 284 start-page: 58 year: 2015 end-page: 68 ident: CR2 article-title: Convergence of Hermite interpolants on the unit circle using two derivatives publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.10.001 – volume: 24 start-page: 547 year: 2004 end-page: 556 ident: CR7 article-title: The numerical stability of barycentric Lagrange interpolation publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/24.4.547 – volume: 234 start-page: 223 year: 2014 end-page: 236 ident: CR4 article-title: Algorithms and convergence for Hermite interpolation based on extended Chebyshev nodal systems publication-title: Appl. Math. Comput. – volume: 9 start-page: 223 year: 1974 end-page: 232 ident: CR12 article-title: The Hermite–Fejér process on the Tchebycheff matrix of second kind publication-title: Stud. Sci. Math. Hungar. – volume: 253 start-page: 274 year: 2015 end-page: 286 ident: CR3 article-title: Gibbs phenomenon in the Hermite interpolation on the circle publication-title: Appl. Math. Comput. – volume: 26 start-page: 195 year: 1979 end-page: 203 ident: CR5 article-title: An upper bound for the rate of convergence of the Hermite–Fejér process on the extended Chebyshev nodes of the second kind publication-title: J. Approx. Theory doi: 10.1016/0021-9045(79)90057-1 – ident: CR10 – year: 1975 ident: CR6 publication-title: Interpolation and Approximation – year: 1930 ident: CR8 publication-title: The Theory of Approximation – volume: 215 start-page: 118 year: 2017 end-page: 144 ident: CR1 article-title: An interpolation problem on the circle between Lagrange and Hermite problems publication-title: J. Approx. Theory doi: 10.1016/j.jat.2016.12.004 – year: 2003 ident: CR9 publication-title: Chebyshev Polynomials – year: 1974 ident: CR11 publication-title: The Chebyshev Polynomials, Pure and Applied Mathematics – year: 1975 ident: CR13 publication-title: Orthogonal Polynomials, American Mathematical Society, Colloquium Publications – volume-title: Interpolation and Approximation year: 1975 ident: 802_CR6 – ident: 802_CR10 – volume: 9 start-page: 223 year: 1974 ident: 802_CR12 publication-title: Stud. Sci. Math. Hungar. – volume: 284 start-page: 58 year: 2015 ident: 802_CR2 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.10.001 – ident: 802_CR14 – volume: 24 start-page: 547 year: 2004 ident: 802_CR7 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/24.4.547 – volume: 253 start-page: 274 year: 2015 ident: 802_CR3 publication-title: Appl. Math. Comput. – volume: 234 start-page: 223 year: 2014 ident: 802_CR4 publication-title: Appl. Math. Comput. – volume-title: The Chebyshev Polynomials, Pure and Applied Mathematics year: 1974 ident: 802_CR11 – volume: 26 start-page: 195 year: 1979 ident: 802_CR5 publication-title: J. Approx. Theory doi: 10.1016/0021-9045(79)90057-1 – volume-title: The Theory of Approximation year: 1930 ident: 802_CR8 – volume: 215 start-page: 118 year: 2017 ident: 802_CR1 publication-title: J. Approx. Theory doi: 10.1016/j.jat.2016.12.004 – volume-title: Orthogonal Polynomials, American Mathematical Society, Colloquium Publications year: 1975 ident: 802_CR13 – volume-title: Chebyshev Polynomials year: 2003 ident: 802_CR9 |
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| Snippet | The aim of this piece of work is to study an interpolation problem on the interval
[
-
1
,
1
]
, which can be considered an intermediate case between the... |
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| Title | Algorithms, Convergence and Rate of Convergence for an Interpolation Model Between Lagrange and Hermite |
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