Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics
| Autor: | Juan F. Mañas-Mañas, Richard Wellman, Juan J. Moreno-Balcázar, Lance L. Littlejohn |
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| Rok vydání: | 2018 |
| Předmět: |
Numerical Analysis
Series (mathematics) Applied Mathematics General Mathematics 010102 general mathematics Eigenfunction Space (mathematics) Differential operator 01 natural sciences 010101 applied mathematics Sobolev space Combinatorics Mathematics::Probability Mathematics - Classical Analysis and ODEs Product (mathematics) Orthogonal polynomials Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics 33C47 42C05 Analysis Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Journal of Approximation Theory. 230:32-49 |
| ISSN: | 0021-9045 |
| DOI: | 10.1016/j.jat.2018.04.008 |
| Popis: | We consider the following discrete Sobolev inner product involving the Gegenbauer weight ( f , g ) S ≔ ∫ − 1 1 f ( x ) g ( x ) ( 1 − x 2 ) α d x + M [ f ( j ) ( − 1 ) g ( j ) ( − 1 ) + f ( j ) ( 1 ) g ( j ) ( 1 ) ] , where α > − 1 , j ∈ N ∪ { 0 } , and M > 0 . Our main objective is to calculate the exact value r 0 = lim n → + ∞ log max x ∈ [ − 1 , 1 ] | Q ˜ n ( α , M , j ) ( x ) | log λ ˜ n , α ≥ − 1 ∕ 2 , where { Q ˜ n ( α , M , j ) } n ≥ 0 is the sequence of orthonormal polynomials with respect to this Sobolev inner product. These polynomials are eigenfunctions of a differential operator and the obtaining of the asymptotic behavior of the corresponding eigenvalues, λ ˜ n , is the principal key to get the result. This value r 0 is related to the convergence of a series in a left-definite space. In addition, to complete the asymptotic study of this family of nonstandard polynomials we give the Mehler–Heine formulae for the corresponding orthogonal polynomials. |
| Databáze: | OpenAIRE |
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