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• W2964221552 abstract "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." @default.
• W2964221552 created "2019-07-30" @default.
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• W2964221552 date "2018-06-01" @default.
• W2964221552 modified "2023-10-16" @default.
• W2964221552 title "Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics" @default.
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• W2964221552 doi "https://doi.org/10.1016/j.jat.2018.04.008" @default.
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