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• W2619714822 abstract "We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=int_{-1}^1f(x)g(x)(1-x^2)^{alpha}dx+Mbig[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)big],$$ where $alpha>-1,$ $jin mathbb{N}cup {0},$ and $M>0.$ Let ${Q_n^{(alpha,M,j)}}_{ngeq0}$ be the sequence of orthogonal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $mathbf{T}. $ We establish the asymptotic behavior of the corresponding eigenvalues. Furthermore, we calculate the exact value $$r_0 = lim_{nrightarrow infty}frac{log left(max_{xin [-1,1]} |widetilde{Q}_n^{(alpha,M,j)}(x)|right)}{log widetilde{lambda}_n},$$ where ${widetilde{Q}_n^{(alpha,M,j)}}_{ngeq0}$ are the sequence of orthonormal polynomials with respect to this Sobolev inner product. This value $r_0$ is related to the convergence of a series in a left--definite space. Finally, we study the Mehler--Heine type asymptotics for ${Q_n^{(alpha,M,j)}}_{ngeq0}.$" @default.
• W2619714822 created "2017-06-05" @default.
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• W2619714822 date "2017-05-23" @default.
• W2619714822 modified "2023-09-22" @default.
• W2619714822 title "Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics" @default.
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