Geodesic
How sharp is Bernstein's inequality for Jacobi polynomials?
Electronic transactions on numerical analysis, Tome 36 (2010).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Bernstein's inequality for Jacobi polynomials P $(\alpha ,\beta )$ n , established in 1987 by P. Baratella for the region R1/2 = $|\alpha | \leq 1$/2, $|\beta | \leq 1$/2, and subsequently supplied with an improved constant by Y. Chow, L. Gatteschi, and R. Wong, is analyzed here analytically and, above all, computationally with regard to validity and sharpness, not only in the original region R1/2, but also in larger regions Rs = - 1/$2 \leq \alpha \leq s$, - 1/$2 \leq \beta \leq s$, s > 1/2. Computation suggests that the inequality holds with new, somewhat larger, constants in any region Rs. Best constants are provided for s = 1 : .5 : 4 and s = 5 : 1 : 10. Our work also sheds new light on the so-called Erdélyi-Magnus-Nevai conjecture for orthonormal Jacobi polynomials, adding further support for its validity and suggesting .66198126 . . . as the best constant implied in the conjecture.
Classification : 33C45, 41A17
Keywords: Bernstein's inequality, Jacobi polynomials, sharpness, erdélyi-magnus-nevai conjecture 
@article{ETNA_2010__36__a12,
     author = {Gautschi, Walter},
     title = {How sharp is {Bernstein's} inequality for {Jacobi} polynomials?},
     journal = {Electronic transactions on numerical analysis},
     publisher = {mathdoc},
     volume = {36},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2010__36__a12/}
}
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Gautschi, Walter. How sharp is Bernstein's inequality for Jacobi polynomials?. Electronic transactions on numerical analysis, Tome 36 (2010). http://geodesic.mathdoc.fr/item/ETNA_2010__36__a12/